Theoretical Ecology a Unified Approach

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‘r Theoretical Ecology A Unified Approach

Second Edition

Luís Soares Barreto

Costa de Caparica 2017 Costa de Caparica 201X 2 © L. S. Barreto, 2017. Theoretical Ecology Theoretical Ecology

Theoretical Ecology

Sem texto

Theoretical Ecologyy

Theoretical Ecology A Unified Approach

Second Edition

4 © L. S. Barreto, 2017. Theoretical Ecology Theoretical Ecology

Sem texto © L. S. Barreto, 2017. Theoretical Ecology 5 Theoretical Ecology

Theoretical Ecology A Unified Approach

Second Edition

Luís Soares Barreto Jubilee Professor of Forestry University of Lisbon Portugal 6 © L. S. Barreto, 2017. Theoretical Ecology © Luís Soares Barreto, 2005, 2017

Theoretical Ecology A Unified Approach

Second Edition

E-book published by the author

Prof. Doutor Luís Soares Barreto Av. do Movimento das Forças Armadas, 41 – 3D 2825-372 Costa de Caparica Portugal

This e-book is free ware but neither public domain nor open source. It can be copied and disseminated only in its totality, with respect for the authorship rights. It can not be sold.

With compliments © L. S. Barreto, 2017. Theoretical Ecology 7 Dedication

Dedication

In memoriam of my parents,

also

to Sandra Isabel, Luísa Maria, ecology students, and ecologists

I am grateful to those who contribute to the existence of ClickCharts, LibreOffice, Maxima, R, Scilab e wxMaxima 8 © L. S. Barreto, 2017. Theoretical Ecology Cover image

Cover image

Cover image Modified photo of an Iberian lynx, obtained from the internet. Cer- tainly from a Spanish author that I was unable to identify. With my gratitude © L. S. Barreto, 2017. Theoretical Ecology 9 The author

The author

Luís Soares Barreto was borne in 1935, in Chinde, a small village in the delta of the Zambezi River, in Mozambique. In this African country, from 1962 till 1974, he did research in forestry, and he was also member of the faculty of the Universidade de Lourenço Marques (actual Universidade Eduardo Mondlane, Maputo), where he started the teaching of Forestry. While member of this university, from 1967 to 1970, he was a graduate student at Duke Univer- sity, Durham, NC, U.S.A.. From this university, he received his Master of Forestry in Forest Ecology (1968), and his Ph. D. in Operations Research applied to Forestry (1970). Since 1975 till March, 2005, he taught at the Instituto Superior de Agronomia, Universidade de Lis- boa. From 1975 to 1982, simultaneously, he taught in the Department of Environmental Sciences of the Universidade Nova de Lisboa. Here, in 1977, he conceived, and created a new five years degree in environmental engineering. He is the only Portuguese who established a scientific theory. Besides the ecological theory presented in this book, he proposed also a unified theory for forest stands of any kind and ecologically sound management practices grounded on it. The later theory is a particular case of the former one. His first contact with mathematical or theoretical ecology occurred in the spring of 1968, when he was graduate student at Duke University, and he attended a course (mathematical ecology, MBA 591) in North Carolina State University, at Raleigh, taught by Professor Evelyn C. Pielou, holding a visitor professorship in this university. The author taught population dynamics for about twenty years. Actually, he is jubilee professor of the University of Lisbon. 10 © L. S. Barreto, 2017. Theoretical Ecology The author © L. S. Barreto, 2017. Theoretical Ecology 11 The author

Other books by the author:

Madeiras Ultramarinas. Instituto de Investigação Científica de Moçambique, Lourenço Marques, 1963. A Produtividade Primária Líquida da Terra. Secretaria de Estado do Ambiente, Lisboa, 1977. O Ambiente e a Economia. Secretaria de Estado do Ambiente, Lisboa, 1977. Um Novo Método para a Elaboração de Tabelas de Produção. Aplicação ao Pinhal. Serviço Nacional de Parques, Reservas e Conservação da Natureza, Lisboa 1987. A Floresta. Estrutura e Funcionamento. Serviço Nacional de Parques, Reservas e Conservação da Natureza, Lisboa, 1988. Alto Fuste Regular. Instrumentos para a sua Gestão. Publicações Ciência e Vida, Lda., Lisboa, 1994. Ética Ambiental. Uma Anotação Introdutória. Publicações Ciência e Vida, Lda., Lisboa, 1994. Povoamentos Jardinados. Instrumentos para a sua Gestão. Publicações Ciência e Vida, Lda., Lisboa, 1995. Pinhais Mansos. Ecologia e Gestão. Estação Florestal Nacional, Lisboa, 2000. Pinhais Bravos. Ecologia e Gestão. E-book, Lisbon, 2005. Theoretical Ecology. A Unified Approach. E-book, Lisbon, 2005. Iniciação ao Scilab. E-book, Lisbon, 2008. Árvores e Arvoredos. Geometria e Dinâmica. E-book, Costa de Caparica, 2010. From Trees to Forests. A Unified Theory. E-book, Costa de Caparica, 2011. Iniciação ao Scilab. Second edition. E-book, Costa de Caparica, 2011. Ecologia Teórica. Uma Outra Explanação. I - Populações Isoladas. E-book, Costa de Caparica, 2013. Ecologia Teórica. Uma Outra Explanação. II - Interações entre Populações. E-book, Costa de Capa- rica, 2014. Ecologia Teórica. Uma Outra Explanação. III – Comunidade e Ecossistema. E-book, Costa de Capa- rica, 2016. 12 © L. S. Barreto, 2017. Theoretical Ecology Quotations

Quotations

“No theory, no science" Mario Bunge. Philosophy of Science. From Problem to Theory. Volume I, page 437.

"I know that most men, including those at ease with problems of the highest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives." L. Tolstoy

Quotation obtained from Bohmian Mechanics,* (Section 15), Stanford Encyclopaedia of Philosophy, in the internet. *Sheldon Goldstein. Bohmian Mechanics. The Stanford Encyclopaedia of Philosophy (Spring 2009 Edition), Edward N. Zanta (Ed.) URL=< http//plato.stanford.edu/archives/spr2009/entriesqm-bohm/>

“We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts”. James B. Conant, 1958. On Understanding Science. Mentor Book, New York. Page 48.

Without text © L. S. Barreto, 2017. Theoretical Ecology 13 Contents

Contents

Cover ...... 1 Theoretical Ecology ...... 2 Theoretical Ecologyy ...... 3 Theoretical Ecology ...... 3 Theoretical Ecology ...... 5 © Luís Soares Barreto, 2005, 2017 ...... 6 Dedication ...... 7 Cover image ...... 8 The author ...... 9 Quotations ...... 12 Contents ...... 13 1. Introduction ...... 18 1.1. The Scope of this Book ...... 18 1.2 The Fundamental Assumptions ...... 18 1.3 The Book ...... 19 1.4 Theory Synopsis ...... 20 1.5 Connection ...... 21 1.6 References ...... 21 PART I ...... 22 ISOLATED POPULATIONS ...... 22 2 Population Descriptors, and other Basic Concepts ...... 23 2.1 Introduction ...... 23 2.2 Basic Concepts of Dimensional Analysis ...... 23 2.3 Organism and Population Variables ...... 24 2.4 References ...... 27 3 Allometry ...... 28 3.1 Introduction ...... 28 3.2 Allometric Equations ...... 29 3.3 Self-Similarity ...... 33 3.4 References, and Related Bibliography ...... 36 4 The Gompertz Equation ...... 38 4.1 Introduction ...... 38 14 © L. S. Barreto, 2017. Theoretical Ecology Contents

4.2 A Unique Pattern for Biological Growth ...... 38 4.3 Assumptions ...... 39 4.4 Model ...... 40 4.5 Analysys of the Model ...... 50 4.6 Discret Models ...... 51 4.7 The EGZ with Time Lag ...... 59

4.8 The Specific Constancy of c, and Ri ...... 62 4.9 Properties of Gompertzian Variables ...... 63 4.10 The Fitting of the EGZ ...... 64 4.11 References, and Related Bibliography ...... 66 5 The Laws of Growth of Isolated Populations ...... 68 5.1 Introduction ...... 68 5.2 The Fundamental Laws ...... 68 5.3 Time-space Symmetry Between a Cohort and its Aged Structured Population ...... 72 5.4 Self-similarity in Population Growth ...... 73 5.5 Empirical Evaluation with an Animal Species ...... 76 5.6 The Total Production of a Cohort ...... 79 5.7 References, and Related Bibliography ...... 80 6 Structured Populations: The Gompertzian Approach ...... 81 6.1 Introduction ...... 81 6.2 Retrieving some Concepts, and Notation ...... 83 6.3 Gompertzian Demography ...... 84 6.4 Finding the Rates of Permanence, Transition, and Mortality ...... 87 6.5 References, and Related Bibliography ...... 93 PART II ...... 95 POPULA TION INTERACTIONS ...... 95 7 Introduction to Part II ...... 96 7.1 Explanation ...... 96 7.2 References ...... 98 8 Models for Predation, Omnivory, and Pantophagy ...... 99 8.1 Introduction ...... 99 8.2 Model SBPRED ...... 99 8.3 Discrete Model SBPRED11-de ...... 113 8.4 Model sbparasit-p ...... 118 8.5 Model sbparasit-bn ...... 122 © L. S. Barreto, 2017. Theoretical Ecology 15 Contents

8.6 Direct, Indirect, and Total Effects ...... 124 8.7 Omnivory ...... 125 8.7 Pantophagy ...... 133 8.6 References, and Related Bibliography ...... 134 9 S ystems of Herbivor y ...... 136 9.1 Introduction ...... 136 9.2 Extension of Model SBPRED11 to Herbivory ...... 136 9.3 Extension of Model SBPRED11-de to Herbivory ...... 142 9.4 References, and Related Bibliography ...... 145 10 Models for Amensalism, Commensalism, and Detritivory ...... 146 10.1 Introduction ...... 146 10.2 Amensalism ...... 146 10.3 Commensalism ...... 152 10.4 Detritivory ...... 161 10.5 References, and Related Bibliography ...... 167 11 Non Predictive Models for Competition ...... 168 11.1 Introduction ...... 168 11.2 Modelo SB-BACO3 ...... 168 11.3 A generalization of model SB-BACO3 ...... 181 11.4 Discrete model SB-BACO4 ...... 184 11.5 References, and Related Bibliography ...... 188 12 Predictive Models for Competition SB-BACO2, and SB-BACO6 ...... 189 12.1 Introduction ...... 189 12.2 Model SB-BACO2 ...... 189 12.3 Model SB-BACO5 ...... 204 12.4 Model SB-BACO6 ...... 210 12.5 Competition, and Total Effects ...... 214 12.6 References, and Related Bibliography ...... 218 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps...... 220 13.1 Introduction ...... 220 13.2 The Characteristic Parameters of the Species ...... 220 13.3 The Relative Variation Rates ...... 221 13.4 Simulations with model SB-BACO2 ...... 224 13.5 The Butterfly Effect ...... 236 13.6 Evaluation of the Simulations ...... 238 16 © L. S. Barreto, 2017. Theoretical Ecology Contents

13.7 References, and Related Bibliography ...... 239 14 Modelling Mutualism ...... 240 14.1 Introduction ...... 240 14.2 A Gompertzian Model for Facultative Mutualism (FM+FM) ...... 240 14.3 A Gompertzian Model for Obligatory Mutualism (OM+OM) ...... 245 14.4 A Gompertziano Model for Facultative, and Obligatory Mutualism ...... 250 14.5 First Gompertzian Model for a System with Three Mutualists ...... 254 14.6 Second Gompertzian Model for a System with Three Mutualists ...... 261 14.7 Third Gompertzian Model for a System with Three Mutualists ...... 267 14.8 References, and Related Bibliography ...... 272 PART III ...... 274 Com munity, and Ecos ystem ...... 274 15 Conceptual, and Mathematical Models for the Community ...... 275 15.1 Introduction ...... 275 15.2 The Nature of Communities, and Ecosystems ...... 276 15.3 A Model of a Community ...... 277 15.4 References, and Related Bibliography ...... 289 16 Modelling, and Analysis of the Ecosystem ...... 291 16.1 Introduction ...... 291 16.2 The Ecosystem to Be Modelled ...... 291 16.3 Modelling, and Simulating the Proposed Ecosystem ...... 293 16.4 The Effects of the Omission of Interactions ...... 310 16.5 Ascendency ...... 311 16.6 The Ascendency of the Matrix of Total Positive Effects ...... 320 16.7 Conclusive Comments ...... 322 16.8 References, and Related Bibliography ...... 323 PART I V ...... 325 Applications ...... 325 17 Identification of Keystone Species, and Controlling Components in the Ecosystem ...... 326 17.1 Introduction ...... 326 1 7. 2 The Proposed Procedure ...... 326 17.3 The Identification of Keystone Species ...... 327 17.4 The Identification of Controlling Components in Ecosystems ...... 328 1 7.5 Conclusive Remarks ...... 329 17.6 References, and Related Bibliography ...... 329 © L. S. Barreto, 2017. Theoretical Ecology 17 Contents

Appendix. R scripts for the simulation of the ecosystem without omissions, and without competition ...... 330 18 Developmental, Structural, and Functional Sensitivities to Initial Values ...... 336 18.1 Introduction ...... 336 18.2 Analysis ...... 336 18.3 Conclusion ...... 338 PART V ...... 339 Theory Evaluation ...... 339 1 9 Theory Evaluation ...... 340 1 9.1 Introdu ction ...... 340 19.2 Semantic Unity ...... 340 19.3 Evaluation ...... 340 19 .4 References ...... 342 18 © L. S. Barreto, 2017. Theoretical Ecology 1. Introduction

1. Introduction

1.1. The Scope of this Book

In Barreto (2005), I presented a first attempt to establish a unified approach to theoretical ecology. Later, written in Portuguese, I expanded the ambit of my former text (Barreto, 2013, 2014, 2016). The scope of this book is to present an English text of the complete, and updated version of my unified theory for organisms, populations, communities, and ecosystems displayed in the previously mentioned references. I assume that the reader is already familiar with mathematical ecology, and has the minimum knowledge of ecology, mathematics, dimensional analysis, and statistics to feel comfortable when reading texts of this discipline. Thus, it is clear that my purpose is not to introduce the reader in the subject of mathematical or theoretical ecology. You can find, elsewhere, several good books dedicated to this aim, such as Berryman, and Kindlman (2008), Case (2000), Hasting (1997), Kot (2001), Rockwood (2006), Roughgarden (1998), Stevens (2009), Vandermeer, and Goldberg (2003). The only reality that exists is the ecosystem. Isolated populations, population interactions, and communities are conceptual abstractions. Thus, a unified mathematical theory for ecology must aim to establish a hierarchy of models, from the organism to the ecosystem. To the hierarchy of conceptual systems (biosystems) frequently covered in the books of theoretical ecology (1-3), let me add two more:

organism (level 1) isolated population (2) population interactions (3) community (4) ecosystem (5)

I present a hierarchy of models such that the model for the biosystem of level n+1 is an expansion of the model for the system of level n, without formal or conceptual discontinuities. The models for interactions also have the capacity to accommodate, simultaneously, more than one type of interaction. After, I will show that communities and ecosystems are linear stochastic dynamic systems (LSDS) that can be modelled by multivariate autoregressive models of order 1 (MARM(1)). The way my theory is constructed, given its sistemicity, and deducibility, the empirical validation of level n+1 implies the validation of all levels of inferior hierarchy. Thus, to validate the all theory, I only have to empirically sustain that ecosystems are linear stochastic dynamic systems that can be modelled by MAR(1). A complete description of the book is displayed in section 1.3. In brief, my ultimate aim is to attempt a conceptual, and formal reconstruction of the reality referred to the structure, and dynamics of ecosystems.

1.2 The Fundamental Assumptions

The elaborations here presented assume two levels of very basic assumptions. The first level comprehends the following philosophical hypothesis (Bunge, 2005: section 5.9):  The world external to my subjectivity has a real, concrete existence. © L. S. Barreto, 2017. Theoretical Ecology 19 1. Introduction

 The reality has a multilevel structure, and it is not a homogeneous block. Each level has its own properties and laws. The levels are not independent of each other, but are related. This concept was already applied in the previous section  The external world it is not lawless, but evince ontological determinism.  The external world can be known.  Logic and mathematics are autonomous formalisms. The second level is clarified by the following explanations: A1. When life appeared, in the Earth, the reality in our planet was already submitted to physical and chemical laws. There is a continuum from physics to life, through chemistry. But each one of these levels of the organization of matter exhibits his particular set of emergent properties and regularities. The existence of life is affected by ecophysiological, chemical, and physical constraints. A2. Organisms live in space: two dimensional space (almost all terrestrial organisms), and three dimensional space (almost all aquatic organisms). The growth of organisms and populations, are related to the physical variables of time and space. Ecology studies the effects of environmental factors (temperature, soil fertility, etc), but pays little attention to the basic physical relation of biology with space and time. This forgotten link can be introduced, in the discourse of ecology, through the metric concepts (variables, and constants) here used (figure 1.1). This attitude facilitates the use of dimension analysis, and allometry, as detailed in the next chapter.

Figure 1.1. The basic triangle of the presence, and growth of organisms, and populations in the physical space. PL= power of the linear dimension

Now we arrived to a set of fundamental assumptions that permeates all the theory. The unifying concepts that bind all biosystems previously mentioned are: • A1. The linkage between the power of the linear dimension of the ecological variables among them, and with the physical space; • A2. The existence of allometric relationships intra, and inter biosystems; • A3. A unique pattern of growth for organisms and isolated populations: the Gompertz equation; • A4. Populations interactions can be modelled as modifications of the Gompertz pattern. Within the perspective of an axiomatic theory, these propositions can be seen as the fundamental axioms of the theory.

1.3 The Book

This book contains five main parts: Part I. Isolated populations. The issues here covered are the following ones: • Organism and population variables 20 © L. S. Barreto, 2017. Theoretical Ecology 1. Introduction

• The dynamic properties of these variables: self-similarity (allometry) • The pattern of growth of organisms and isolated populations: the Gompertz model • The laws that isolated populations abbey • Demography • Modelling structured populations • The time self-similarity of biological growth

Part II. Population interactions. This part covers the following issues: • Amensalism • Commensalism • Competition • Detritivory • Epidemiology • Herbivory, fitophagy • Mutualism • Pantophagy • Parasitism e parasitoidism • Predation, and zoophagy

Part III. Community, and ecosystem. Here, I sustain that communities, and ecosystems are linear stochastic dynamic systems that can be modelled as multivariate auto-regressive models of order 1. The chapters are the following: • Modelling, and simulating the community • Modelling, and simulating the ecosystem

Part IV. Applications of the theory • A procedure to identify keystone species, and controlling factors in the ecosystem-based • Developmental, Structural, and Functional Sensitivity to Initial Conditions

Part V. Evaluation of the theory.

When considered useful, I introduce scripts of Maxima, R, Scilab, and wxMaxima to sustain statements, conclusions, and to illustrate how they can be obtained, using free software.

1.4 Theory Synopsis

The theory that will be explained in this book can be concisely described by the following propositions:

Organism ➢ The variables describing the organism as a whole follow the Gompertz equation, and are alllometricaly related. ➢ The weight of the organs of the body follow the Gompertz equation.

Isolated populations ➢ The variables describing the dynamics of a cohort (number, and biomass) as a whole follow the Gompertz equation. © L. S. Barreto, 2017. Theoretical Ecology 21 1. Introduction

➢ The mean values of the descriptors of a population (e.g., individuals height) follow the Gompertz equation. ➢ All these variables are related by allometric equations. ➢ The powers of the linear dimension of the ecological variables relate their growth to the physical space.

Community ➢ Populations interactions are modelled by modified Gompertz equations. ➢ Communities are linear stochastic dynamic systems that can be modelled by MAR(1).

Ecosystem ➢ Ecosystems are also linear stochastic dynamic systems that can be modelled by MAR(1).

As any scientific theory, the theory that you will read ahead is partial (its approach to its system of reference – the ecosystem – involves a high degree of abstraction, and it is dominated by its mathematization), and approximate (it is not exempted of error).

1.5 Connection

In this same CD there is a folder named ‘Conexos’. In this folder there is a file named Trees2- Forests (T2F) where I display my unified theory for trees, and forests. I will refer to this file to let you know where there are available supplementary applications, and illustrations of the issues approached in this book. In the same folder ‘Conexos’, there is a file named Theoeco containing the first edition of this book.

1.6 References

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. E-book. Costa de Caparica. Barreto, L. S., 2013. Ecologia Teórica. Uma outra Explanação. I. Populações Isoladas. E-book. Costa de Caparica. In- cluded in the CD. Barreto, L. S., 2014. Ecologia Teórica. Uma outra Explanação. II. Interações entre Populações. E-book. Costa de Capar- ica. Included in the CD. Barreto, L. S., 2016. Ecologia Teórica. Uma outra Explanação. III. Comunidade e Ecossistema. E-book. Costa de Capar- ica. Included in the CD. Berryman, A. A., and P. Kindlmann. Population Systems. A General Introduction. Springer, Berlin. Case, T. J., 2000. An Illustrated Guide to Theoretical Ecology. Oxford University Press, Oxford, U. K. Hasting, A., 1997. Population Ecology. Concepts and Models. Springer, Berlin. Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, Cambridge, United Kingdom. Rockwood, L. L., 2006. Introduction to Population Ecology. Blackwell Publishing, Oxford, UK. Roughgarden, J., 1998. Primer of Ecological Theory. Prentince-Hall, Inc., Upper-SaddleRiver, New Jersey. Stevens, M. H., 2009. A Primer of Ecology with R. Springer, Berlin. Vandermeer, J. H. e D. E. Goldberg, 2003. Population Ecology. First Principles. Princeton University Press, Prince- ton. 22 © L. S. Barreto, 2017. Theoretical Ecology PART I

PART I

ISOLATED POPULATIONS

In the first part of the book, I will approach: • Organism and population variables • The dynamic properties of these variables: self-similarity (allometry) • The pattern of growth of organisms and isolated populations: the Gompertz model • The laws that isolated populations abide • Modelling structured populations • Demography • The time self-similarity of biological growth

2 Population Descriptors, and other Basic Concepts

2.1 Introduction

The assumptions previously stated require a more detailed annotation for the descriptors or variables used to represent the entities here considered, and some of their properties. The proposed symbols will allow:  To make more clear the relations among the different variables of the same biosystem;  To establish the relation between the dynamics of biosystems organism, and population;  To apply in a more straightforward way dimensional analysis to the variables;  To clearly establish allometric relationships that emphasize the self-similarity of the geometry of these subsystems. Thus, the dynamics of the variables will emerge harmonized, and integrated.

2.2 Basic Concepts of Dimensional Analysis

Often, ecology students are not familiar with dimensional analysis (DA). Given this situation I present a short section dedicated to this issue. DA assumes: • Physical laws do not depend on the units used. • The definition of fundamental quantities, such as time (T), mass (M), length (L). Systems of units can be constructed on sets of these fundamental quantities. Since 1960, there is an internationally accepted system of units, called the International System of Units (SI, from the French name Système international d’unités). The SI has seven quantities, to which are associated seven base units (e.g., Legendre, and Legendre, 1998: Table 3.1). In ecology, the more often used units are, L, M, T, temperature (Θ), luminous intensity (J), and amount of substance (N). • The fundamental quantities can be combined to construct derived units (such as units for area, work, and force). Derived units are not only simple products of the fundamental units, but that they are often powers and combinations of powers of these units (e.g., Le- gendre, and Legendre, 1998: Table 3.2). Simple examples of derived units will be introduced ahead. The theoretical support of DA is the π theorem, which is also known as the Buckingham theorem. DA can be seen as a process to eliminate spurious information in order to obtain dimensional sets. It is used in physics, and engineering to plan experiments, and also to establish equations. In the context of this book, the concept of dimensional homogeneity has particular relevance. An equation is correct when both sides have he same dimension. If this happens the ratio of both sides is dimensionless or has dimension zero. Let us introduce a few examples, applying a notation that will be clarified ahead.

The dimension of length (linear extension) is represented by y1 within square bracket:

1 [y1]=L =L (2.1) 24 © L. S. Barreto, 2017. Theoretical Ecology 2 Population Descriptors, and other Basic Concepts

The area A of a square of side y is A =y x y=y2. Thus, the dimension of area is:

[A]=L1 x L1= L1+1 = L2 (2.2)

The volume of a cube of edge y, V, is:

[V]=L x L x L= L3 (2.3)

The volume per area unit, VA, has dimension 1:

[VA]=L3 / L2 =L3-2 = L (2.4)

Variables referred to an area per area unit are dimensionless, this is, they are constant:

2 2 2-2 0 [y0]= L / L = L = L (2.5)

Let the number of individual be dimensionless, thus density (the number of individuals per area unit) has dimension:

[Density]= L0 L-2= L-2 (2.6)

Similarly, concentration or the number of individuals per volume unit has dimension L-3 ([Concentration]= L0 L-3= L-3). Velocity is the space moved in the period of time. Thus we can write:

[Velocity]=space time-1= L T-1 (2.7)

DA will be used to establish allometric equations, and related relationships. Free software for DA is Dimensions by Dr. John Kummailil (2009), and the library dimension.mac, in software Maxima. A text related to the issues approached in this chapter, and in the next one is Pennycuick (1992). Burton (2001) presents a, and accessible introduction to DA in the biological context.

2.3 Organism and Population Variables

I will use the following notation to refer the descriptor of organism, and isolated populations :

yi,j,t

i = power of the linear dimension (PL) j = identify a variable among the ones with the same i t = age. It can be omitted if not necessary

The values of i here admitted are presented in Box 2.1. © L. S. Barreto, 2017. Theoretical Ecology 25 2 Population Descriptors, and other Basic Concepts

Box 2.1. Powers of the linear dimension of population descriptors or variables

i=-3. Associated to a population that occupies a 3 dimension space (concentration. For example, phytoplankton. i=-2. Associated to a population that occupies a 2 dimension space (density). For example, sessile organisms. i=0. Associated to variables that do not change with time. For example, foliar area per area unit. I=0,666666…. Related to the total biomass or volume per area unit of a sessile population. For example, plants (2,666666+(-2)=0,666666). i=1. Associated to variables with linear dimension, such as height, diameter, and length. i=2. Associated to the growth of the number of organisms that use the resources of a 2 dimension space and to the components of the biomass of tree crowns. Variables of organisms that are constant when referred to area unit (2-2=0). i=2,666666… .Refers to the total biomass or volume of organisms. It indicates fractal geometry. i=3. Associated to the growth of the number of organisms that use the resources of a 3 dimension space. Variables of organisms that are constant when referred to volume (3-3=0). i=4,666666…. Related to the total biomass or volume of a non sessile population, per area unit E.g., terrestrial animals. (2,666666+2=4,666666). i=5,666666…. Related to the total biomass or volume of a non sessile population, per volume unit. E.g., fishes. (2,666666+3=5,666666).

I illustrate the application of this concept to trees, and forests in Table 2.1. 26 © L. S. Barreto, 2017. Theoretical Ecology 2 Population Descriptors, and other Basic Concepts

Table 2.1. The description of the variables of trees, and a population of trees

A. Trees

As the reader can verify, the value of i of a variable referred to stand is equal to the homologous referred to the tree minus 2:

istand = itree-2, as supported by dimensional analysis. The value of PL for root biomass probably needs a confirmation or eventual improvement. The non integer values of i (0,6666, 2,6666) let us assume that trees, and forests have fractal geometry. © L. S. Barreto, 2017. Theoretical Ecology 27 2 Population Descriptors, and other Basic Concepts

2.4 References

Kummailil, J., 2009. Dimensions. Available in www.baixaqui.com.br. Legendre, L. e P. Legendre, 1998. Numerical Ecology. Second edition. Elsevier, Amsterdam. Pennycuick, C. J., 1992. Newton Rules Biology. A Physical Approach to Biological Problems. Oxford University Press, Oxford. 28 © L. S. Barreto, 2017. Theoretical Ecology 3 Allometry

3 Allometry

3.1 Introduction

As allometry plays an important role in my theoretical construction, I dedicate a chapter to this issue. Allometry is seen as a matter with increasing scientific relevance in ecology (Brown, West, and Enquist, 2000; Anderson-Teixeira, Savage,and Gillooly, 2009). I also take advantage of this chapter to evaluate the correctness of the information displayed in Box 2.1. Allometry deals with the relative sizes of the components of a given entity. During growth, the parts of the body of an organisms are simultaneously conditioned by physical, chemical, and biological restrictions that harmonize the whole development of the organism. Allometry establishes relationships between: • The sizes of the components of the body of an organism; • Physiological functions and the sizes of the components of the body of an organism; • The sizes of the components of the body of an organism and attributes of its population. Allometric equations are used for a long time in biology. Already in the XIX century, Rub- ner (1883) verified that the rate f metabolism of the organisms changes with the size of the body. Also, in the first half of the XX century several authors approached allometric relationships, as can be verified in Savege et al. (2004). Here we must emphasize the contribution, in 1927, of de D’Arcy Thompson (Thompson, 1994), who used the principle of geometrical similitude. This principle is justified by the fact that most organisms are incompressible, and its mass per volume unit is almost equal to the one of sea water. In this particular situation we can use without distinc- tion, in allometric equations, the biomass or volume of the body or one of its parts. In the eighties of the last century, McMahon, and Bonner (1983), Petters (1983), Calder (1984), Schmidt-Nielsen (1984) published four books about allometric relationships that had a great impact on biological theory, and research, in a broad sense. For instance, the book written by Petters, in the appendices, contains about one thousand allometric equations, obtained from about five hundred references. After, the collective work Scaling in Biology (Brown, and West, 2000) reinforced the interest for allometry, in biology, and ecology. During many years, allometry was seen as a numeric curiosity with little explaining power, but since the last decade of XX century, an increasing, and lasting interest in allometric relationships and fractal geometry emerged in biology, and ecology. Today, it is admitted that allometric relationships contribute to deepen our knowledge in several areas of biology, and to establish fruitful, and unifying synthesis among them. As stated by West, Brown, and Enquist (2000:91), the allometric relationships: • Evince systematic simplicity in the most complex systems that exist – living organisms; • They are rare examples of universal quantitative laws in biology; • They suggest the existence of a set of fundamental principles common to all life; • Their interpretations, and clarifications open a wide field for research in biology, and ecology; • Allometricc equations also have a relevant role in other sciences, as physics, hidrology, geology, and economics. For more information on this subject see Brown e West (2000). In Niklas (1994), besides theoretical explanations, there is a rich assemblage of information about plant allometry. © L. S. Barreto, 2017. Theoretical Ecology 29 3 Allometry

Ginzburg e Colyvan (2004: chapter 2) present a set of allometric equations related to animal populations that they see as candidates to become ecological laws. Brose (2010) shows that allometric equations are indispensable to clarify the structure, and dynamics of food webs. Allometry is located in the broader issue of scale models (scaling; e.g., Barenblatt, 2003).

3.2 Allometric Equations

The simplest allometric equations has the form:

β ya,j = β0 yb,h 1 (3.1)

that can be transformed as:

log (ya,j) = log (β0) + β1 log (yb,h) (3.2)

Some authors call β0 allometric coefficient, and β1 allometric power.

The way the variation ybj affects yaj is regulated by β1. There are for cases:

 β1<0 → yaj decreases when ybj increases

 0<β1<1 →yaj increases slower than ybj increases

 β1=1→ yaj is proportional to ybj (isometry)

 β1>1→yaj increases faster than ybj increases These cases are illustrated in figure 3.1. The dimensional homogeneity of equation (3.1) requires that the following relationship is satisfied:

β1=a/b (3.3)

This basic relationship can be used to establish the desired allometric equations. I will present several applications using the variables of Table 2.1. A very often mentioned allometric equation is the 3/2 power law. This equations relates the growth of the average tree stem or biomass with stand density, in pure even-aged forests. Let me write it:

-3/2 y31= β0 y-21 (3.4)

The consequence of the absolute value of the power being greater than 1 (3/2=1,5>1) is the increasing of the standing volume with age, because the growth of the stem volume over- compensate the loss of volume caused by self-thinning. If the same value was equal to 1 the standing volume would be constant; It would decrease if the same volume was less than 1. Self-thinning is the result of intraspecific competition. The growth of self-thinned pure even aged stands (SPES) goes through three phases. In phase I, there are intraspecific competition between the small trees, and interspecific competition with the other vegetation. In phase II (FII), when the trees became taller with age, the trees compete dominantly among them. This intra specific competition for space and resources (mainly light, nutrients, and 30 © L. S. Barreto, 2017. Theoretical Ecology 3 Allometry water) is responsible for the changes that occur in the stand: decreasing of the number of trees (self-thinning), growth of the trees that remain, standing volume and biomass increasing. When the trees attain maturity, the growth becomes residual, and the same happens to intra specific competition. This is phase III. See figure 3.2.

Growth of yb Alllometric growths of ya

Figure 3.1.Allometric variations given by equation (3.1). Left graphic represents the variation of y b, and the right

graphic shows the variation of ya for several values of β1. Β0=0.5 © L. S. Barreto, 2017. Theoretical Ecology 31 3 Allometry

Figure 3.2. The continuous line represents the logarithmic form of the 3/2 power law. During its growth, the SPES is not always in this line, but occupies position in the space between the continuous line and the line with points (amplitude). The SPES moves from right to left, this is from larger density, and smaller trees to lower densities, and larger trees

Allometric equations show that the growths of the tree and the stand follow patterns that are interrelated, and harmonized.

Allometric equations can also be written using the symbol of proportionality:

We introduce two more allometric equations:

These equations are examples of the action of physical restrictions on the tree growth because they can be proved using engineering concepts (e.g., Niklas e Enquist, 2001:2926). These findings sustain the correctness of the way we defined the variables in Box 2.1, and Table 2.1. An allometric equation that had been intensively investigated is the one that relates the number of individuals of a given species (N), and the mean biomass of its individuals (W) per area unit:

(3.8)

As this equation is written, and being N dimensionless we can not use DA to estimate β 1. We still have two alternatives if we use the concepts of density, and concentration (Box 2.1):

(3.9) 32 © L. S. Barreto, 2017. Theoretical Ecology 3 Allometry

In this equation the power is close to -3/4 (-2/2.66667). This vale is supported by many authors but we think that it only applies to animals, and plants that use the resources of a two dimension space. Equation (3.9) is known as the allometry or law of Damuth (1987). If the species uses the resource of a volume we find:

In this equation the power is -1,125 (-3/2.66667). Let me introduce some empirical evidence that support equations (3.9), and (3.10). Equation (3.9) was first proposed by John Damuth (Damuth, 1981, 1987, 1991) whose research was mainly concentrated in mammals populations. Another corroboration of the same equation can be found, for instance, in Dobson, Bertram, and Silva (2003). Cyr (2000) analysed 240 populations of phytoplankton, zooplankton, and fishes of 18, natural, and artificial, well investigated lakes, and she found the overall value for β1=0.93±0.02.

Only for algae she found β1=0.95±0.10, and for the fishes β1=1.24±0.20. As each community has its own history, it is my understanding that one equation should had been fit for each lake. Values between -0.97, and -1.29 for phytoplankton in tropical, and sub-tropical zones of the Atlantic Ocean (Huete-Ortega, Cermeño, Calvo-Diaz, and Marañon, 2011). This example embraces an allometric relationship between an attribute of the organism, and a population parameter. Anderson-Teixeira, Savage, Allen, and Gillooly (2009) proposed allometric relationships involving not only organisms, and populations, but also the community, and the ecosystem. It must be emphasized that the deduced alllometric equations refer to the dynamics of isolated populations.

Population interactions (e.g., competition, predation) change the values of β1. This occurrence had already been empirically depicted in several studies, such as Cyr (2000), and will be illustrated with tree populations. The discrepancies of the value of β1 relatively to the theoretical values are smaller in communities whose populations had co-evolved. Equation (3.3), and the values in Table 2.1 provide allometric equations already empirically corroborated. For instance, Sprugel (1984) for SPES of Abies balsamea verified yhe con- b stancy of basal area. Referring to the allometric equation yij= β0 y-21 he found b=-1.04 for the foliage biomass of the mean tree, being the theoretical value here expected b=-1. This value sustains i=2 for the crown biomass. For the stem biomass we obtain b=-1.43, and for the above ground biomass b=-1.24. These values do not imply the rejection of Table 2.1. Osawa e Allen (1993) in SPES of Pinus densiflora, and Nothofagus solandri verified the constancy of the foliage biomass. In animals, the verification of equation (3.6) is difficult because animals can temporary lose weight, and regain it after, without changing their length or height. In the ecology of fisheries, the equation that relates the fish weight to its total length (equation (3.6)) is established for the main commercial fish species of many countries. The fitted equations are used to: a) obtain an estimated value of the biomass from the fish length; b) scrutinize the health state of the population; c) compare the behaviour of populations in differ-

ent places (e. g., Binohlan, and Pauly, 2000). Here, the fitted values β 1 have great variation. For example, Karachle, and Stergiou (2012) mention a study where for 3929 fittings, related to 1773 © L. S. Barreto, 2017. Theoretical Ecology 33 3 Allometry

species, the values of β1 are in the range 1.96 to 3.94, being 90% of these values in the interval

2.7-3.4. In this situation, β0 mirrors the fish form. These deviations are caused by three main reasons: • The species are not isolated, and thus their allometries are modified, as already stated; • The values are not obtained from a time series of a single cohort, but from several cohorts simultaneously sampled;

• During the year, for a given population, the value of β1 varies (e.g., Lima-Junior, Cardone, and Goitein, 2002). Refering to equation (3.6), in the context of the stated theory, in normal conditions of the environment for a given species:

• β1<2,6667 means that the species has restrictions on its growth caused by the presence of other species;

• β1=2,6667 suggests that the population behaves as isolated due to processes of coevolution;

• β1>2,6667 indicates that the species benefits from the presence of other species. Karachle, and Stergiou (2012: figure 1), fitted equation (3.6) to 60 species of the North Aegean Sea, and grouped the species according to the magnitude of the power, trophic function, and type of habitat. This resulted in he adjustment of eleven allometric equations. They found three values smaller then 2, one equal to 2.307, six values between 2.592 and 2.743, and one value equal to 3.034. Physiological stress, caused by very unfavourable environmental conditions, can origin values of β1 smaller than 2.6667. These authors verified that the allometric equations are the best models to describe the morphometry of fishes, and mention several authors that arrived to the same conclusion. Another important allometric equation is found in community ecology, and is known as species-area relation. It is written as

S=cAz (3.11) where S is the number of species in a patch of area A, and c and z are fitted constants. The value of c depends on the taxonomic group, but the value of z is in the interval 0.2 to 0.3 (May, Crawley, and Sugihara, 2007:125). Beside equation (3.9), and Kleiber’s law, Ginzburg e Colyvan (2004: chapter 2) see as ecological laws several allometric equations, they describe, and analyse. From Table 2.1 I conclude:  The allometric equations between variables of the same biosystem (organism, and population) mirror the internal order prevailing in the biosystem;  The allometric equations between variables of two different biosystems (organism, and population) guaranties that their growth follow a harmonized pattern.

3.3 Self-Similarity

An issue related to allometry is self-similarity or geometric similitude. To explain this concept I will use SPES. One important concept in forestry is how the trees are using the available resources for their growth. For instance, in sites of average fertility for a given species, it is highly probable 34 © L. S. Barreto, 2017. Theoretical Ecology 3 Allometry

that in their SPES with low density (small number of trees per area unit) the trees are not taking advantage of all resources locally available for their development. One way to evaluate the fertility of a site for a give species is the forest dominant height.

Dominant height is the mean height of the 200 trees with larger diameter at breath height (y12d, Table 2.1). Dominant height has a favourable characteristic: shows little sensibility to forest

density (y-2). Now, we can introduce an index of density relative to the local fertility. It is known as

Wilson´s index (Fw), is given by he ratio of the tree spacing (y18) and dominant height (y12d):

y Fw= 18 (3.12) y 12d

y1,8= Fw y1,3 (3.13)

This equation can be interpreted as “tree spacing measures Fw dominant heights". It is formally equivalent to the statement dominant height measures 24 meters. The difference lies on the fact that dominant height changes with time. But simultaneously, due to self-thinning, y18 also increase in a way that its size is constant, and equal to Fw. Thus, we say that the SPES maintains its self-similarity or geometric similitude. Using DA, [Fw]=L1L-1=L0. It is confirmed that Fw is constant. Let the spacing of trees be square and N is the number of trees per hectare (dimensionless). We obtain:

100 Fwsq= (3.14) √N y12 d

Equation (3.13) can be generalized as:

ya,j= ka,j,h ya,h (3.15)

This equation is sad isometric because PL=a for both variables, and in the allometric equa-

tion β1=1. I this notation, we write Fw as:

y18 k 182d= (3.16) y12d

This relationship is named isometric because the two variables has the same PL(=a), being

β1=1. In this notation, the isometric relationship of Fw is:

y1,8 k1,8,3= (3.17) y1,3

β In the equation (3.1) let us make yb,h 1=z. The same equation can be written as: © L. S. Barreto, 2017. Theoretical Ecology 35 3 Allometry

yaj=β0 z (3.18)

As time changes, the size of yaj is constant when measured with the variable metric z. This

constancy mirror the geometric similarity between the two variables we used. Here, β 1 is called the scale factor.

We can summarize the previous results. Between two elements of the two subsets Yi ex-

ists a binary isometric relation. Given two sets Ya, and Yb (for instance, columns of Table 2.1.A, and Table 2.1B) and two elements belonging to one of the sets, there is a binary relation of a al-

lometry between them. The relationships of allometry within each set Ya, and Yb ensure the geometric similarity of the tree, and of the forest during their growth. The allometric relationships between elements of each set guaranties the harmonized growth of the tree, and the stand. The consequences of these allometric relationships are the nomological structure of the dynamics of both trees, and stands. Let us insert an illustration.

Consider two ages T1 and T2. At age T1 variables yajT1 e ybjT1 were measured. At age T2 variable ybjT2 was measured. The allometric equation let us write:

y y a a, j , T 2 =( b, j ,T 2 )b (3.19) y a, j , T 1 yb, j ,T 1

then:

a yb , j ,T 2 b ya, j ,T 2=( ) ya, j ,T 1 (3.20) yb , j ,T 1

Let yb,j,T1 be the density, and assume that age T1 we measured all variables ya,j,T1 with interest. At any other age of the SPES, we only have to count the number of trees, to evaluate the

new values of the variables ya,j,T1 previously measured, as the following equation can be used:

a y−2, j , T 2 −2 ya, j ,T 2=( ) ya, j ,T 1 (3.21) y−2, j , T 1

The relation between the structure of a SPES at an age T>t0, and its structure at age t0, is the same that exists in engineering between a model and its prototype. The rule to scale the results of experiments with a model to a prototype is an extension of equation (3.21) as explained in Barenblatt (2003:38-39). Isolated populations are geometrically similar in time, and in space given the space-time symmetry. Populations with interactions with other populations do not evince self-similarity or isomorphism. The body of the majority of the species also shows isomorphism, maintaining the same shape during growth. In support of the previous statements, let me quote Gregory I. Barenblatt (Professor-in-Resi- dence at the University of California at Berkeley, and Lawrence Berkeley National Laboratory, Emeritus G. I. Taylor Professor of Fluid Mechanics at the University of Cambridge, Adviser, Institute of Oceanology, Russian Academy of Science, Honorary Fellow, Gonville and Caius College, Cambridge):

36 © L. S. Barreto, 2017. Theoretical Ecology 3 Allometry

“One may ask, why is that scaling laws are of such distinguished importance? The answer is that scaling laws never appear by accident. They always manifest a property of a phenomenon of basic importance, “self-similar” intermediate asymptotic behaviour: the phenomenon, so to speak, repeats itself on changing scales. This behaviour should be discovered if it exists, and its absence should also be recognized. The discovered of scaling laws very often allows an increase, sometimes even a drastic change, in the understanding of not only a single phenomenon but a wide branch of science. The history of science of the last two centuries knows many such examples”. (Barenblatt, 2003:xiii; italics in the original).

Thus, scalability is a very important characteristic of natural entities, and phenomena. More information, at an introductory level, can be found in Schneider (1994: chapters 13 and 14). Scaling is a recurrent topic in papers published in the most prestigious journals in the area of ecology, and biology. Before I close this chapter, let me recall a result from Brajzer (1999). This author showed that similarity, and allometry are characteristic of the growth that follows the Gompertz equation. This is, if we accept the allometry, and self-similarity of living organisms, coherently, we must admit the possibility that their growth follow the Gompertzian pattern.

3.4 References, and Related Bibliography

Anderson-Teixeira, K. J., V. M. Savage, A. P. Allen, and J. F. Gillooly, (dezembro de 2009) Allometry and Metabolic Scaling in Ecology. In Encyclopedia of Life Sciences (ELS). John Wiley & Sons, Ltd. Chichester. DOI: 10.1002/9780470015902.a002122 Bajzer, Z., 1999. Gompertzian Growth as a Self-Similar and Allometric Process. Growth Dev Aging, 63:3-11. Barenblatt, G. I., 2003. Scaling. Cambridge University Press, Cambridge. Barreto, L. S., 2007. The Changing Geometry of Self-Thinned Mixed Stands. A Simulative Quest. Silva Lusitana, 15(1):119-132. Barreto, L S., 2010. Árvores e Arvoredos. Geometria e Dinâmica. E-book, Costa de Caparica. Included in the CD. Binohlan, C., and D. Pauly, 2000. The length-weight table, In: Fishbase 2000: Concepts, design and data sources, Froese R. & D. Pauly, (Editors), 121-123, ICLARM, ISBN 971-8709-99-1, Manila, Philippines. Brose, U., 2010. Body-Mass Constraints on Foraging Behaviour Determine Population and Food Web Dynamics. Functional Ecology, 24:28-34. Brown, J. H., and G. B. West, (Editors), 2000. Scaling in Biology. Oxford University Press, Oxford. Brown, J. H., G. B. West, and B. J. Enquist, 2000. Scaling in Biology: Patterns, Processes, Causes and Consequences. In J. H. Brown, and G. B West, (Editors), 2000. Scaling in Biology. Oxford University Press. Pages 1-24. Burton, R. F., 2001. A Biologia Através dos Números. Um Encorajamento ao Pensamento Quantitativo. Editora Replicação, Lda., Lisboa. Calder, W.A., 1984. Size, Function, and Life History. Harvard University Press, Cambridge, MA. McMahon, T. A. e J. T. Bonner, 1983. On Size and Life. Scientific American Library, New York. Cyr, H., 2000. Individual Energy Use and the Allometry of Population Density. In J. H. Brown. and G. B. West, ( Edi- tors), Scaling in Biology. Oxford University Press. Pages 267-295. Damuth, J., 1981. Population Density and Body Size in Mammals. Nature 290, 699-700. Damuth, J., 1987. Interspeciﬁc Allometry of Population Density in Mammals and other Animals: the Independence of Body Mass and Population Energy Use. Biol. J. Linn. Soc. 31, 193-246. Damuth, J., 1991. Of Size and Abundance. Nature 351, 268-269. Dobson, F. S., Z. Bertram, and M. Silva, 2003. Testing Models of Biological Scaling with Mammalian Populations Densities. Canadian Journal of Zoology 81(5):844-851. Ginzburg, L., and M. Colyvan, 2004. Ecological Orbits. How Planets Move and Populations Grow. Oxford University Press, Oxford. © L. S. Barreto, 2017. Theoretical Ecology 37 3 Allometry

Huete-Ortega, P. Cermeño, A. Calvo-Diaz, and E. Marañon, 2011. Isometric Size-scaling of Metabolic Rate and the Size Abundance Distribution of Phytoplankton. http://rspb.royalsocietypublishing.org/content/early/2011/12/08/rspb.2011.2257.full Huxley, J. S., 1932.Problems of Relative Growth. Methuen, London. Mentioned in Burton (2001). Karachle, P. K., and K. I. Stergiou, 2012. Morphometrics and Allometry in Fishes, Morphometrics, Prof. Christina Wahl (Ed.), ISBN: 978-953-51-0172-7, InTech. Disponível em: http://www.intechopen.com/books/morphometrics/morphometrics-and-allometry-in-fishes Lima-Junior, S.E., I. Braz Cardone, and R. Goitein, 2002. Determination of a Method for Calculation of Allometric Condition Factor of Fish. Acta Scientiarum, 24(2):397-400. May, R. M., M. J. Crawley, and G. Sugihara, 2007. Communities:Patterns. Em R. May e A. McLean, (Editors), Theo- retical Ecology. Principles and Applications, Oxford University Press. Pages 111-131. Niklas, K. j., 1994. Plant Allometry. The Scaling of Form and Process. The University of Chicago Press. Niklas K. J., B. J. Enquist, 2001. Invariant Scaling Relationships for Interspecific Plant Biomass Production Rates and Body Size. Proc Nat Acad Sci U S A 98: 2922–2927. Owen-Smith, N., 2007. Introduction to Modeling Wildelife and Resource Conservation. Blackwell Publishing, Ox- ford. Osawa, A. e R. B. Allen, 1993. Allometric Theory Explains Self-thinning Relationships of Mountain Beech and Red Pine. Ecology, 74(4):1020-1032. Peters, R. H., 1983. The Ecological Implications of Body Size .Cambridge Univ. Press, Cambridge, U.K.. Ritchie, M. E., 2010. Scale, Heterogeneity, and the Structure and the Diversity of Ecological Communities. Prin- ceton University Press. Rubner, M., 1883. Ueber den Einﬂuss der Körpergrösse auf Stoff-und Kraftwechsel. Zeitschrift für Biologie 19, 535– 562. Reference obtained in Savage et al. (2004). Savage, V.M., J. F. Gillooly, W. H. Woodruff, G. B. West, A. P. Allen, B. J. Enquist, and J. H. Brown, 2004. The predom- inance of quarter-power scaling in biology. Functional Ecology, 18, 257–282. Schmidt-Nielsen, K., 1984. Scaling: Why Is Animal Size So Important? Cambridge Univiversity Press, Cambridge, U.K. Sprugel, D. G., 1984. Density, Biomass Productivity, and Nutrient Cycling Changes During Stand Development in Wave Regenerated Balsam Fir Forests. Ecological Monographs, 54(2):165-186. Anderson-Teixeira, K. J., V. M. Savage, A. P. Allen, and J. F. Gillooly, (Dezembro de 2009) Allometry and Metabolic Scaling in Ecology. In Encyclopedia of Life Sciences (ELS). John Wiley & Sons, Ltd. Chichester. DOI: 10.1002/9780470015902.a0021222 Thompson, D. A. W., 1994. Forme et Croissance. Éditions du Seuil/Éditions du CNRS. French translation of an Eng- lish text. Schneider, D. C., 1994. Quantitative Ecology. Spatial and Temporal Scaling. Academic Press. West, G. B., J. H Brown, and B. J. Enquist, 2000. The Origin of Universal Scaling Laws in Biology. In J. H. Brown, and G. B West, (Editors), 2000. Scaling in Biology. Oxford University Press. Pages 87-112. West, G.B., e J.H. Brown. 2004. Life's Universal Scaling Laws. Physics Today 57:36-42. 38 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

4 The Gompertz Equation

4.1 Introduction

In this chapter I sustain that there is only a unique pattern for organism and population growth: the Gompertz equation (EGZ). First, I present arguments to sustain my statement, and after I will deduce the EGZ under the perspectives of the organism growth (ageing effect), and of the population growth (density effect). In the remaining of the chapter, I will characterize the Gompertzian growth, considering both continuous, and discrete models.

4.2 A Unique Pattern for Biological Growth

Only after the third decade of the XX century, the EGZ started to be successfully used to reproduce the dynamics of systems in several domains (astrophysics, biology, medicine, ecology, and economy). In some literature, the EGZ is simply referred as the ‘natural law’ (Lauro, Mar- tino, Siena e Giorno, 2010). Its vast application and the ubiquity of the logarithm structure, that underlies it, led various authors to seek for proofs and justifications for the Gompertzian pattern, using several, and more basic principles approaches (synergistic and saturate systems, sen- escence in biological hierarchies, entropy, cellular kinetic). A review of these works can be found in Bajzer, Vuk-Pavlovic, and Huzak (1997). The organism that actually exist, representing an evolutionary process of many hundreds thousand years, have in common that their growth are conditioned by the same physical, and chemical laws. This situation led me to two conjectures: • The growth pattern is the same for all organisms, and their populations; • The unique pattern is the EGZ. To reinforce the verisimilitude of these conjectures I list some more arguments: • Several authors formally deduced that biological growth follows the EGZ; • Several authors verified that the EGZ is the most adequate to model bilogical phenomena. For instance, Zullinger et al. (1984) fitted the logistic, von Bertalanffy, and Gom- pertz equtions to 331 species of mammals of 19 order. The best fittings, in 49 species, were obtained with von Bertalanffy equation, and EGZ. Among others, see also Caution, and Vénus (1981); • For fisheries, Pradhan, and Chaudhuri (1998) showed that the EGZ was supperior to the other available alternatives; • Sibly et al. (2005) analised 1780 time series of populations of mammals, birds, fishes, and insects. They verified that EGZ had was the one with the best adherence to the rate of growth per capita. It is also verified: 1. The EGZ has the flexibility to reproduce empirical data from several origins, and apparently with different patterns: 2. EGZ is consistent with temporal, and spatial allometric variation; 3. EGZ is able to generate patterns of growth that mirror the bionomic strategies (life- histories) of the species; © L. S. Barreto, 2017. Theoretical Ecology 39 4 The Gompertz Equation

4. EGZ is able to generate consistent results for the growth of organisms, and populations considering different time scales (e.g., years, months, weeks); 5. Most organisms show allometric growth. As already stated, Bajzer (1999) showed that self-similarity and allometry are characteristics of the Gompertzian dynamics; 6. EGZ let we develop models for population interactions with desirable properties, as we will show ahead; 7. Lauro, Martino, Siena, and Giorno (2010) ‘show that the macroscopic, deterministic Gom- pertz equation describes the evolution from the initial state to the final stationary value of the median of a log-normally distributed, stochastic process. Moreover, by exploiting a stochastic variational principle, they account for self-regulating feature of Gompertzian growths provided by selfconsistent feedback of relative density variations. This well defined conceptual framework shows its usefulness by allowing a reliable control of the growth by external actions’ (adapted from the abstract of the paper). The adoption of the EGZ let us have an integrated, and unified approach for the biosystems organisms, populations, communities, and ecosystems. For a poorly unified science as ecology, this is an extremely valuable achievement. Two final examples. EGZ reproduces the regeneration of lost organs, as registered in Baranowitz, Maderson, and Connely (1979). Kritzinger (2011), verified in ostrich (Struthio camelus var. domesticus) not only the body variables had Gompertzian dynamcs, but also its chemical composition, and the biomasses of the body components (feathers, skin, other tissues, organs, and bones) exhibit the same pattern. The adoption of the EGZ has also implications in other sciences, as genetics where the logistic growth is assumed in several equations.

4.3 Assumptions

In the biosystem population, the following assumptions underlay the EGZ: A1. There is no migration, this is, the variations on the number of individuals are only due to births, and deaths of identical individuals. A2. The resources for the population growth, and maintenance are limited. Thus the population is affected by its density: the mortality, and birth rates change linearly with the logarithm of the density. A3. The effect of density upon mortality, and natality is instantaneously. A4. The probability of matting is independent of density. A3. Growth is continuous. A5. The age distribution is stable. A6. The environment does not change. If we accept the results in Lauro, Martino, Siena, and Giorno (2010) we can formulate the two following premisses: P1. In several scientific domains, the study of natural systems shows that they are stochastic, and evince dynamic with lognormal distribution. This is, lognormality is common in a variety of natural systems. P2. It can be proved that the Gompertzian dynamics deterministically mirrors the behaviour of the median of these stochastic processes. The inferred conclusion is: 40 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

C. Being organisms, and populations natural stochastic systems, it is highly probable that they can be correctly modelled by the EGZ.

4.4 Model

As we assume that the EGZ models both the individual, and population growth, we present two deductions of the EGZ, approached from two different perspectives: The effect of ageing upon organism growth, and the effect of the population density on its growth. Let us start with the population perspective.

4.4.1 The Gompertzian Dynamics of Populations Let us recall assumptions A2, and A3. Also it is convenient to remember that since the last quarter of the XIX century, several authors, following the pioneer statements of Galton (1879), and McAllister (1879), registered the ubiquity of the lognormal distribution to describe the ran- domness of natural, and social phenomena. In this conjecture, natality, and mortality are not simply affected by density, but by the logarithm of density. An illustration of this mechanism is inserted in figure 4.1, where Ns is the number of births, and Mo the numbert of deaths.

Figure 4.1. Graphical representation of the Gompertzian population growth

Assuming that the instantaneous rate of variation of the number of individuals is equal to the balance between births, and deaths multiplied by the number of individuals, we write:

dy =(n−m) y (4.1) dt

where

n=n0-n1 ln y (=Ns) (4.2)

m=m0+m1 ln y (=Mo) (4.3)

Thus, © L. S. Barreto, 2017. Theoretical Ecology 41 4 The Gompertz Equation

dy =(n −n ln( y)) y−(m −m ln( y)) y (4.4) dt 0 1 0 1

When the population attains its maximum size it is verified dy/dt=0. This implies n0-n1 ln yf

= m0+m1 ln yf, and we can write:

(4.5)

From equation (4.4):

(4.6)

(4.7)

Recalling equation (4.5), we obtain:

(4.8)

Let be:

(4.9) to obtain the differential form of the EGZ:

(4.10)

Let us use the free software wxMaxima to solve this ordinary differential equation (ODE), as presented in Box 4.1. 42 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Box 4.1. The integration of the differential equation of the EGZ, using wxMaxima

As exp(ln(x))=x, and being g=ab, mg=mab, the last output in Box 4.1 let us write the Gom- pertz equation in a form suitable for our posterior developments, in the book:

exp(-ct) y= yf R (4.11)

-1 as R=y0/yf, yf=y0 R , we can also write:

(exp(-ct)-1) y= y0 R (4.12)

where exp refers the exponential function(exp(x)=ex). I use a graphic to illustrate the deduction we just obtained. Consider the following EGZ (*=multiplication),

y= 2*0.1128 (exp(-0.059*i)-1) the two following equations for natality , and mortality m=0.9122425+0.9*ln(y) n=4.3625858-0.3*ln(y) and the recursive equation:

yt+1=yt+nt-mt

Using these equations with software R, we obtained the graphics in figure 4.2. In the upper left graphic we show the sizes of the population with the continuous model, and the recursive one (red dots). The differences are due to the employment of a discrete model to simulate a continuous variable. In the upper right graphic we display the values of n (red), and m. © L. S. Barreto, 2017. Theoretical Ecology 43 4 The Gompertz Equation

When the asymptotic size is attained occurs m=n. In the lower graphic we exhibit the population increments, this is, the difference n-m that converges to zero.

Comparing the two simulations Natality, and mortality dynamics

Natality minus mortality

Figura 4.2. Simulation of a Gompertzian variable using the explicit form of the EGZ, and using the effect of the density on natality, and on mortality. For more details see the text

In figure 4.3, it is exhibited an example of the EGZ simulating a population. 44 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Biomass growth Current growth, dy/dt Growth per capita

Figure 4.3. An example of the application of the EGZ. Simulation of the total biomass of an hypothetical small forest of common oak (Quercus robur) that at age 20 has 74 megagrams (Mg). ub= units of biomass; ut= units of time. The age of the maximum current increment is 22 years. Obtained with software R

4.4.2 The Ageing Perspective Let assume that the resources do not constrain the growth of an organism, and it is proportional to the size of the population. This assumption leads to the differential equation of the exponential growth:

As a complementary hypothesis we assume that with age the organisms becomes less ef- ficient. This ageing effect causes the exponential declining of r:

(4.13)

with the initial condition r=r0 , at age t=0

r=r0 exp(-ct) (4.14)

Replacing equation (4.14) in equation (4.13), we obtain: © L. S. Barreto, 2017. Theoretical Ecology 45 4 The Gompertz Equation

(4.15)

To solve this equation we write::

(4.16)

integrating gives

(4.17)

Thus y=y(0) exp(r(0)(1-exp(-c t))/c) (4.18)

Coherently, the limit of equation (4.18( when c→0 is an exponential:

y= y0 exp(r0t) (4.19)

Given equation (4.16), when t→∞ we get:

yf=y0 exp(r0/c) (4.20)

Replacing equation (4.19) in equation (4.18) we obtain a alternative form of the EGZ:

y=yfexp(-r0/c exp(-ct)) (4.21)

As exp(-r0/c)=1/exp(.r0/c)=y0/(y0 exp(.r0/c))=y0/yf=R, we write:

exp(-ct) y= yf R (4.22)

Given the two different deductions of the EGZ, we may admit that both organisms, and populations have Gompertzian dynamics. For small values of t (initial growth) as exp(-ct)≈1-ct, thus 1-exp(-ct)=ct. This implies exponential growth, and thus:

y=y0 exp(-r0/c) (4.23)

Let us do a=-ln R, and b=e-c To obtain a continuous alternative form of the EGZ: 46 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

t yt=yf exp(-a b ) (4.24)

Remember the definitions of variables introduced in chapter 2. PL=0 implies R=1; PL<0 implies R>1 ( the variable decreses with age); PL>0 implies R<1 (the variable increases with age). We made several fittings of the EGZ, with biological data, and we always found 0

exp(-ct) yi,j= yi,j,f Ri (4.25a)

(exp(-ct)-1) yi,j=yi,j,0 Ri (4.25b)

Ri, and c are real positive numbers. For a given species they are constant.

As it was sad in the introduction, several authors proposed other deductions, and justifications for the EGZ, within the biological context. For instance, Ling,and He (1993) used an entropic argument to prove that the EGZ is preferable to the logistic, and to the von Bertalanffy equations. Bajzer, and Vuk-Pavlovic (2000) deduced the EGZ from the existence, in organisms, of both cells that reproduce, and others that do not reproduce. Probably, Medawar (1940) was the first to demonstrate that biological growth has a Gompertzian pattern. In the exponentials of equations (4.11), (4.12), and (4.25) it is assumed that the first values correspond to age (t0=0). For instance, if the age of the first value is n (t 0=n), the exponential of these equations must be write as exp(-c(t-n)). Thus:

exp(-c(t-n)) yt=yfR

If we want to maintain the form of the mentioned equations, we must use a corrected value of R as:

exp(cn) Rcorrected=R

4.4.3 The Behaviour of the EGZ Differentiating eq.(4.11) , equating to zero, I obtain the age of the inflection point, t*:

(4.26)

Replacing equation (4.26) in equation (4.21), we obtain the value of y at age t*, y*:

* y = yf exp(-1) (4.27)

* y = 0,368 yf (4.28)

From equation (4.21) it is obtained r0=-c ln R. replacing in equation (4.26) we find: © L. S. Barreto, 2017. Theoretical Ecology 47 4 The Gompertz Equation

(4.29)

The relative variation rate (RVR) of yi is defined as:

d yi −ct RVRi= =−c ln(Ri)e (4.30) y i dt

Equation (4.10) let us write:

RVR=c (ln yf-ln y) (4.31)

In the EGZ, the variation of RVR, with age, is not linear as it is associated to a negative exponential (equation (4.31)). exp(-ct) In equation y=yf R only the exponential varies with age. The exponential is equal to 1

when t=0, and converge to zero as t →∞. As R=y0/yf, at t=0, y=y0, and when t→∞, y→yf. This is illustrated in figure 4.4.

Figure 4.4. The anatomy of equation y=555 x 0.00891exp(-0.041 t). Mean annual increments represented by the line in black colour. Maximum current annual increment= 8.370; maximum mean annual increment= 6.180

Now, we insert several figures (4.5 to 4.8) to show the sensitivity of the EGZ to y0, yf, R, and c. They are reproduced from Barreto (2013), and although the titles of each graph is in Por- tuguese we assume they are easily interpreted. 48 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.5. Sensitivity of y, dy/dt 1/y e dy/dt to variations of y0, being c=0.05, and yf=200. As yf is maintained con-

stant, R change with y0, and is equal to 0.05, 1.05, 2.05. R>1 implies decreasing of the variable © L. S. Barreto, 2017. Theoretical Ecology 49 4 The Gompertz Equation

Figure 4.6. Sensitivity of y, dy/dt 1/y e dy/dt to variations of yf, being c=0.05. As R is constant, y0 has the values 60,

100 e 160. Now, dy/dt changes ad yf, but dy/dt 1/y is not altered

Figure 4.7. Sensitivity of y, dy/dt 1/y e dy/dt to variations of R, being c=0.05, and y f=500. The value of R>1 implies decreasing y, negative values for dy/dt, and dy/dt 1/y, being y 0=1000. When R<1, as yf does not change, y0 has values 60, and 6. The smaller is R, the greater is RVR 50 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.8. Sensitivity of y, dy/dt 1/y e dy/dt to variations of c, being R=0.2, and y0=10. The larger is c, the larger is dy/dt, thus the growth is faster. The larger is c the larger is the RVR

4.5 Analysys of the Model

4.5.1. Graphical Analysis Let us use wxMaxima to find the equilibrium or fixed points of the ODE of the EGZ:

There are two fixed points: y=0 e y=yf. As the logarithm of 0 does not exist, we ignore this equilibrium point It can be verified the following:

 For values of y between y0, and yf,y occurs y’>0, and y increases . (y0≤y0, y grows)

 For values of y larger than yf, y’<0, y decreases, and return to yf. (y> yf → y’<0, y decreases)

 The value of y is stationary when y= yf. These results can be illustrated in the graphic of figure 4.9. © L. S. Barreto, 2017. Theoretical Ecology 51 4 The Gompertz Equation

Figure 4.9. The interpretation of the curve dy/dt(=y’) under the perspective of the stability of the fixed points. See the text

4.5.2. Formal Analysis

Let us differentiate equation (4.10) with respect to y:

(4.32)

When y=yf it comes:

(4.33)

As c>0, yf is a stable equilibrium.

4.6 Discret Models

4.6.1 Recursive Equations for the EGZ

Equation (4.10) let us write:

(4.34)

(4.35) 52 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

(4.36)

Thus the recursive equation for EGZ is:

(4.37)

To obtain a general solution, I write the model using yt+1, and yt as:

(4.38)

If c is enough small, it occurs 1-c)t ≈ e-ct , and the solution is the same as for the continuous model (eq. (4.11)). I introduce to more recursive models for the EGZ:

yt+1= yt exp(c ln yf-c ln yt) (4.39)

yt+1=yt + yt c ln(yf/yt) (4.40)

c If in equation (4.40) we do λ=yf , and θ=1-c, it can be written as:

θ yt+1= λ yt (4.41)

In figure 4.10, I compare the data generated with discrete models (4.37), and (4.41) with the continuous model for EGZ (equation (4.12)). Equation (4.41) has a better performance. © L. S. Barreto, 2017. Theoretical Ecology 53 4 The Gompertz Equation

Figure 4.10. Comparing the discrete dynamics of equations (4.37) in the left graphic, and (4.41) in the right graphic, with the continuous form of the EGZ (line). We used y0=1, yf=50, and c=0,5

A known discrete model for population growth, used in fisheries, is the Beverton-Holt model:

(4.42)

It is possible to fit the Beverton-Holt model to data with Gompertzian dynamics. Let me illustrate. The two following models simulate the total biomass of forests of maritime pine (Pinus pinaster; yp), and beech (Fagus sylvatica; yf):

exp(-0.05*(t-10)) yp= 350*0.549742 exp(-0.043*(t-10)) yf= 981*0.101843

The adjustments give the following values for the parameters: Maritime pine: λ=1.062317; a= 0.0001792 Beech: λ=1.0631373; a= 0.0000650 The values given by the EGZ, and the Beverton-Holt model, for both species, are virtually equal, as illustrated in figure 4.11. The values of λ are insensitive to the amount of resources available for the forest growth, but the larger are this amount the smaller is a. The parameter a can be interpreted as a coefficient of intraspecific competition. 54 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.11. Comparing the fitting of the Beverton-Holt model (line) with data with Gompertzian dynamics. Total biomass, Mg, in the y axis; age-10 years in the x axis; maritime pine in the left graphic. Initial data ***

Satoh (2000) proposed a discrete form for the EGZ that has an exact solution. Satoh’s equation is:

(4.43) with the form of a function of time, as exact solution:

(4.44)

In this equation it is verified 1+δ ln b= e-c. To illustrate the Gompertzian dynamic generated by equation, put t=1,2,3…55, a=0.08,

δ=0.5, b=0.8, yf=50. Tothe values generated by equation (4.44) we fitted equation (4.11), and obtained yf=50, R=0.08 e c=0.1183014. The simulations made with equations (4.25a), and (4.46) are exhibited in figure 4.12. The two time series are virtually coincident.

Figure 4.12. Comparing the dynamics of equations (4.11), and (4.44). anos means years © L. S. Barreto, 2017. Theoretical Ecology 55 4 The Gompertz Equation

4.6.2 Analysis of the stability of the discrete EGZ Recall equations (4.37),(4.40). Put:

(4.45)

f2= yt + yt c ln(yf/yt) (4.46)

f3= yt exp(c ln yf - c ln yt) (4.47)

When yt=yf occurs:

i=1,2,3 (4.48)

The fixed point ŷ is stable if the condition |df(ŷ)/dy|<1 is verified. In equation (4.48) this implies 0

c>2: yf is unstable; oscillations start in the neighbourhood of yf In figure 4.13, using cobweb graphics, the sensitivity of equation (4.39) to the values of c is exhibited. The following occurs: c<2 Stable equilibrium point 2≤2,71845 Extinction, y=0

In figure 4.14, I summarize the information about the stability of equation(4.39), where Lyapunov exponents, and the bifurcation diagram give consistent results. As already stated, I fitted the EGZ to data of several species (micro-organisms. plants, and animals), and I did not find any value greater than 1 (c<1). It is highly unprovable that isolated populations with Gompertzian dynamics show chaotic behaviour. 56 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.13. Six situations of the stability of equation (4.43). Stable:c=0.5 ; 2-point cycles:c=2,2 ; 4-point cycles: c=2,46 ; 16-point cycles: c=2,488 ; 32-point cycles: c=2,4901 ; chaos: c=2,491 © L. S. Barreto, 2017. Theoretical Ecology 57 4 The Gompertz Equation

Figure 4.13. Conclusion 58 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figura 4.14. Lyapunov exponents, and bifurcation diagram associated to equation (4.43) © L. S. Barreto, 2017. Theoretical Ecology 59 4 The Gompertz Equation

Figure 4.15. Sensitivity of equation (4.42) to the values of c, being y0=1, and yf=20

4.7 The EGZ with Time Lag

To study the behaviour of the EGZ with time lag, we modify equation (4.37) as follows:

(4.49) being s=1,2,.. The following dynamics are observed, dictated by the product cs: cs≤0,36 monotonic convergence for the equilibrium 0,361.56 extinction of the population In figure 4.16, I illustrate these patterns of behaviour. 60 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.16. Behaviour of equation (4.49) for several values of the product cs

Let us introduce the following equation:

(4.50)

We used numeric simulations to find: cs≤0,45 monotonic convergence for the equilibrium 0,45

Figure 4.17. Several behaviours of equation (4.50). When cs=1.854, y starts having very small values, and close to period 5000 explodes with a sequence of very high values

It is worth to mention that the high flexibility of the EGZ let us successfully fit this equation without time lag to data obtained from an EGZ with time lag. Let us introduce an example. We use the following model with 3 periods of lag:

yt+h=yt+h*0,05*yt*(ln 400-ln yt-3) (4.51)

At the beginning of the simulation we have yt-3=2, yt-2=5, yt-1=10, yt=18, and h=0.01. To the data geneated with this model we fit the following equation:

exp(-0,0458759*t) yt=402,88175*0,0050296 (4.52)

with an absolute percent error equal to 0.1155045. In figure 4.18, we display the values obtained with equation (4.51), and those generated by equation (4.52). They are virtually coincident. 62 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

Figure 4.18. Data generated by equation (4.51), red, and those generated by the fitted equation (4.52), black

This verification strengths the truthlikeness of the assumptions, and results of the paper by Lauro, Martino, Siena e Giorno (2010). Probably, the use of models with time lag is a consequence of the low flexibility of the logistic equation.

4.8 The Specific Constancy of c, and Ri

Today, it is accepted that the growth rates of the species is an attribute of ecological paramount relevancy. For instance, in 2002, the Royal Society dedicate a complete issue of its Philosophical Transactions to the subject. The introductory text starts with the following statement:

“The thesis put forward in this issue of Philosophical Transactions is that population growth rate is the key unifying variable linking the various facets in population ecology.” (Sibly, Hone, Clutton-Brock, 2002:1149).

This statement is clearly corroborated, and illustrate in this book. This unifying role of the growth rate is only possible if the following is verified: • Each species has a pattern of growth with an underlying constancy, mirroring the species life-history, and genetic identity; • This pattern of growth evinces a limited variability. The EGZ can reproduce these characteristics. The correctness of these conjectures contributes to improve the predictive capacity of ecology, a desirable feature in a world of increasing disruptive impact of human actions. If we accept the upper presented arguments, it is highly probable that the values of c, and Ri are constant for a given species. The adoption of this premiss is a fruitful decision, as it will be seen through this book. © L. S. Barreto, 2017. Theoretical Ecology 63 4 The Gompertz Equation

I characterized the dynamics of microorganisms, plants, and animals. The fitting of the EGZ to the growth of several species can be found in the literature, as for instance in Zullinger et al. (1984). I used yield tables of diverse origin to obtain the characteristic values of forest tree species, as displayed in Barreto (2010: Quadro 4.1). The EGZ obtained successfully reproduce the yield tables. For all this species it is verified 0

In Table 4.1, I display the characteristic values of c, and Ri for several species.

Table 4.1. Characteristic parameters of six species. t0=0 Paramecium aurelia (ciliate); time unit: day; data from Roughgarden (1998:110)

c R-3 R-2 R1 R2 R2.67 R3 R5.67 0.2970 300.9307 44.9069 0.1492 0.0223 0.0062 0.003 0.00002 Acipenser stellatus (fish); year; Bertalanffy (1973:236) 0.132 1361.7350 122.8145 0.09023* 0.00814 0.0016 0.0007 10-5 Quercus robur (common oak); year; Hamilton, and Christie (1979:179-180)

c R-2 R0.7 R1 R2 R2.67 R3 0.041 1461.1201* 0.08813 0.02616 0.00068 6*10-5 2*10-5 Eulata fruticum (snail); month; Banks (1994:227)

c R-2 R1 R2 R2.67 R3 R4.677 0.0935 114.7350 0.09336 0.00871 0.00179 0.00081 1*10-5 Odocoileus hemionus hemionus (mule-dear); year; Medin , and Anderson (1979:40) 0.2239 15.70039* 0.25237 0.06369 0.02544 0.01607 0.00162 Homos sapiens (man), year; Banks(1994:71) 0.14515 11.50963 0.29795 0.08690 0.03849 0.02562 0.00345 Lynx pardinus (Iberian lynx); year; Barreto (2009) 0.3920 9.7730 0.3199 0.1023 0.0478 0.0327 0.0049

From here on, the notation for the variables, introduced in chapter 2, will be used. Environmental factors affects the size of the individuals, the sustaining capacity, the standing total biomass (yi,f), phisiological aspects as sexual maturity, but do not change the characteristic values of c, and Ri, of a given species. For instance the increasing of CO2 in the atmosphere is increasing the yield of Scots pine (Pinus sylvestris) forests in Spain (Martinez-

Vilalta, Lopez, Adell, Badiella, and Ninyerola, 2008), but do not change the values of c, and Ri of the pine. In the next chapter, from the accepted Gompertzian pattern of growth, we deduce the laws of the dynamics of organisms, and isolated populations.

4.9 Properties of Gompertzian Variables

It easy to prove that variables that follow the EGZ have the following properties:

 If variable ya has an allometric relationship wit a Gompertzian variable yb, variable ya is also Gompertzian.  The product of two Gompertzian variables is also Gompertzian.  The power of a Gompertzian variable is also Gompertzian. 64 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

4.10 The Fitting of the EGZ

We use software R to introduce a simple method to fit the EGZ to data.

Let us have a data Y=[y1,y2..yn]. Create a variable S such thatl X=log(Y). Fit the autoregressive model

Xt+1=a+bXt (4.55)

where a=c*log(yf). Thus:

c=-(b-1), yf=exp(a/c), and R=y1/yf.

Now, successfully we illustrate the application of the method with R:

t<-c(0:90) y0←c(400*0.4076^exp(-0.05*t)) ← Generate hypothetical data

x<-log(y0[-91]) y<-log(y0[-1])

model0<-lm(y~x) summary(model0) Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -1.123e-15 -4.081e-16 -1.320e-17 2.317e-16 4.031e-15

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.922e-01 1.607e-15 1.819e+14 <2e-16 *** x 9.512e-01 2.773e-16 3.430e+15 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.065e-16 on 88 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.177e+31 on 1 and 88 DF, p-value: < 2.2e-16

We obtained a good fitting. Let us illustrate this statement.

> plot(x,y) > cf<-coef(model0) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv) > © L. S. Barreto, 2017. Theoretical Ecology 65 4 The Gompertz Equation

Figure 4.19. Graphic originated by the fitting of the EGZ with R. The two data are virtually coincident. The scales of the axis are the logarithm of the initial data

We can obtains the values of c, yf, and R:

> #Obtain a > a=c*log(yf) > > #Obtain c > c=-(b-1) > c x 0.04877058 > #Obtain yf > yf=exp(a/c) > yf x 400 > #Obtain R > y0[1]/yf x 0.4076

In Barreto (2013:chapter 14), I show that models for exploitation of populations using the EGZ give more ecological sound results then those using the logistic equation. In Barreto (2016: section 36.4), I demonstrate that the EGZ is an adequate model for the species abundance distribution, and the rank abundance diagram.

66 © L. S. Barreto, 2017. Theoretical Ecology 4 The Gompertz Equation

4.11 References, and Related Bibliography

Bajzer, Z., 1999. Gompertzian Growth as a Self-Similar and Allometric Process. Growth Dev Aging, 63:3-11. Bajzer, T., S. Vuk-PavloviC, and M. Huzak, 1997. Mathematical modelling of tumor growth kinetics. Em J . A. Adam and N. Bellomo (Editors) A Survey of Models for Tumor-Immune System Dynamics, Birkhauser, Boston, pp. 89-132. Bajzer, Z., and S. Vuk-Pavlovic, 2000. New dimensions in Gompertzian growth. J. Theor. Med. 2, 307-315. Banks, R. B., 1994. Growth and Diffusion Phenomena. Mathematical Frameworks and Applications. Springer-Ver- lag, New York. Baranowitz, S. A., P. F. A. Maderson, and T. G. Connely, 1979. Lizard and Newt Tail Regeneration: A Quantitative Study. J. Exp. Zool. 210: 17-38. Barreto, L.S., 1989. The 3/2 power law: a comment on the specific constancy of K. Ecological Modelling, 45:237-242. Barreto, L. S., 1991. SPESS - a simulator for pure even-aged self-thinned stands. Ecological Modelling, 54:127-132. Barreto, L. S., 1994. The clarification of the 3/2 power law using simulators SANDRIS and PINASTER. Silva Lusitana, 2(1):17-30. Barreto, L. S., 1995. Povoamentos Jardinados. Instrumentos para a sua Gestão. Publicações Ciência e Via, Lda., Lis- boa. Barreto, L. S., 2000. Pinhais Mansos. Ecologia e Gestão. Estação Florestal Nacional, Lisboa. Barreto, L. S., 2004. Pinhais Bravos. Ecologia e Gestão. E-book. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. E-book. Costa de Caparica. Included in the CD. Barreto, L. S., 2009. Caracterização da Estrutura e Dinâmica das Populações de Lince Ibérico ( Lynx pardinus). Uma digressão Exploratória. Silva Lusitana, 17(2):193-209. Barreto, L. S., 2013. Ecologia Teórica. Uma outra Explanação. I. Populações Isoladas.E-Book- Costa da Caparica. Bartlett, S. E., 1960. Stochastic Population Models in Ecology and Epidemiology. Methuen and Company, Ltd, Lon- don. Bauer, F., and C. Castillo-Chávez, 2001. Mathematical Models in Population Biology and Epidemiology. Springer, New York. Bertalanffy, L. von, 1973. Teoria Geral dos Sistemas. Editora Vozes, Petrópolis, Brasil. Caustion, D. R., and J. C. Venus, 1981. The Biometry of Plant Growth. Edward Arnold, London. Deléage, J-P., 1993. História da Ecologia. Uma Ciência do Homem e da Natureza. Publicações Dom Quixote, Lisboa. France, J.,and J. H. M. Thornley, 1984. Mathematical models in agriculture. Butherworths, London. Gentle, J. E., 1998. Random Number Generation and Monte Carlo Methods. Springer, New York. Gillespie, D. T. ,1977. Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry, 81, 2340–2360. Gompertz, B., 1825. On the nature of the function expressive of the law of human mortality and on a new method of determining the value of life contingencies. Philos. Trans. R. Soc., 36, 513–585. Available in the internet. Hamilton, G. C., and J. M. Christie, 1971. Forest Management Tables (Metric). Forest Commission Booklet Nº 34, Her Majesty’s Stationery Office, London. Hammer, Ø., Harper, D.A.T., and P. D. Ryan, 2001. PAST: Paleontological Statistics Software Package for Education and Data Analysis. Palaeontologia Electronica 4(1): 9pp. Hooker, P. F., 1965. Benjamin Gompertz. Journal of the Institute of Actuaries, 91:203-312. Hutchison, G. E., 1978. An Introduction to Population Ecology. Yale University Press, New Haven. Karkach, A. S., 2006. Trajectories and models of individual growth. Demographic Research, 15:347-400. Kingsland, S. E, 1985. Modeling Nature. Episodes in the History of Population Ecology. The University of Chicago Press, Chicago. Khilmi, . G. F., 1962. Theoretical Forest Biogeophysics. National Science Foundation, Washington, D. C. Kot, M., 2001. Elements of Mathematical Ecology. Cambrige Universisty Press, Cambridge. Kritzinger, W. J, 2011. Allometric Description of Ostrich (Struthio camelus var. domesticus) Growth and Develop- ment. Department of Animal Sciences, Faculty of AgriScience, Stellenbosch University. Master thesis. Lauro, E. De, S. De Martino, S. De Siena, and V. Giorno, 2010. Stochastic origin of Gompertzian growths. http://arxi.org/abs/1012.4978v1 [q-bio.QM] 22 Dec 2010. Li, Y., 2011. Stability and bifurcation analysis of a discrete Gompertz model with time delay. World Academy of Sci- ence, Engineering and Technology 80 : 1441-1444. © L. S. Barreto, 2017. Theoretical Ecology 67 4 The Gompertz Equation

Martinez-Vilalta, J., B.C. Lopez, N. Adell, L. Badiella, and M. Ninyerola, 2008. Twentieth century increase of Scots pine radial growth in NE Spain shows strong climate interactions. Global Change Biology 14: 2868-2881. Medawar, P. B., 1940. Growth, growth energy, and ageing of the chicken's heart. Proc. Roy. Soc. B. London 129:332- 355. Michaelis, L., and M. Menten, 1913. Die kinetic der invertinwirkung . Biochemische Zeitung, 49:333-369. Medin, D. E., and A. E. Anderson, 1979. Modelling the Dynamics of a Colorado Mule Deer Population. Wildlife Mo- nographs Nº 68. The Wildlife Society, Inc. Monod, J., 1958. Recherches sur la Croissance des Cultures Bacteriennes. Herman, Paris. Morris, W. F., and D. F. Doak, 2002. Quantitative Conservation Biology. Theory and Practice of Population Viability Analysis. Sinauer Associates, Inc. Publishers, Sunderland, MA. Peng, T., 2008. A Bioeconomic Model of Continuous Harvesting for a Single Species Fishery. International Journal of Pure and Applied Mathematics, 48(1): 21-31. Poole, R. W., 1974. An Introduction to Quantitative Ecology. McGraw-Hill, New York. T. Pradhan, and K.S. Chaudhuri, 1998. Bioeconomic Modeling of a Single-species Fshery with Gompertz Law of Growth. J. Biol. Syst., 6:393-409. Pütter, A., 1920. Studien über physiologische ähnlichkeit. VI. Wachstumsähnlichkeiten. Pﬂüg. Arch. ges. Physiol., 180:298–340. Ricker, W. E., 1975. Computation and Interpretation of Biological Statistics of Fish Populations. Bulletin of the Fish- eries Research Board of Canada. 191: 382 pp. Roughgarden, J., 1998. A Primer of Theoretical Ecology. Prentice Hall. Satoh, D., 2000. A discrete Gompertz equation and a software reliability growth model, IEICE Trans., E83-D-7: 1508– 1513. Scudo, F. M., and J. R. Ziegler, 1978. The Golden Age of Theoretical Ecology, 1923-1940. Springer-Verlag, Berlin. Soetaert, K. e P. M. J. Herman, 2009. A Practical Guide to Ecological Modelling Using R as a Simulation Platform. Springer, Berlin. Sibly, R. M., D. Barker, M. C. Denham, J. Hone, and M. Pagel, 2005. On the regulation of populations of mammals, birds, ﬁsh, and insects. Science, 309:607–610. von Bertalanffy, L., 1973. Teoria Geral dos Sistemas. Editora Vozes Lda., Petrópolis, Brazil. Waring, R. H., and Schlesinger,W. H., 1985. Forest Ecosystems. Concepts and Management. Academic Press, Orlando. Winsor, C. P., 1932. The Gompertz curve as growth curve. Proc. Nat. Acad. Sci., 1-8. Por citação. Yoda, K.. et al., 1963. Self-thinning in overcrowded pure stands cultivated and natural conditions. (Intraspecific competition among higher plants. XI). J. Biol. Osaka City Univ., 14:107-126. Por citação. Zullinger, E. M., Ricklefs, R. E., Redford, K. H., and Mace, G. M. (1984). Fitting sigmoidal equations to mammalian growth rates. J. Mamm., 65 (4):607–636. As mentione in Karkach (2006). 68 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

5 The Laws of Growth of Isolated Populations

5.1 Introduction

The material exposed in this chapter is applicable to organisms, isolated populations, and to a population that, in community, has a dominance enough strong to be only subject to intra-specific competition, as the individuals of a unique tree species in a forest (pure stand). Let us recapitulate some accepted assumptions, attributes of organisms, and populations: 1. Organisms, and populations have deterministic dynamics. 2. Biological, and ecological variables have allometric relationships. 3. The variables have Gompertzian dynamics.

4. Ri, and c are constant for a given species. 5. Populations have a time-space symmetry. 6. The fractal geometry of individuals, (i=2,66667), and populations (i=0,66667, 4,66667, 5,77777). We admit two more initial conditions: 7. Individuals of a population are randomly mixed in space. 8. The environment is constant in all space occupied by a population. As individuals of all species have the same the same basic pattern of growth (Gompert- zian), the implications of this pattern, at population level, are the same for all species, and the variables of isolated populations have also Gompertzian variation. The allometric relationships harmonize the dynamics of the organism variables and the population variables (section 4.9). The values of R and of RVR of the variables, are harmonized within each biosystem (organism, and population), and between the variables of these biosystems.

5.2 The Fundamental Laws

Let us sustain the last proposition of the previous section. The RVR of variables with PL=i will be

referred as RVRi. For a given species, Y is the set of all organism, and population variables. According to the conclusions of chapter 4, all the variables of set Y follow EGZ. Consider

two variables of the same set, ya e yb, being a/b=m. The following allometric equation is satisfied:

m ya=kab yb (5.1)

-ct E Put E=e , and yb=ybf Rb (equation (6.10)), then this equation can be written:

m mE ya =kabybf Rb (5.2)

being

(5.3) © L. S. Barreto, 2017. Theoretical Ecology 69 5 The Laws of Growth of Isolated Populations

(5.4) given equation (6.33), we write:

RVRa=mRVRb (5.5)

Given two RVR, RVRa e RVRb, RVRa>RVRb occurs if a>b.

The RVRi, i=-2, -3, is also named relative mortality rate. In the case of SPES, this is the relative rate of self-thinning. It is negative, as the following relationship is verified:

yijt+1=yijt+ yijt RVRit (5.6)

a Obviously, it is verified Ra=R1 . In Box 5.1, we display the main characteristics of the dynamics organisms, and populations, and how they are related to each other.

Box 5.1. Summary of the main characteristics of the dynamics organisms, and populations, and how they are related to each other 70 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

These are the equations that guarantee the harmonized dynamics of the individuals, the allometric relationships of their bodies, and the coherent dynamics of the variables of the organism, and the population.

For any given species, and for any variable yi, at ages t, and t+n, the ratio yit+n/yit is con- exp (-c((t-t0)+n))-exp(-c(t-t0)) stant, and equal to Ri . For instance in all SPES of Pinus pinaster the ratio of the B B -0.05*(40-10) -0.05*(10-10). standing volume at ages 40 years, and 10 years is R1 =0.4076 , being B=e -e Thus B R1 =2. This is, from age 10 to 40, the forest duplicates its standing volume, its mean height, and its mean dbh (diameter at breast height). -2 What happens with the density (y-2)? It is straight as 2 =0.25, the self-thinning reduces the number of trees to one quarter of the density at age 10.

We can verify these predictions with wxMaxima. For P. pinaster it is verified R1=0.4076, c=0.05, t0=10. The script in wxMaxima is:

←

exp (-c((t-t0)+n))-exp(-c(t-t0)) Now, let us verify the coherence of the equation Ri . To satisfy this propose we make explicit calculations to confirm the results obtained with this equation: © L. S. Barreto, 2017. Theoretical Ecology 71 5 The Laws of Growth of Isolated Populations 72 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

5.3 Time-space Symmetry Between a Cohort and its Aged Structured Population

All we know that natural, and undisturbed forests of a single tree species are of two types: • In the forest, all trees have virtually the same age, already identified as SPES. • In the forest, there are trees of all ages till the maximum longevity of the species. This type is called self-thinned pure uneven-age stands (SPUS). We also observe that younger SPES have greater density than older stands. We wrote in chapter 3 that the reduction of the number of trees is due to the process of self-thinning. This tree mortality is a consequence of a greater necessity of space, and resources evinced by the surviving, and growing trees. The dynamics of the number of surviving trees is regulated by R-2. We assume that there is a time-space symmetry between SPES, and SPUS. In one hectare of a SPES, what happens in time has reification in the same hectare in a SPUS. Consider a SPES of a given species with the area of 81 ha. The longevity of the species is 90 years, being t 0=10 years. Its symmetrical SPUS is a mixture of 81 areas of 1 hectare, of even-aged trees, with ages from 10 to 90 years. In its wholeness, the frequencies of each age follow the GZE. Trees with age less than 10 years are under the cover of the other trees. This concept is graphically exhibit in figure 5.1. This symmetry is not confined to y-21, but is extensible to the other variables of the tree population. In a SPUS the number of trees converges to a stable age distribution. Now, let us consider an animal population. Assume a cohort that starts with 1000 newborn

Iberian lynxes. In table 4.1, for this species we wrote c=0.392, and R-2=9.773. The model for the number of individuals of this cohort from age 0 to 13 years is (equation 4.25b; *=multiplication):

(exp(-0,392*t)-1) y-2=1000*9,773 t=0,1,….13

We obtain the figures in table 5.1.

Table 5.1. Number of individuals of a cohort of Iberian lynx that starts with 1000 animals (y-2)

Age 0 1 2 3 4 5 6 7 8 9 10 11 12 13

y-2 1000 477 290 207 164 141 127 118 113 109 107 105 104 103

In an area 13 times large than the area occupied by the cohort, an aged structured population, without reproduction restrictions, in each age, has the same number of individuals as the ones in table 5.1. This time-space symmetry will be explored in the next chapter. © L. S. Barreto, 2017. Theoretical Ecology 73 5 The Laws of Growth of Isolated Populations

Figure 5.1. Graphical illustration of the concept of the time-space symmetry between SPES, and SPUS

5.4 Self-similarity in Population Growth

The fractal geometry of organisms, and populations, the material displayed in the previous section lead us to the following conjecture: In the time continuum the growth of biological variables is self-similar. This is, the growth using a shorter time unit is embedded in the Gompert- zian growth using a larger time unit. Thus, it is possible to use the same EGZ in all time scales, to simulate the Gompertzian variation of a given biological or population variable. To evaluate this hypothesis, first I will introduce model KHRONOSKHABA (Barreto, 2004b), and after I will use the model to evince simulations that sustain the hypothesis.

5.4.1 Model KHRONOSKABA Model KHRONOSKHABA (MK) is derived from the concept of RVR. Let assume the year as the standard time unit. Let me write the RVR for a time unit smaller then the year:

RVRf=f(-ct ln Ri exp(-ct(t-t0)) (5.7)

where: f=1/(number of time units in the growth season)

ct is the new value for c, in the new time unit, during a year. MK is written as:

yijt+d= yijt (1+RVRf) (5.8)

being d=1/(number of time units in a year). 74 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

As it easier to detect the growth season in plants, I will use a forest example. Let us con-

sider a SPES of a hypothetical species that evinces t0=10 years, R0.666=0.0891, and c (referred to -1 the year) is 0.05. At age 10, the total standing biomass is measured as y 0,6661=50 Mg ha . The

growth season spans from March to July. Now we simulate the variation of y0,6661.

In the year, the vector of the monthly values of ct is [0, 0, 0,03 0,06, 0,08, 0,05, 0,03, 0, 0, 0, 0, 0], with a maximum in May. In figure 5.2, I display the simulations using MK, and EGZ, for a period of time of 200 years (10-210 years). To emphasize the pattern of variation associated to MK, in figure 5.3, I exhibit the same simulation for the period from 10 to 15 years. Now, let us use the week as time unit, for the period from week 9 to week 30, for this

period ct is eqqual to [0,029, 0,033, 0,0345, 0,0412, 0,0493, 0,0579, 0,0661, 0,0731, 0,0781, 0,0803, 0,0787, 0,0736, 0,066, 0,057, 0,0477, 0,0393, 0,0328, 0,0325, 0,0324, 0,0313, 0,031, 0,029]. The results we obtain are in figures 5.4 e 5.5. The results displayed in figures 5.2 a 5.5, sustain our hypothesis. More confirmations of the hypothesis can be found in the detailed applications of MK to Pinus pinaster stands displayed in Barreto (2004a: chapter 8).

Applies MK, uses the month as time unit, and covers the period from March to July

Age, months

Applies the EGZ, and the time unit is the year

Age, yearss

Figure 5.2. Simulation of the growth of the biomass per hectare, of an SPES, with a hypothetical species. The upper graphic applies MK, uses the month as time unit, and covers the period from March to July. The lower graphic applies the EGZ, and the time unit is the year. At age 210 years, the percent error of the MK simulation is 0.3350315% © L. S. Barreto, 2017. Theoretical Ecology 75 5 The Laws of Growth of Isolated Populations

Growth during 5 years with MK

Age, months

Growth from ages 10 to 15 with EGZ

Age, years (10-15)

Figure 5.3. First 5 years of the simulation in figure 5.2

Simulaion with MK being the week the time unit

Age, weeks

Simulaion with EGZ

Age, years

Figure 5.4. The same simulation in figure 5.2, with week as time unit. At age 210 years, the percent error of the MK simulation is 1.3443862% 76 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

Simulaion with MK being the week the time unit

Age, weeks

Growth from ages 10 to 15 with EGZ

Age, years

Figure 5.5. First 5 years of the simulation in figure 5.4

5.5 Empirical Evaluation with an Animal Species

This section is related with what we wrote in section 3.2, about equation (3.6) when applied to animals, particularly to fishes. In Barreto (2010a, 2011) I displayed several confirmations of my theory when applied to plants. Now I appraise the theory applied to an animal species. Ricker (1975), published measurements of cisco or lake herring (Coregonus artedi). He measured the length (mm), and the weight (g), from ages 2 to 11 years. This is the data we use.

I fit the EGZ to both the length measurements (y1), and the weights (y2,7). After I will verify 2.7 if the equality R2.7=R1 is confirmed. The fittings are done in R, using the method presented in section 4.10. Here are the fittings:

> #cisco length > y0<-c(172, 210, 241, 265, 280, 289, 294, 302, 299, 306) > > x<-log(y0[-10]) > y<-log(y0[-1]) > > model0<-lm(y~x) > summary(model0)

Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -0.018219 -0.004986 0.001234 0.007993 0.011424

Coefficients: Estimate Std. Error t value Pr(>|t|) © L. S. Barreto, 2017. Theoretical Ecology 77 5 The Laws of Growth of Isolated Populations

(Intercept) 2.00016 0.10190 19.63 2.22e-07 *** x 0.65118 0.01835 35.49 3.66e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.00997 on 7 degrees of freedom Multiple R-squared: 0.9945, Adjusted R-squared: 0.9937 F-statistic: 1260 on 1 and 7 DF, p-value: 3.661e-09

> split.screen(c(1,2)) [1] 1 2 > screen(1) > plot(x,y, main='Length fitting') > cf<-coef(model0) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv) > > c<--(b-1) > k<-exp(a/c) > c x 0.3488225 > k (Intercept) 309.2148 > > R1<-y0[1]/k > R1 (Intercept) 0.5562477 > > #cisco weight > y0<-c(99,193,298,383,462,477,505,525,539,539) > > x<-log(y0[-10]) > y<-log(y0[-1]) > > model0<-lm(y~x) > summary(model0)

Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -0.040045 -0.010800 0.002173 0.005070 0.041085

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.50777 0.08607 29.14 1.44e-08 *** x 0.60301 0.01467 41.11 1.31e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02383 on 7 degrees of freedom Multiple R-squared: 0.9959, Adjusted R-squared: 0.9953 F-statistic: 1690 on 1 and 7 DF, p-value: 1.314e-09 78 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

> screen(2) > plot(x,y, main='Weight fitting') > > cf<-coef(model0) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv) > > c<--(b-1) > k<-exp(a/c) > c x 0.396992 > k (Intercept) 553.8646 > > R2<-y0[1]/k > R2 (Intercept) 0.1787441 > #The desired verification > R1^2.7-R2 (Intercept) 0.02647808 >

Figura 5.6. Fittings of the EGZ (line) to the cisco data © L. S. Barreto, 2017. Theoretical Ecology 79 5 The Laws of Growth of Isolated Populations

Let us calculate the relative error associated to difference:

> d=R1^2.7-R2 > d (Intercept) 0.02647808 > d/R2 (Intercept) 0.148134

The difference is about 15% of R2.7. Let us remember the constrictions displayed in section 3.2, and the following ones: • There are measurement errors; • The theory is applicable to natural isolated populations, not disturbed by exogenous factors, and the lake where the measurements were made ( Vermilion, Minnesota, USA) do not accomplish these conditions. Given these facts we can not peremptory reject the theory.

Given the deductibility, and systemic structure of my theory, the non rejection of the analysed equality implies the validation of the relations in box 5.1 to animals.

5.6 The Total Production of a Cohort

In Table 5.1, we verified that a cohort of Iberian lynx that starts with 1000 animals simultaneously came into existence, at the beginning of year 13 it has only 103 animals that will die in this year, because they attained the limit of their longevity. Let us assume that, in average, it is verified 12 kg of biomass per individual, being the total final biomass of the cohort 1236 kg (12*103). To obtain a correct estimation of the total production we must add to this value the biomass of the dead animals. Thus, the total production of the cohort (PT) will be:

PT=Biomass of mortality (BM) + Final mortality (BF)

Let us assume that in each year the mean biomass of the dead animals is equal to the mean biomass of the existing animals, that is, it occurs neutral mortality. If so, in a given instant the biomass of mortality is equal to the existing living biomass times the relative mortality rate(TRM):

BMt=TRM-2t y4,6667t (5.8)

With a continuous mortality process, the total biomass of mortality, from age 0 to 13, will be:

(5.9)

For the Iberian lynx, the TRM is estimated as: 80 © L. S. Barreto, 2017. Theoretical Ecology 5 The Laws of Growth of Isolated Populations

TRM-2=-0.392 * ln 9.773 * exp(-0.392 *t) (5.10)

and the growth of total biomass at time t is:

exp(-0,392*t) y4,7=y4,7f * 0.0049 (5.11)

So, the biomass of mortality is written as:

(5.12)

As y4,7f = 1236 kg, we write:

(5.13)

BM=1104.505* 0.462=510.281 kg

The total production of the cohort described in table 5.1 is:

PT=510.281+1236=1746.281 kg

This procedure can be applied to any animal or plant population, as the process of self- thinning is neutral (see Barreto, 2011).

5.7 References, and Related Bibliography

Barreto, L. S., 2004a. Pinhais Bravos. Ecologia e Gestão. E-book. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. Barreto, L. S., 2004b. SB-KHRONOSKHABA. A Gompertzian Model for the Seasonal Growth of Forest Trees and Stands An Application to Pinus pinaster Ait. Research Paper SB-05/04. Departamento the Engenharia Florestal, In- stituto Superior de Agronomia, Lisboa. Barreto, L. S., 2009. Caracterização da Estrutura e Dinâmica das Populações de Lince Ibérico (Lynx pardina). Uma Digressão Exploratória. Silva Lusitana, 17(2):193-209. Barreto, L. S., 2010a. Árvores e Arvoredos. Geometria e Dinâmica. E-book. Edição do autor. Costa de Caparica. In- cluded in the CD. Barreto, L. S., 2010b. Simulação do Carbono Retido no Pinhal Bravo e da sua Acreção. Silva Lusitana, 18(1):47-58. Barreto, L. S., 2011. From Tress to Forests. A Unified Theory. E-book. Costa de Caparica. Included in the CD. Ricker, W.E. 1975. Computation and interpretation of biological statistics of fish populations. Bulletin of the Fish- eries Research Board of Canada. 191: 382 pp. © L. S. Barreto, 2017. Theoretical Ecology 81 6 Structured Populations: The Gompertzian Approach

6 Structured Populations: The Gompertzian Approach

6.1 Introduction

I start this chapter with a figure (6.1) that remember the three classical types of survivorship curves.

Figure 6.1. Types of survivorship curves

In table 4.1, we assume that survivorship curve of type III (red line in figure 6.1), the most ubiquitous, is the pattern in situations where the species attained a stabilized state of coevolution, in mature ecosystems, where the principal factors that cause changes are constrained variations of the physical environment that do not provoke great changes in the structure, and composition of the community. Let me sustain this assumption with some arguments. The existence of the other types of survivorship curves represent recent situations in communities that received important changes that displaced them away from their pristine conditions characterized by the coevolution of the species, or the species that has type different from III, is a newcomer in the community. As we will see ahead, in some circumstances, competition, and predation can change the survivorship curve. In the literature, we can find cases of survivorship curve changes caused by the elimination of predators. Animals captive in zoos, and animals raised in laboratories, have survivorship curve of type I, but in their natural environment they have type III (e.g., Hutchinson, 1978:74). B. Gompertz formulated his equation from the analysis of survivorship curves of human populations in several European countries, and named it ‘the law of human mortality’ (Banks, 1994:151), this is, at its inception the EGZ was a survivorship curve. At that time, two centuries ago, humans have a survivorship curve of type III. 82 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

Our survivorship curve is an example of the changing from type III to type I, due to the scientific, and technological progress. High child mortality rate is seen as an indication of underdevelopment. It is admitted that crocodiles, and close species, exist since more then one hundred million years. With the extinction of their pristine predators, today the adult crocodiles have survivorship curve of type I, because no animal species predate them. In figure 6.2, I use a diagram to resume these arguments.

Captive animals. Animal in new Adults without predators habitat without predators. Example: crocodiles Humanity receiving the benefits of science , and technology

Figure 6.2. Diagrammatic representation of changes of survivorship curve type III into other types. The reversal of these changes is a possibility that can not be discarded

Sessile populations of organisms easy to individualize, count, and measure are preferred for many kinds of research. Because forests evince these attributes I will use undisturbed self-thinned forests to illustrate some of the issues I will approach ahead. I assume that the reader is already familiar traditional demography, that admits an exponential pattern of growth. Although this assumption I will define most of the notation here used. Here, I introduce a Gompertzian demography. I will present a procedure to estimate the rates of permanence, transition, and mortality of the classes in a structured population. This information is used to model the population. For structured populations, I describe a matrix model, © L. S. Barreto, 2017. Theoretical Ecology 83 6 Structured Populations: The Gompertzian Approach and a model using ODE. To close the chapter I introduce a model for maritime pine that uses the trimester as time unit instead of the year.

6.2 Retrieving some Concepts, and Notation

We assume that the births occur at the beginning of the time unit adopted (year, month, week, …). We clarify this choice in figure 6.3.

Age class(i)

Age (x)

Births

Figure 6.3. Time convention for ages, births, and age classes

Age class i has individuals with age from i-1 to i. For the reader’s convenience, in this section, I will remember some common notation that will be used ahead.

Sx is the number of survivors with age x. Corresponds to y-2 or y-3.

lx is the number of survivors with age x referred as a fraction of the initial numbers at instant 0. We writte:

S x lx= (6.1) S0

For an individual, lx is the probability of survival from age 0 to age x.

For an individual, gx is the probability of survival from age x to age x+1:

l x+1 Sx+1 gx= = (6.2) lx Sx

Some authors reprent Sx+1/Sx by sx.

S15 is equal to g14*S14. Thus, gx ou sx can be seen as the relative rate of survival or the survival rate per capita, at age x.

The number of dead dx at age x is

dx=Sx-Sx+1

As alll individual die, the sum of all dx is equal to 1. As the fraction of the survivors, at any x−1 age, is equal to 1 minus the sum of the fractions of the previously dead, we write lx=1−∑d j . j=0 If 30% survived, 70% had die. We can introduce the relative mortalility rate or the mortality rate per capita, at age x, qx 84 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

qx=1-gx =1-sx (6.3)

Now, Sx+1 is equal to the number of survivors at age x minus the number of those who died:

Sx+1=Sx-qx Sx=Sx(1-qx) (6.4)

6.3 Gompertzian Demography

In this section, I will introduce lx in the equations, and related attributes, defined in section 4.6. Thus, we wll consider the demography in the situation of discrete time, and, following other authors, the maximum longevity as infinite. -c c Equation (4.36) is now written as lx+1=lx lx yf . From this equation, applying recursion, and substitution, we arrive to a general equation:

c(x-1) -c lx=l0 lf (l0l1…lx-1) (6.5)

6.3.1 Discrete Demography We consider homogeneous age classes, this is, the values of the attributes are the same for all its individuals.

Put l0=1. The survival rate is:

c -c sx= lx+1/lx = lf lx (6.6)

The mortality rate is written as:

c -c dx=1- lf lx (6.7)

The net reproductive rate has the form:

(6.8)

The generation time is expressed as:

(6.9)

The life expectancy at age 0 (birth) is given by (see figure 6.4):

(6.10)

cx -c c(x-1) -c Lx=( l0 lf (l0l1…lx) + l0 lf (l0l1…lx-1) )/2 (6.11) © L. S. Barreto, 2017. Theoretical Ecology 85 6 Structured Populations: The Gompertzian Approach

6.3.2 Continuous Time Demography

Again, let us assume l0=1. From equation (4.12)

(exp(-cx)-1) lx= yix=Ri i=-2, -3 (6.12)

The rate of survival is:

-cx sx= 1-c ln(Ri) e (6.13)

The mortality rate is:

.cx dx= -c ln(Ri) e (6.14)

Survivorship curve

Age Figure 6.4. The area Lx

The net reproductive rate assume the form:

(6.15)

The equation for time generation is now:

(6.16)

The life expectancy at age 0 (birth) is given by:

(6.17)

At age x, life expectancy is:

(6.18) 86 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

6.3.3 Life Table In the Gompertzian context (see chapters 2 and 4), the survival in a cohort is given by: lx=yi,x=yi,0 Ri(exp(-ct)-1) i=-2, -3 (6.19)

As the model is continuous, we can calculate instantaneous mortality rates at the beginning of each age, and average values for the time span of a year. In Box 6.1, I display a script written in Scilab, tha calculates the net reproductive rate (Ro), generation time (T), the life table, and establish two graphics (figure 6.5). One for the cohort, and the other for the life expectancy of the ages. It is assumed l0= 100.

Caixa 6.1. Scilab script to establish the life table of Iberian lynx

//Life table of L. pardina //tiMia=instantaneous moratalityrate at the beginning of the year //dx=number of deaths in a year //tmS=average survivorship rate of the year //tmM=average mortality rate of the year //E(x) = life expectancy of the year clear r=9.7719828;c=0.392;a=[];t1=[];E=[];z=1;k=[]; bx=[0 0 2 2 2 2 2 2 2 0 0 0 0 0]; t=[0:1:13]';t5=[1:1:14]; //calculation of lx and dx slx='100*r^(exp(-c*t)-1)';lx=[eval(slx)]'; slx2='100*r^(exp(-c*t5)-1)';lx2=[eval(slx2)]'; dx=[lx]'-[lx2];mrt='-c*log(r)*exp(-c*t)'; tiMia=eval(mrt); // Calculation of Ro slx3='r^(exp(-c*t)-1)';lx4=eval(slx3); s1=lx4'.*bx; Ro=intsplin([t],s1);ro=string(Ro); //Duração de uma geração, T To=t'.*s1;iRo=1/Ro; T1=intsplin([t],To);T=T1*iRo;tt=string(T); //Calculation of E for i=0:13 x=i;[t1]=[x:15];j=t1;s3=r^(exp(-c*j)-1); s2=intsplin([j],s3); E=s2/(r^(exp(-c*i)-1)); a(z,1)=E; z=z+1; end uns=ones(14,1); tms=lx2./lx';tmm=[uns]-[tms]; a(14,1)=0; disp(["Ro=" ro "Female/female/longevity"]) disp(["T=" tt "yearss"]) disp(["Age l(x) tiMia dx tmS tmM E(x)"]) M=[t lx' tiMia dx tms tmm a]; disp(M) clf subplot(1,2,1) plot2d(t,lx) xtitle('Survivorship, lx','Age, years','lx') subplot(1,2,2) plot2d(t,a) © L. S. Barreto, 2017. Theoretical Ecology 87 6 Structured Populations: The Gompertzian Approach

xtitle('Life expectancy','Age, years', 'Expectancy, years')

Figure 6.5. Graphics established for the Iberian lynx by the script in Box 6.1

The output is:

!Ro= 2.2437203 Female/female/longevity !

!T= 4.4522078 years !

Age l(x) tiMia dx tmS tmM E(x)

0. 100. - 0.8935716 52.252275 0.4774773 0.5225227 2.7582533 1. 47.747725 - 0.6037900 18.772975 0.6068300 0.3931700 4.3195558 2. 28.97475 - 0.4079834 8.3001467 0.7135386 0.2864614 5.8382659 3. 20.674603 - 0.2756761 4.2160833 0.7960743 0.2039257 7.0043596 4. 16.45852 - 0.1862754 2.3505439 0.8571838 0.1428162 7.6832286 5. 14.107976 - 0.1258671 1.3951385 0.9011099 0.0988901 7.8873866 6. 12.712837 - 0.0850489 0.8637295 0.9320585 0.0679415 7.7025312 7. 11.849108 - 0.0574679 0.5501531 0.9535701 0.0464299 7.2303118 8. 11.298955 - 0.0388313 0.3572057 0.9683859 0.0316141 6.5597729 9. 10.941749 - 0.0262385 0.2349500 0.9785272 0.0214728 5.7587407 10. 10.706799 - 0.0177294 0.1558938 0.9854397 0.0145603 4.8748876 11. 10.550905 - 0.0119799 0.1040510 0.9901382 0.0098618 3.940023 12. 10.446854 - 0.0080948 0.0697261 0.9933256 0.0066744 2.9746207 13. 10.377128 - 0.0054697 0.0468506 0.9954852 0.0045148 0.

6.4 Finding the Rates of Permanence, Transition, and Mortality

I assume that the reader is already familiar with the Leslie model (e.g., Case, 2000) for structured populations, that is written as:

The notation is the usually used. 88 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

The individuals are counted immediately after birth, and the births occurs instantaneously at a given time point, as illustrated in figure 6.6.

Reproduction

Figure 6.6. Graphical representation of the after reproduction census, when all individuals of all ages are counted

The time-space symmetry (section 5.3, and figure 5.1), and the relative mortality rate already defined as:

dy TRM= i =−cln R e−ct i=-2, -3 (6.19) ydt i

underpins the constructions of models for structured populations, under a Gompertzian perspective. Let us introduce two figures (6.7, and 6.8) that clarify the dynamic of an age structured population.

Figure 6.7. Schematic illustration of the dynamics of the number of individuals in each age class of a structured population

Figure 6.8. Detail of figure 6.6 © L. S. Barreto, 2017. Theoretical Ecology 89 6 Structured Populations: The Gompertzian Approach

Now we introduce a procedure to establish the rates of permanence P, transition T, and exp (-ct)-1 mortality M, using the EGZ (yi=y0Ri , i=-2, -3), and equation (6.19). For each age (or stage) class it is verified T+P+M=1. With periodicity of projection equal to 1, let us consider the calculations for the next year. The mortality rate of class j, with individuals with ages from j to j+n, is equal to the geometric mean of the relative mortality rates (RMR) that occurs in these ages:

j+n 1/n

M j a j+n= ∏ RMRx (6.20) ( x= j )

The rate of transition of class j to class j+1 is equal to the number of individuals in the first age of the next age class, j+1, divided by the total number of individuals in class. The first age in class j+1 is j+n+1, and the number of individuals given by:

exp(-c(j+n+1))-1 yj+n+1= y0Ri , i=-2, -3 (6.21)

Thus, the transition rate is:

y j+n+1 T j= j+n (6.22)

∑ y x x= j

Finally, the permanence rate of class j is:

Pj=1-Mj-Tj (6.23)

We omitted the index i associated to variable y. Now, I use Scilab to introduce the application of equations (6.20) to (6.23) to Iberian lynx. The stage matrix has four classes: class 1: females with age less then 1 year; class 2: females with age from 1 to 2 years; class 3: reproductive females with ages from 2 to 9 years; class 4: females in post reproductive age. The periodicity of the projections is 1 year. The script is the following one: clear //Characteristic parameters of the lynx r=9.773;c=0.392; //Periodicity of projection h=1; //Longevity of the species w=13;

//Initial number of individuals for simulation n0= 1000/w; //Final density f=n0/r;

Let us define the age classes: 90 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

//Ages in each class t1=(0:0.1:0.9); t2=(1:0.1:1.9); t3=(2:0.1:8.9); t4=(9:0.1:12.9); a1=size(t1);a1=a1(2);n1=1/a1; a2=size(t2);a2=a2(2);n2=1/a2; a3=size(t3);a3=a3(2);n3=1/a3; a4=size(t4);a4=a4(2);n4=1/a4;

Applying equation (6.20):

//rates of mortality for periodicity of 1 year m1=h*(prod(c*log(r)*exp(-c*t1)))^n1;m2=h*(prod(c*log(r)*exp(-c*t2)))^n2; m3=h*(prod(c*log(r)*exp(-c*t3)))^n3;m4=h*(prod(c*log(r)*exp(-c*t4)))^n4;

M=[m1,m2,m3,m4];

Now, we create a hypothetical population to apply equation (6.22). We display the number of animals in each class of the created population:

//The frequencies of each class is equal to the sum of the individuals with the ages of its age range f1=sum(f*rêxp(-c*t1));f2=sum(f*rêxp(-c*t2));f3=sum(f*rêxp(-c*t3)); f4=sum(f*rêxp(-c*t4));

//Integer values F=floor([f1,f2,f3,f4]/10); disp(F)

55. 29. 82. 32. Scilab output

We apply equation (6.22) to obtain the transition rates of the classes:

//Fequency of the first age of the next class/f*h=fraction that transits t21=t2(1);t31=t3(1);t41=t4(1);t51=13; ft1=h*(f*rêxp(-c*t21))/F(1);ft2=h*(f*rêxp(-c*t31))/F(2);ft3=h*(f*rêxp(- c*t41))/F(3); ft4=h*(f*rêxp(-c*t51))/F(4); //Classes 1, and 2 have only one year, T+M=1 T=[1-m1,1-m2,ft3,ft4]; disp(T)

0.2509003 0.4938302 0.1026327 0.2494244 ilab outputScilab output

Is time to apply equation (6.23), and display the permanence rates of the classes:

//Fração que permanece na classe p1=0;p2=0;p3=1-m3-ft3;p4=1-m4-ft4;

P=[p1,p2,p3,p4]; disp(P)

0. 0. 0.7918503 0.7383581 Scilab output © L. S. Barreto, 2017. Theoretical Ecology 91 6 Structured Populations: The Gompertzian Approach

Let us construct a table, whose columns, from left to right, are: class number (Cl), class frequency (F), permanence rate (P), transition rate (T), mortality rate (M), first year of the class (idin).

//idin=first year of the class Cl=1:4;idin=[0,1,2,9]; resumo=[Cl;F;P;T;M;idin]; disp([' Cl F P T M idin']) disp(resumo')

Cl F P T M idin

1. 55. 0. 0.2509003 0.7490997 0. 2. 29. 0. 0.4938302 0.5061698 1. Scilab output 3. 82. 0.7918503 0.1026327 0.1055170 2. 4. 32. 0.7383581 0.2494244 0.0122175 9.

We assume that the dominant eigen value o the Leslie matrix is one (λ1=1), and calculate f3. The matrix is established with the script:

//To the matrix with lambda1=1 //corresponds F3=1,68 a=[0 0 1.68 0;T(1) 0 0 0;0 T(2) P(3) 0;0 0 T(3) P(4)]; disp('Leslie matrix') disp(a)

Leslie matrix

0. 0. 1.68 0. 0.2509003 0. 0. 0. 0. 0.4938302 0.7918503 0. 0. 0. 0.1026327 0.7383581

We can accomplish an analysis of the matrix: valpr=spec(a); [ab,x,bs]=bdiag(a,1/%eps); [ic,ir]=find(ab'==max(ab')); w=conj(inv(x)); v1=x(:,ic); w1=real(w(ic,:))'; s=w1*v1'; VR=w1*1/w1(1,1); eee=v1/sum(v1); h=gsort(spec(a)); el=s.*a/h(1,1); disp("Eigenvalues") disp(h) disp("Reproductive values") disp(VR) disp("Stable age distribution") disp(eee) disp("Sensitivity of the dominant eigenvalue") disp(s) disp("Elasticity of the dominant eigenvalue") disp(el)

Eigenvalues

1.0000042 λ1 92 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

0.7383581 - 0.1040769 + 0.4442102i - 0.1040769 - 0.4442102i

Reproductive values

1. 3.985664 8.0709532 0.

Stable age distribution

0.4808554 0.1206462 0.2862246 0.1122738

Sensitivity of the dominant eigenvalue

0.1469689 0.0368744 0.0874819 0.0343154 0.5857688 0.1469689 0.3486734 0.1367698 1.1861794 0.2976115 0.7060621 0.2769583 0. 0. 0. 0.

Elasticity of the dominant eigenvalue

0 0 0.1469689 0 0.1469689 0 0 0 0 0.1469689 0.5590932 0 0 0 0 0

We use the establish matrix, to present a projection for 50 years, for the initial lynx population and present its graph (figure 6.9).

//Projection with the Leslie matrix //Iniatial population p=[10 30 60 12]; //Vector to receive the projection P=[p']; for i=1:50 p=a*p'; p=p'; P=[P p']; end xset('window',1) clf plot(P') xset('window',1) clf plot(P') xtitle(' ','Years','Frequencies of the classes') legend(['Newborns';'Juveniles';'Adults';'Seniles']) © L. S. Barreto, 2017. Theoretical Ecology 93 6 Structured Populations: The Gompertzian Approach

Figure 6.9. Projection of the lynx population with the matrix we established. For more details see the text

I suggest the reading of Barreto (2011), section 7.5, to find models of difference and differential equations for age-structured populations. In this same reference an intra-annual model for a forest can be found.

6.5 References, and Related Bibliography

Banks, R. B., 1994. Growth and Diffusion Phenomena. Mathematical Frameworks and Applications. Springer- Verlag. Barreto, L. S., 1979. Modelos Matriciais em Ecologia. Série Estudos, n.º 10, Direcção Geral do Fomento Florestal, Lisboa. Barreto, L. S., 1991. SPESS - A simulator for pure even aged self-thinned stands. Ecoll. Modelling, 54:127-132. Barreto, L. S., 1995a. Povoamentos Jardinados. Instrumentos para a sua Gestão. Publicações Ciência e Vida, Lda., Lis- boa. Barreto, L. S., 1995b. The fractal nature of the geometry of self-thinned pure stands. Silva Lusitana, 3(1):37-52. Barreto, L. S., 1996. Modelling and managing uneven-aged pure forests of Corsican pine and beech. Silva Lusitana, 4(2):185-198. Barreto, L. S., 2000. Pinhais Mansos. Ecologia e Gestão. Estação Florestal Nacional, Lisboa. Barreto, L. S., 2002. Uneven-aged Stands of Pinus pinaster Ait. for Protection. Research Paper SB-01/02. Departa- mento de Engenharia Florestal, Instituto Superior de Agronomia. Barreto, L. S., 2003. A Unified Theory for Self-Thinned Pure Stands. A Synoptic Presentation. Research Paper SB-03/03. Departamento de Engenharia Florestal, Instituto Superior de Agronomia. Barreto, L. S., 2004. Pinhais Bravos. Ecologia e Gestão. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Lisboa. Barreto, L. S., 2010. Árvores e Arvoredos. Geometria e Dinâmica. E-book. Costa de Caparica. Included in the CD. Barreto, L. S., 2011. From Trees to Forests. A Unified Theory. E-book. Costa de Caparica. Included in the CD. Barreto, L. S., 2013. SB-PRISM - An Algorithm to Unveil the Structure of a Population Simulated by the Gompertz Equation. Pre-print with restricted dissemination. Burel, A., 1888. Étude sur la Constitution Normal des Forêts Jardinées. Paris. Mentioned by Patronne (1944). Caswell, H., 2001. Matrix Population Models. Construction, Analysis, and Interpretation. Second edition. Sinauer As- sociates, Inc. Publishers, Sunderland, Massachusetts. Hubbell, S P, 2001. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton. Hutchinson, G. E., 1978. An Introduction to Population Ecology. Yale University Press. Murray, J. D., 2002. Mathematical Biology. I. An Introduction. Third edition. Springer, New York. Oliveira, A. C., 1985. Tabela de produção geral para o pinheiro bravo nas regiões montanas e submontanas. Direcção Geral das Florestas/Centro de Estudos Florestais, Lisboa. 94 © L. S. Barreto, 2017. Theoretical Ecology 6 Structured Populations: The Gompertzian Approach

Owen-Smith, N., 2007.Modeling in Wildlife and Resource Conservation. Blackwell Publishing, Oxford, United King- dom. Patronne, G., 1944. Lezioni de Assestamento Florestale. Tipografia Mariano Rici, Firenze. Perona, V., 1885. Di alcuni metodi per determinar ela ripresa dei boschi de taglio saltuario. Ecco dei Campi e dei Bos- chi, pp. 5 e seguintes. Referido por Patronne (1944). Puton, A., 1890. Traité d’Économie Forestière (Aménagement). Paris, páginas 201 e seguintes. Referido por Patronne (1944). © L. S. Barreto, 2017. Theoretical Ecology 95 PART II

PART II

POPULATION INTERACTIONS

In this second part we will explore models (mainly with 2 populations) of population interactions. We approach both ODE, and discrete models. The interactions covered are the following ones:  Amensalism  Comensalism  Competition  Detritivory  Herbivory  Mutualism  Pantophagy  Parasitism e parasitoidism  Predation or zoophagy 96 © L. S. Barreto, 2017. Theoretical Ecology 7 Introduction to Part II

7 Introduction to Part II

7.1 Explanation

We present species interactions in an unusual way. Our treatment of the subject is different of the common presentation in three different aspects: ● We use a more rich set of interactions, and thus of terminology; ● We respect the fact that the sizes of populations, and individuals are non negative numbers; ● The models of two ordinary differential equations are positive systems, this is, if the initial

values are positive all subsequent solutions are non negative (yi≥0, i=1,2); ● This positiveness allows the use of more simple and direct analytical processes to deal with the solutions of the systems of two ODE. Let us start by summarizing the populations interactions, in a conventional way, in table7.1.

Table 7.1. Types of species interactions, and their brief, and symbolic description

Designation Interaction Description (A B) Amensalim 0 - Species A is not affected Species B is harmed Comensalism 0 + Species A is not affected Species B is benefited Detritivory 0 + The resource A is not affected Species B is benefited Symmetric competition - - Both species B are harmed Mutualism + + Both species are benefited Asymmetric competition - + Species A is harmed Herbivory, Parasitism Species B is benefited Parasitoidism,Predation Neutralism 0 0 Neither species is affected by the other

Generically we will consider two state variables y1, and y2, and a system of two ODE:

y’1=f1(y1,y2) (7.1a)

y’2=f2(y1,y2) (7.1b)

The positiveness of the variables of biological, ecological, chemical systems allow the in- sertion of the analysis of their ODE systems in the context of cooperative, and competitive systems. Models of biological, and ecological systems that generate negative solutions must be re- jected. They are impinging on the real systems proprieties that in reality they do not possess. Systems of two ODE are named cooperative if the following is verified: © L. S. Barreto, 2017. Theoretical Ecology 97 7 Introduction to Part II

δ f δ f 1 ≥ 0 and 2 ≥ 0 (7.2) δ y2 δ y1

Systems of two ODE are named competitive if the following is verified :

δ f δ f 1 ≤ 0 and 2 ≤ 0 (7.3) δ y2 δ y1

In the models of population interactions presented in the following chapters their parameters are positive numbers, unless the contrary is stated. Now let us see what happens with the solutions of cooperative, and competitive systems: ● The interaction mutualism satisfies equation (7.2). The solutions of cooperative systems are only stable nodes, and periodic solutions do not occur. ● The interaction symmetric competition (- -) satisfies equation (7.3). A competitive system can be converted to a cooperative system, and has identical solution (Smith, 2012, and references herewith). ● If the Jacobian matrix of a cooperative system is irreducible the trajectories of almost all initial values converge to a unique equilibrium. The solutions of cooperative systems are monotone. This is, the order of the size of the variables of the system do not change. If for the initial values it is observed y1>y2, this inequality is never reversed. See figure 7.1.

Figure 7.1. The dynamics of a monotone system representing facultative mutualism. It is always verified y1

The solutions of the interaction [- +] can be a stable stable node, neutral equilibrium or centre, and limit cycle. Chaos does not occur. We summarize the solutions of the systems of our interest in figures 7.2. 98 © L. S. Barreto, 2017. Theoretical Ecology 7 Introduction to Part II

Biological,Sistemas and biológicos ecological e ecológicos systems (y (yi i≥0) ≥0) withde duastwo ODE EDO

Symmetric systems Asymmetric sys- Asymmetric tem system [+ +] and [- -] [0 -] and [0 +] [- +]

Symmetric competition, Amensalism, Asymmetric competition, mutualism commensalism, herbivory, detritivory parasitism, parasitoidism predation

Stable node Stable node, and cycles

Figure 7.2. Preliminary characterization of positive systems of two ODE

An accessible, and under an ecological perspective, text of cooperative, and competitive systems can be found in Ramos-Jiliberto, Hoecker-Escuti, and Mena-Lorca (2004; see references herewith). This article has broader mathematical scope then the one exhibited in this chapter. For a more formal, and embracing approach see Hirsch, and Smith (2005).

7.2 References

Hirsch, M. W., and H. Smith, 2005. Monotone Dynamical Systems. In A. Cañada, P. Drábeck, and A. Fonda, (Editors), 2005. Handbook of Differential Equations. Volume 2. Elsevier. Pages 239-257. Ramos-Jiliberto, R., F. Hoecker-Escuti, and J. Mena-Lorca, 2004. Why the dimension matters in ecological models?. Revista Chilena de Historia Natural, 77: 711-723. http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716- 078X2004000400012 Smith. H., 2012. Dynamical Systems in Biology. School of Mathematical and Statistical Sciences. Arizona State Uni- versity, Temple. http://math.la.asu.edu/~halsmith/MDS-2012.pdf © L. S. Barreto, 2017. Theoretical Ecology 99 8 Models for Predation, Omnivory, and Pantophagy

8 Models for Predation, Omnivory, and Pantophagy

8.1 Introduction

The prey-predator models available evince three serious shortcomings (e.g., Murdoch, Briggs e Nisbet, 2003:11,71; Turchin, 2003:96,133-136): ● The unstable equilibria evinced by the models seem relatively rare in nature. ● In nature, predators routinely maintain the prey population in equilibrium, at numbers far below the carrying capacity. The models do not mimic this situation. ● Contrary to what happens in nature, they do not allow the existence of stable food chains with more than two trophic levels. The models proposed in this chapter overcome these disadvantages. The models here presented for prey-predator are the following ones: • Model sbpred (Barreto, 2005;2011) • Model sbparasit-p • Model sbparasit-bn These models can be extended to the interaction host-parasit.

8.2 Model SBPRED

Model SBPRED (MSP) was first proposed in Barreto (2005:123-133). Let us introduce the model assumptions.

8.2.1 Assumptions MSP assumes the following:  Both prey, and predator show Gompertzian dynamics;  It applies the hyperbolic or Holling type 2 functional response. If it is justified it can accept type 3.

 The carrying capacity of the predator (y2f or K2) is equal to the number of prey multiplied by the parameter b, that mirrors the contribution of each prey to the carrying capacity of the predator It depends on the parameters of the functional response, and the efficiency of the predator on transforming the preys in its own growth.

8.2.2 The Model The model can accommodate any plausible number of trophic levels but, as usual, we consider here only one prey, and one predator. I call this model SBPRED(1,1) (MSP11), and it is written as:

(8.1)

(8.2)

It is not possible to find an explicit solution for the model. In Barreto (2005:123-133) it is presented a general formulation of the model. 100 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

8.2.3 Model Analysis Let us start introducing the zero growth lines (y’=0; ZGL) of the two variables, obtained with wxMaxima:

The prey has a non linear ZGL (y1=- ((a*c1*h*x+c1)*log(x)-a*c1*h*log(k)*x-c1*log(k))/a); The predator a line with slope 1/b. Thus, the ZGL of the predator can not be orthogonal to the ZGL of the prey, and the occurrence of a neutral equilibrium is precluded. These ZGL are exhibited I figure 8.1. Contrary to what happens in the models Lotka-Volterra, and Rozenzweig e Macarthur for predation, the ZGL of the predator is not constant, what may be more realist. Let us comment in figure 8.1: • If b is high, 1/b is low (point C) occurring a stable equilibrium; • For moderate values of b (point B), to the right of the maximum value of the ZGL of the prey, the equilibrium is also stable; • For low values of b (point A), the equilibrium occurs to the left of the maximum value of the ZGL of the prey, the equilibrium is now unstable. © L. S. Barreto, 2017. Theoretical Ecology 101 8 Models for Predation, Omnivory, and Pantophagy

Unstable equilibrium y2’=0 y1’=0 Stable equilibrium y2’=0

Predator

Prey Figure 8.1. Zero growth lines of MSP11

MSP11 has two carrying capacities: K for the prey, and by1 for the predator. We will analyse the eventual occurrence of Hopf bifurcations caused by variations of K, and b. We will use Oscill8. The parameters of the system we analyse are: c1=0.9, c2=0.02, a=1, h=0.5, K=25, b=0.6. The analysis for K is displayed in figure 8.2. For K>44.97, the system becomes unstable. 102 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Prey

Figure 8.2. Bifurcation analysis of MSP11 associated to K. A Hopf bifurcation (point HB 6) occurs when K=44.97. For values K>44.97 the system becomes unstable (thinner line)

The bifurcation analysis related to b is exhibited in figure 8.3.

Predator

b Figure 8.3. Bifurcation analysis of MSP11 associated to b. A Hopf bifurcation (point HB 2) occurs when b=0.79. For values b>0.79 the system becomes unstable (thinner line)

Let us use wxMaxima to simulate the system, being with K=40, and using values of b equal to 0.1, 0.6, 0.9. The graphics are in figure 8.4. © L. S. Barreto, 2017. Theoretical Ecology 103 8 Models for Predation, Omnivory, and Pantophagy

Stable node; b=0.1

Attractive focus; b=0.6

Limit cycle; b=0.9

Figure 8.4. Simulations of MSP11, with three different values of b. See text for details 104 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

In figure 8.5, we illustrate the effect of the value of K on the stability of the model. The transition of the value from K=44 to K=45 has a conspicuous effect on the stability of the solution.

Figure 8.5. Simulations of MSP11, with three different values of K. See text for details

8.2.4 Testing the Model In section 8.1 we annotated that the available models for predation have undesirable attributes. In this section I we show that MSP11 has the desired qualities. MSP11 can have stable fixed points with low levels of prey I already showed this property of MSP11 in Barreto (2005). Let us consider the model with the following parameters: c1=0.05; k=20; a=5; h=0.5; c2=0.256; b=0.5 Let us find the fixed points of its system of ODE, in R, and analyse its stability (I omit some irrelevant output) :

> > library(BB) > library(numDeriv) > > c1=0.05; k1=20; a=5; h=0.5; c2=0.256; b=0.5 > #Finding fixed points © L. S. Barreto, 2017. Theoretical Ecology 105 8 Models for Predation, Omnivory, and Pantophagy

> sbpred <- function(x) { + n <- length(x) + F <- rep(NA, n) + + + F[1]<- c1*x[1]*(log(k1)-log(x[1]))-a*x[1]*x[2]/(1+a*h*x[1]) + F[2]<- c2*x[2]*(log(b*x[1])-log(x[2])) + + F + } > p0 <- c(0.12,0.05) > > ans<-BBsolve(par=p0, fn=sbpred) > ans $par [1] 0.1336305 0.0668156 ← Fixed points ……....…..…..…..…... ………………….

> > #Jacobian matrix > x <- ans$par > x [1] 0.1336305 0.0668156 > J<-jacobian(sbpred, x) > J [,1] [,2] [1,] 0.01271079 -0.5008352 [2,] 0.12800072 -0.2560014 > #Eigenvalues > L<-eigen(J)$values > L [1] -0.1216453+0.2146059i -0.1216453-0.2146059i > > if (Re(L[1])<0 && Re(L[2])<0) print("Stable equilibrium") [1] "Stable equilibrium" ←← Stable fixed points > >

At equilibrium, the size of the prey is about 0.007 of its carrying capacity, and the predator has half of the size of the prey. The real part of the eigenvalues are negative (-0.1216), thus the solution is stable. The desired attribute is confirmed. To the output of the model we can fit MAR(1) models I use the numerical solution of the same model to show that the data satisfies MAR(1) models:

> ##sbpred > library(deSolve) > library("mAr") > > sbpred<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + a<-parms[3] 106 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

+ h<-parms[4] + + c2<-parms[5] + b<- parms[6] + + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1]))-a*n[1]*n[2]/(1+a*h*n[1]) + dn2.dt<- c2*n[2]*(log(b*n[1])-log(n[2])) + return(list(c(dn1.dt,dn2.dt))) + } > > #parms<-c(c1=0.05, k1=40, a=1, h=1, c2=0.5, b=0.04) > parms<-c(c1=0.05, k1=20, a=5, h=0.5, c2=0.256, b=0.5) > > initialn<-c(10, 3) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, sbpred, parms=parms) > > > matplot(out[,1], out[,-1], type="l", xlab="Tempo", ylab="N") > title("Predation SBPRED(1,1)") > > > h<-matrix(c(out[,2],out[,3]),2991,2) > x=h[2991,] #fixed point > x [1] 0.13363207 0.06681604 > #Fitting MAR(1) > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > N1<-dat0[,1] > N2<-dat0[,2] > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > ########### > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.90895 0.00207 0.00207 0.00207 0.68815

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.08861 0.00441 20.09 <2e-16 *** N1b 0.52390 0.01173 44.66 <2e-16 *** © L. S. Barreto, 2017. Theoretical Ecology 107 8 Models for Predation, Omnivory, and Pantophagy

N2b -0.40503 0.02648 -15.30 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.07228 on 296 degrees of freedom Multiple R-squared: 0.9196, Adjusted R-squared: 0.9191 F-statistic: 1693 on 2 and 296 DF, p-value: < 2.2e-16

> > summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.212392 0.001697 0.001697 0.001697 0.284644

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.014078 0.001396 10.08 <2e-16 *** N1b 0.230230 0.003715 61.98 <2e-16 *** N2b 0.303447 0.008385 36.19 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02289 on 296 degrees of freedom Multiple R-squared: 0.9892, Adjusted R-squared: 0.9891 F-statistic: 1.35e+04 on 2 and 296 DF, p-value: < 2.2e-16

The quality of the fittings confirms that the data adhere to MAR(1). To the output of each populationl we can fit a Gompertz equation Now we show that each population follows a EGZ.

> #Fitting Gompertz equations > > x1=h[,1] > x<-log(x1[-2991]) > y<-log(x1[-1]) > > model0<-lm(y~x) > summary(model0)

Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -0.78574 0.00224 0.00224 0.00224 0.02388

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0137389 0.0019903 -6.903 6.19e-12 *** x 0.9942858 0.0008603 1155.766 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03997 on 2988 degrees of freedom 108 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Multiple R-squared: 0.9978, Adjusted R-squared: 0.9978 F-statistic: 1.336e+06 on 1 and 2988 DF, p-value: < 2.2e-16

> > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > plot(x,y, main='Prey fitting') > cf<-coef(model0) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv) > > > x2=h[,2] > x<-log(x2[-2991]) > y<-log(x2[-1]) > > mode20<-lm(y~x) > summary(mode20)

Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -0.166294 0.001806 0.001806 0.001806 0.036693

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0179373 0.0013356 -13.43 <2e-16 *** x 0.9940383 0.0004646 2139.63 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01706 on 2988 degrees of freedom Multiple R-squared: 0.9993, Adjusted R-squared: 0.9993 F-statistic: 4.578e+06 on 1 and 2988 DF, p-value: < 2.2e-16

> > screen(2) > plot(x,y, main='Predator fitting') > cf<-coef(mode20) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv) >

Now we present the two graphics of the output, in figure 8.6. © L. S. Barreto, 2017. Theoretical Ecology 109 8 Models for Predation, Omnivory, and Pantophagy

Prey fitting Predator fitting 1 2 0 0 1 - 2 - 2 - y y 3 - 4 - 4 - 6 - 5 - 6 - 8 -

-8 -6 -4 -2 0 2 -6 -5 -4 -3 -2 -1 0 1

x x

Figure 8.6. Graphics of the logarithms of the data, and of the simulated values with the fitted EGZ

We can accept the the sizes of each population (prey, and predator) follows EGZ. The model allows the existence of stable food chains with more then 2 trophic levels Two sustain this statement I present a short script in R that analysis a food chain with four trophic levels, assuming that the variables are measured in grams of biomass:

> > ############ Parâmetros > a1=1;h1=1;b1=0.8 > c1=0.1;c2=0.4;c3=0.3;c4=0.15 > k=60 > a2=0.8;h2=1;b2=.8 > a3=0.5;h3=1;b3=.8 > > #################################### > > > #Finding fixed points > panto4b <- function(x) { + n <- length(x) + F <- rep(NA, n) + + + F[1]=c1*x[1]*(log(k)-log(x[1]))-a1*x[1]*x[2]/(1+a1*h1*x[1]); + F[2]=c2*x[2]*(log(b1*x[1])-log(x[2]))-a2*x[2]*x[3]/(1+a2*h2*x[2]); + F[3]=c3*x[3]*(log(b2*x[2])-log(x[3]))-a3*x[3]*x[4]/(1+a3*h3*x[3]); + F[4]=c4*x[4]*(log(b3*x[3])-log(x[4])); + + F + } > p0 <- c(2, 1, 0.5, 0.4) > > ans<-BBsolve(par=p0, fn=panto4b) > ans $par [1] 2.304369 1.080320 0.498944 0.393880 ← Fixed points

………...…....…..…....….. > > #Jacobian matrix > x <- ans$par > x [1] 2.304369 1.080320 0.498944 0.393880 110 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

> J<-jacobian(panto4b, x) Warning message: In jacobian(panto4b, x) : bytecode version mismatch; using eval > J [,1] [,2] [,3] [,4] [1,] 0.1270130 -0.6973704 0.0000000 0.0000000 [2,] 0.1875255 -0.3010875 -0.4635930 0.0000000 [3,] 0.0000000 0.1385545 -0.2613356 -0.1996619 [4,] 0.0000000 0.0000000 0.1184141 -0.1480044 > #Eigenvalues > L<-eigen(J)$values > L [1] -0.1806916+0.3909994i -0.1806916-0.3909994i -0.1110157+0.1106798i [4] -0.1110157-0.1106798i

The real part of all eigenvalues are negative, thus the equilibrium is stable. We can present a graphic of the dynamics of the system (figure 8.7), and fit a MAR(1) to the trajectory of the system:

> panto4b<-function(times,y,parms) { + n<-y + dn1.dt=c1*n[1]*(log(k)-log(n[1]))-a1*n[1]*n[2]/(1+a1*h1*n[1]); + dn2.dt=c2*n[2]*(log(b1*n[1])-log(n[2]))-a2*n[2]*n[3]/(1+a2*h2*n[2]); + dn3.dt=c3*n[3]*(log(b2*n[2])-log(n[3]))-a3*n[3]*n[4]/(1+a3*h3*n[3]); + dn4.dt=c4*n[4]*(log(b3*n[3])-log(n[4])); + + return(list(c(dn1.dt,dn2.dt,dn3.dt,dn4.dt))) + } > > > > initialn<-c(3, 1, 0.4, 0.3) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, panto4b, parms=parms) > > > > g=out[,-1] > g=ts(g[seq(1,3000,by=10),]) > > plot(g, plot.type = "single", col = 1:4) > title("Linear food chain") > r<-c('Prey','Predator1 ','Predator 2','Predator 3') > legend('topleft',paste(rev(r)),lty=1,col=4:1, bty='n') > © L. S. Barreto, 2017. Theoretical Ecology 111 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.7. The dynamics of a linear food chain with four trophic levels. The system is monotone. See text for the script

> #Fitting MAR(1) > > h<-matrix(c(out[,2],out[,3], out[,4],out[,5]),2991,4) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,4) > > library(MASS) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > N4<-dat0[,4] > > ###################################### > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > N4a<-N4[-1] > N4b<-N4[-m[1]] > > > > fit1=lm(N1a ~ N1b+N2b+N3b+N4b) > fit2=lm(N2a ~ N1b+N2b+N3b+N4b) > fit3=lm(N3a ~ N1b+N2b+N3b+N4b) > fit4=lm(N4a ~ N1b+N2b+N3b+N4b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b + N4b) 112 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Residuals: Min 1Q Median 3Q Max -0.0032735 -0.0000318 -0.0000318 -0.0000318 0.0038612

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.4790061 0.0033092 144.749 < 2e-16 *** N1b 1.0775435 0.0007449 1446.652 < 2e-16 *** N2b -0.6878113 0.0021730 -316.531 < 2e-16 *** N3b 0.1866842 0.0033602 55.558 < 2e-16 *** N4b -0.0255371 0.0054690 -4.669 4.6e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0004888 on 294 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 2.544e+06 on 4 and 294 DF, p-value: < 2.2e-16

> > summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b + N4b)

Residuals: Min 1Q Median 3Q Max -0.0042371 0.0000354 0.0000354 0.0000354 0.0034302

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.1138478 0.0031852 35.74 <2e-16 *** N1b 0.1590595 0.0007169 221.86 <2e-16 *** N2b 0.6945712 0.0020915 332.09 <2e-16 *** N3b -0.3702672 0.0032342 -114.48 <2e-16 *** N4b 0.0837229 0.0052640 15.90 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0004705 on 294 degrees of freedom Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998 F-statistic: 3.692e+05 on 4 and 294 DF, p-value: < 2.2e-16 > > summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b + N4b)

Residuals: Min 1Q Median 3Q Max -5.472e-04 6.050e-06 6.050e-06 6.050e-06 5.352e-04

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0628045 0.0006959 90.25 <2e-16 *** N1b 0.0043324 0.0001566 27.66 <2e-16 *** N2b 0.1196623 0.0004569 261.88 <2e-16 *** N3b 0.7270662 0.0007066 1029.01 <2e-16 *** N4b -0.1592792 0.0011500 -138.50 <2e-16 *** --- © L. S. Barreto, 2017. Theoretical Ecology 113 8 Models for Predation, Omnivory, and Pantophagy

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0001028 on 294 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.642e+06 on 4 and 294 DF, p-value: < 2.2e-16 > > summary(fit4)

Call: lm(formula = N4a ~ N1b + N2b + N3b + N4b)

Residuals: Min 1Q Median 3Q Max -2.172e-04 3.707e-06 3.707e-06 3.707e-06 1.850e-04

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.685e-03 2.040e-04 27.87 <2e-16 *** N1b 5.715e-04 4.592e-05 12.45 <2e-16 *** N2b 2.118e-03 1.340e-04 15.81 <2e-16 *** N3b 9.101e-02 2.071e-04 439.37 <2e-16 *** N4b 8.632e-01 3.371e-04 2560.27 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.014e-05 on 294 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.006e+07 on 4 and 294 DF, p-value: < 2.2e-16

Indeed, to the trajectory of the four populations we can fit successfully a MAR(1). We call this discrete model SBPRED11-dl (MSP11-dl). It can be formally written:

y1,t+1=a1+b1 y1,t+c1 y2,t (8.3)

y2,t+1=a2+b2 y1,t+21 y2,t (8.4)

As it will be illustrated in the following chapters, to the output of all models for populations interactions here introduced it is possible to fit MAR(1). This property is relevant because it let us obtain the community matrix from the coefficients of the MAR(1). From this matrix we can obtain the matrix of total effects (MTE). It is the MTE that allow us to get more insight on the dynamics of the interactions in a system with more then two species. This issue will be clarified ahead, when we will approach omnivory. We will show that the matrix of total effects is an indispensable instrument do identify keystone species, and controlling factors in ecosystems (chapter 17).

8.3 Discrete Model SBPRED11-de

8.3.1 Model Assumptions

In chapter 4, equation (4.39), we introduced a discrete form of the EGZ, with an exponential (yt+1= yt exp (c ln K -c ln yt)). Now we will extend this model to predation, This new model for predation is named SBPRED11-de (MSP11-de). It is formulation assumes the following: 114 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

● The dynamics of the populations are better represented by a discrete model;

● The characteristics parameters of the populations ( c1, c2, y1f=K) are the same if they are isolated or not; ● Equation (4.39) assumes an isolated population. Predation must affect the intraspecific competition of the prey population, and this new situation must be mirrored in the model

(a1 instead of c1 in equation (8.5)).

8.3.2 The Model Model MSP11-de is written as:

y1,t+1= y1,t exp(c1ln K-a1 ln y1,t+ m1 ln y2,t) (8.5)

y2,t+1= y2,t exp(c2 a2+m2 ln y1,t+ c2 ln y2,t) (8.6)

8.3.3 Model Analysis It can be shown that the model has the following fixed point:

y1*=exp(c2*c1*log(K)-m1*c2*a2)/(m1*m2+a1*c2)) (8.7)

y2*=exp-(a1*c2*a2+m2*c1*log(K)/(m1*m2+a1*c2)) (8.8)

We start by fitting the model to the output of model MSP11, using R:

> ##sbpred11b > rm(list=ls(all=TRUE)) > library(deSolve) > > sbpred<-function(times,y,parms) { + n<-y + + c1<-parms[1] + k1<- parms[2] + a<-parms[3] + h<-parms[4] + + c2<-parms[5] + b<- parms[6] + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1]))-a*n[1]*n[2]/(1+a*h*n[1]) + dn2.dt<- c2*n[2]*(log(b*n[1])-log(n[2])) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.9, k1=20, a=1, h=0.5, c2=0.02, b=0.6) > initialn<-c(10, 3) > t.s<- seq(1,90, by=0.1) > > > out<- ode(y=initialn, times=t.s, sbpred, parms=parms) > > matplot(out[,1], out[,-1], type="l", xlab="Tempo", ylab="Biomassa") © L. S. Barreto, 2017. Theoretical Ecology 115 8 Models for Predation, Omnivory, and Pantophagy

> title("Predação SBPRED(1,1)") > rm(list=ls(all=TRUE)) > ##sbpred11b > rm(list=ls(all=TRUE)) > library(deSolve) > > sbpred<-function(times,y,parms) { + n<-y + + c1<-parms[1] + k1<- parms[2] + a<-parms[3] + h<-parms[4] + + c2<-parms[5] + b<- parms[6] + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1]))-a*n[1]*n[2]/(1+a*h*n[1]) + dn2.dt<- c2*n[2]*(log(b*n[1])-log(n[2])) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.9, k1=20, a=1, h=0.5, c2=0.02, b=0.6) > initialn<-c(12, 4) > t.s<- seq(1,90, by=0.1) > > > out<- ode(y=initialn, times=t.s, sbpred, parms=parms) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass") > title("Predation, SBPRED(1,1)") >

We obtain the graphic of figure 8.8.

Figure 8.8. Simulation of predation with the previous script. Prey population is the blue line. The fixed point is stable 116 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Now we fit equations (8.5), and (8.6) to the simulated data:

> yp<-out[,2] > y1<-yp[-1] > x1<-yp[-91] > yq<-out[,3] > y2<-yq[-1] > x2<-yq[-91] > x1g<-log(x1) > x2g<-log(x2) > > fit1<-nls(y1~x1*exp(0.9*log(20)-a1*x1g+m1*x2g), start=list(m1=0.1,a1=0.1)) > summary(fit1)

Formula: y1 ~ x1 * exp(0.9 * log(20) - a1 * x1g + m1 * x2g)

Parameters: Estimate Std. Error t value Pr(>|t|) m1 -1.606273 0.001686 -953.0 <2e-16 *** a1 0.190129 0.001207 157.6 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03668 on 888 degrees of freedom

Number of iterations to convergence: 7 Achieved convergence tolerance: 9.318e-09

> > fit2<-nls(y2~x2*exp(0.02*a2+m2*x1g+0.02*x2g), start=list(a2=0.1,m2=0.1)) > summary(fit2)

Formula: y2 ~ x2 * exp(0.02 * a2 + m2 * x1g + 0.02 * x2g)

Parameters: Estimate Std. Error t value Pr(>|t|) a2 -1.849e+00 3.543e-03 -522.0 <2e-16 *** m2 4.113e-03 3.564e-05 115.4 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.000396 on 888 degrees of freedom

Number of iterations to convergence: 4 Achieved convergence tolerance: 7.215e-07

The two good fittings sustain model MSP11-de. Now, we made a graph with the data generated by the two models:

> bv1<-predict(fit1,list(x1g,x2g)) > plot(y1, main='Prey') > lines(bv1) > > bv2<-predict(fit2,list(x1g,x2g)) > plot(y2, main='Predator') > lines(bv2) © L. S. Barreto, 2017. Theoretical Ecology 117 8 Models for Predation, Omnivory, and Pantophagy

The graphics are exhibited in figure 8.9. The data generated by the two models are coincident.

Figure 8.9. Simulated data obtained with models MSP1, and MSP11-de. The data generated by the two models are coincident

The equilibrium happens when yi,t+1=yi,t, this is when the exponential are equal to 1. This implies the solution of the following system:

c1ln K-a1 ln y1,t+ m1 ln y2,t=0

c2 a2+m2 ln y1,t+ c2 ln y2,t=0

With the values obtained in the fittings, and using the following variable transformations x=ln y1, y=ln y2, we obtain a system of two linear equations. The equilibrium of MSP11-de is the exponential of the solution of the linear system. The system to solve is the following one:

-0.190129 x -1.606273 y = -2.69615 0.004113 x +0.02 y = 0.03698

The solution can be obtained in R, with the following script:

> ## Finding the fixed point ## > #Matrix of the linear system > A=matrix(c(-0.190129, -1.606273, + 0.004113, 0.02), nrow=2,byrow=TRUE) > #The second member of the system > b=matrix(c(-2.696159046198592,0.03698),nrow=2) > #Solving the system > ans=solve(A,b) > #The exponentials of the solution > x=exp(ans) > x [,1] [1,] 7.051168 [2,] 4.251729 > 118 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

> #Find the Jacobian matrix > #Call library numDeriv > library(numDeriv) > > sbpred <- function(x) { + n <- length(x) + F <- rep(NA, n) + + + F[1]<- 1-exp(0.9*log(20.0)-0.190129*log(x[1])-1.606273 *log(x[2])) + F[2]<- 1-exp(0.02*(-1.849)+0.004113*log(x[1])+0.02*log(x[2])) + + F + } > J<-jacobian(sbpred, x) > J [,1] [,2] [1,] 0.0269641857 0.377792862 [2,] -0.0005833076 -0.004703968 > L<-eigen(J)$values > L [1] 0.016639059 0.005621158 Eigenvalues of the Jacobian matrix >

As the absolute values of the eigenvalues are less then 1 the solution is stable. Obviously, the fixed point can be immediately obtained from equations (8.7), and (8.8).

8.4 Model sbparasit-p

8.4.1 Assumptions The model sbparasit-p stabilizes the Nicholson-Bailey model, because it assumes that when isolated the host populations do not have exponential growth but abide the EGZ. The assumptions of the model can be enumerated as: • When isolated, the host population has Gompertzian growth; • Each time interval corresponds to a generation; • The space is homogeneous for the hosts and they are regularly distributed in the space; • The parasitoids search the hosts randomly; • Independently of the number of eggs received, each infected host gives origin only to one new parasitoid. This is, in the next generation the number of new parasitoids is equal to the number of infected hosts in the previous generation: • The parasitoids produce an unlimited number of eggs; • The host density does not affect the area of discovery of the parasitoids; • Each parasitoid find a constant fraction of the hosts that occur in its area of discovery.

8.4.2 The Model The model sbparasit-p is written as:

Ht+1= Ht * exp(c1*(log(KH)-log(Ht))-a * Pt) (8.9)

Pt+1= Ht * (1 - exp(-a * Pt)) (8.10) where: © L. S. Barreto, 2017. Theoretical Ecology 119 8 Models for Predation, Omnivory, and Pantophagy

H t = Host density at period t

KH= Carrying capacity of the host population;

P t = Parasitoid density at period t; a=area of discovery of the parasitoid. It is constant for a given species.

8.4.3 Model Analysis The model has not an explicit solution, thus we will use a numerical approach. To simulate the model I use the following Scilab script:

//sbparasit-p

clear clf time=90 a=0.006 KH=200; c1=log(2);

P=20; H=20;

Y=[H P;];

for n=1:time Ha= H * exp(c1*(log(KH)-log(H))-a * P); Pa=H * (1 - exp(-a * P)); H=Ha;P=Pa; Y=[Y;H P]; end

t=0:1:90; subplot(1,2,1) plot2d(t,[Y]) xtitle('Time series','Time','Number') legend(['Host';'Parasitoid']) subplot(1,2,2)

plot2d4(Y(:,1),Y(:,2)) xtitle('Phase portrait','Host','Parasitoid')

//////////////////

deff('x=f(y)','x=[ H * exp(c1*(log(KH)-log(H))-a * P);y(1) * (1 - exp(-a * y(2)))]') y0=[Y(91,1);Y(91,2)]; [J]=derivative(f,y0); disp('Fixed point') disp(y0) disp("Jacobian matrix") disp(J) z=poly(J,"x"); v=roots(z); disp('Absolute eigenvalues') disp(abs(v))

Its output is: 120 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Fixed point

174.6261 15.673244

Jacobian matrix

0. 0. 0.0897532 0.9537171

Absolute eigenvalues

0.9537171 0.

As the absolute values of the eigenvalues are less then 1 the solution is stable. We obtain also figure 8.10.

Figure 8.10. The graphical output of the Scilab script for model sbparasit-p

The possible dynamics of model sbparasit-p are illustrated in figure 8.11. Very high values of the carrying capacity of the host destabilize the solution. The upper left graphic shows that the existence of parasitoid needs a minimum critical number of hosts. © L. S. Barreto, 2017. Theoretical Ecology 121 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.11. Possible dynamics of model sbparasit-p, with increasing carrying capacity (K) of the host. Both populations start with 20 individuals, a=0.006, and c1=ln 2. ‘Hospedeiro’=host; ‘Parasitoide’=parasitoid. Reproduced from Barreto (2016)

In figure 8.12, I exhibit the effect of the size of the area of discovery on the behaviour of the system. The effect is analogous to the effect of increasing K.

Figure 8.12. Possible dynamics of model sbparasit-p, with increasing area of discovery (a) of the parasitoid. Both populations start with 20 individuals, a=0.006, and c1=ln 2. ‘Hospedeiro’=host; ‘Parasitoide’=parasitoid. Reproduced from Barreto (2016)

Both the increasing of K, and a cause the emergence of Hopf bifurcations. 122 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

8.5 Model sbparasit-bn

8.5.1 Assumptions To formulate model sbparasit-bn, we introduce a unique modification on the assumptions of the previous model. Now, the probability of a host being attacked is given by a negative binomial distribution, instead of a Poisson distribution.

8.5.2 Model Model sbparasit-bn is described by the two following equations:

(8.11)

(8.12)

8.5.3 Model Analysis Also, this model has not an explicit solution, thus we will use a numerical approach. To simulate the model I use the following Scilab script:

//sbparasit-bn

clear clf time=90 a=0.006 KH=200; c1=log(2); k=5; P=20; H=20;

Y=[H P;];

for n=1:time Ha= H * exp(c1*(log(KH)-log(H)))*(1 + a * P/k)^(-k); Pa=H-H*(1 + a * P/k)^(-k); H=Ha;P=Pa; Y=[Y;H P]; end

t=0:1:90; subplot(1,2,1) plot2d(t,[Y]) xtitle('Time series','Time','Number') legend(['Host';'Parasitoid']) subplot(1,2,2)

plot2d4(Y(:,1),Y(:,2)) xtitle('Phase portrait','Host','Parasitoid')

////////////////// © L. S. Barreto, 2017. Theoretical Ecology 123 8 Models for Predation, Omnivory, and Pantophagy

deff('x=f(y)','x=[ y(1) * exp(c1*(log(KH)-log(y(1))))*(1 + a * P/k)^(- k);y(1)-y(1)*(1 + a * y(2)/k)^(-k)]') y0=[Y(91,1);Y(91,2)]; [J]=derivative(f,y0); disp('Fixed point') disp(y0) disp("Jacobian matrix") disp(J) z=poly(J,"x"); v=roots(z); disp('Absolute eigenvalues') disp(abs(v))

The output is the following:

Fixed point

174.6261 15.673244

Jacobian matrix 0. 0. 0.0897532 0.9537171

Absolute eigenvalues

0.9537171 Absolute eigenvalues<1. The solution is stable 0.

Comparing with model sbparasit-p, model sbparasit-bn has its stability reinforced by the adoption of a negative binomial distribution. The Scilab script generate the graphics in figure 8.13.

Figure 8.13. Simulation of the system host-parasitoid with equations (8.11), and (8.12). Initial values of both populations equal to 20, a=0,006, KH=200, k=5, c1= ln 2 124 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

In figure 8.14 the effect of k (power of the negative binomial) on the stability of the model is illustrated.

In Barreto (2014), I showed that the stability of the model depends on c1, KH, a, and k. The destabilizing effect of each parameter is conditioned by the values of the other parameters.

Figure 8.14. The effect of variation of k on the dynamics of the system of equations (8.11), and (8.12). Initial values of both populations equal to 20, a=0,006, KH=800, c1= ln 2. ‘Hospedeiro’=host; ‘Parasitoide’=parasitoid. Reproduced from Barreto (2016)

8.6 Direct, Indirect, and Total Effects

In the social, and financial sphere, we are all familiar with chain effects triggered by a single event or measure adopted by a government. The European Central Bank maintains a low rate of interest, and buy national debt of some countries because it assumes that this measures trigger a net of interrelated effects that favours the economies of the European countries. We are all familiar with the following causal chain: increasing the income of citizens → increases consumption → increases investment in production of goods for the consumers → increase employment …. This example is a very simple causal chain. Real systems are much more complex and the task of economists are much more difficult and plagued with disagreement. What makes the understanding of the behaviour of complex systems (as ecosystems, and national economies) intric- ate, and their dynamics almost unpredictable beyond a short period of time, are the rich network of direct, and indirect effects (those mediated by a third component). Let us introduce an example that will be numerically illustrated ahead. Consider the following chain of omnivory: © L. S. Barreto, 2017. Theoretical Ecology 125 8 Models for Predation, Omnivory, and Pantophagy

Resource Consumer

Figure 8.15. Diagrammatic representation of omnivory

The omnivore y3 has a negative direct effect on plant y1, and a positive indirect effect because it diminishes the number of herbivores y2 that also consume the plant.

The total effect (TE) of y3 on y1 is the sum of the direct, and indirect effects:

Total effect of y3 on y1= Direct effect of y3 on y1 + indirect effect of y3 on y1

If y3 has a very small consumption of y1, y2 grazes y1 intensively, and y3 consumes intensively y2, the total effect of y3 on y1 may be positive.

On the other side, If y3 has a heavy consumption of y1, y2 grazes y1 lightly, and y3 consumes lightly y2, the total effect of y3 on y1 may be negative. At the community level we can conjecture the following chain:

Populations→direct (interactions) effects→indirect effects→total effects →community behaviour

It is the network of TE that controls the community. We can add the effects of the environmental factors on the community, and arrive to a similar conclusion: the network of TE controls the ecosystem. Thus, if we want to understand ecosystems we must find a procedure to measure the TE. It is at this point that MAR(1) reveal their utility. If we construct a matrix with the coefficients of the multivariate linear models, we have the community matrix A. To obtain the matrix of TE (e.g., Case, 2000:357-360): • We calculate the inverse of matrix A; • The inverse matrix is multiplied by -1 to obtain the matrix of total effects E.

Let eij be the element of line i, and column j of E. The element eij is the total effect of species j (column) on species i (row). Now we apply these concepts to a system of omnivory.

8.7 Omnivory

8.7.1 Assumptions Let us approach the interaction of omnivory represented in figures 8.15. We assume the following:

 The basal species (y1) is a plant that only plays the role of resource;

 The middle species (y2) is a herbivore that plays both the role of consumer, and resource;

 The top species (y3) is an omnivore that consumes the previous two species  Model SBPRED is applicable to the interactions of the system of omnivory 126 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

8.7.2 The Model The model is written as:

(8.13)

(8.14) (8.15)

For easier reference I call this model PANT3.

8.7.3 Model Analysis PANT3 has not an explicit solution, thus we will use a numerical approach. This model evinces stable fixed points, and periodic solutions. Now let us illustrate the concepts of the previous section, using R:

> > library(deSolve) > library(rootSolve) Warning message: package ‘rootSolve’ was built under R version 3.3.2 > library(MASS) > > ############ Parameters > c1=0.05; k1=80; a1=1; h1=1; a2=.8; h2=1; c2=0.1 > b1=.4; a3=0.6; h3=0.7; c3=0.15; b2=.3; b3=.2 > #**************** para obter comensalismo presa-omniv subir a1 de 1 para 8 > > #*************** Model, and solution > baco3<-function(times,y,parms) { + n<-y + + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1]))-a1*n[1]*n[2]/(1+a1*h1*n[1])- a2*n[1]*n[3]/(1+a2*h2*n[1]+a3*h3*n[2]) + dn2.dt<- c2*n[2]*(log(b1*n[1])-log(n[2]))-a3*n[2]*n[3]/ (1+a2*h2*n[1]+a3*h3*n[2]) + dn3.dt<- c3*n[3]*(log(b2*n[1]+b3*n[2])-log(n[3])) + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > > > initialn<-c(2, 0.5, 0.8) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, baco3, parms=parms) > > > matplot(out[,1], out[,-1], type="l", col=1, xlab="Time", ylab="Biomass",ylim=c(0,3)) > title("Omnivory") > r<-c('Plant','Herbivore','Omnivore') © L. S. Barreto, 2017. Theoretical Ecology 127 8 Models for Predation, Omnivory, and Pantophagy

> legend('topright',paste(rev(r)),lty=3:1,col=1, bty='n') >

We obtain figure 8.16.

> #Fixed point > y<-initialn > ST2 <- runsteady(y=y,func=baco3,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 1.0750756 0.1435188 0.3512265 > #Preparing data to fit MAR(1) > h<-out[,-1] > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > ###################################### > > ## Functional responses of plant consumption > > rf1=a1*N1*N2/(1+a1*h1*N1) > rf2=a2*N1*N2/(1+a2*h2*N1+a3*h3*N2) > rf=matrix(c(rf1,rf2),nrow=300, ncol=2) > > matplot(rf, type="l", col=1:2, xlab="Tempo", ylab="Biomassa",ylim=c(0,0.5)) > > title("Fuctional response of plant consumption") > r<-c('Herbivore','Omnivore') > legend('topright',paste(r),lty=1,col=1:2, bty='n')

We obtain figure 8.17.

> ###### MAR(1) > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.0187033 -0.0005764 -0.0005748 -0.0005692 0.0176170

Coefficients: 128 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Estimate Std. Error t value Pr(>|t|) (Intercept) 0.147109 0.002704 54.398 < 2e-16 *** N1b 1.031816 0.002032 507.672 < 2e-16 *** N2b -0.984843 0.024103 -40.859 < 2e-16 *** N3b -0.112166 0.013577 -8.261 4.93e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.003742 on 295 degrees of freedom Multiple R-squared: 0.999, Adjusted R-squared: 0.999 F-statistic: 1.025e+05 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.0030700 0.0000033 0.0000037 0.0000049 0.0040778

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0113767 0.0003498 32.53 <2e-16 *** N1b 0.0283827 0.0002629 107.98 <2e-16 *** N2b 0.7648236 0.0031173 245.35 <2e-16 *** N3b -0.0231806 0.0017560 -13.20 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0004839 on 295 degrees of freedom Multiple R-squared: 0.9997, Adjusted R-squared: 0.9997 F-statistic: 3.044e+05 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.0041753 0.0001649 0.0001662 0.0001670 0.0023129

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0020775 0.0006634 3.131 0.00191 ** N1b 0.0563517 0.0004986 113.016 < 2e-16 *** N2b -0.0001651 0.0059132 -0.028 0.97775 N3b 0.8211913 0.0033309 246.535 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0009179 on 295 degrees of freedom Multiple R-squared: 0.9997, Adjusted R-squared: 0.9997 F-statistic: 2.95e+05 on 3 and 295 DF, p-value: < 2.2e-16

> > > L1=coef(fit1)[-1] > L2=coef(fit2)[-1] © L. S. Barreto, 2017. Theoretical Ecology 129 8 Models for Predation, Omnivory, and Pantophagy

> L3=coef(fit3)[-1] > > #Community matrix > A=(matrix(c( L1,L2,L3), nrow=3,ncol=3,byrow=T)) > A [,1] [,2] [,3] [1,] 1.03181611 -0.9848425491 -0.11216580 [2,] 0.02838268 0.7648236499 -0.02318058 Community matrix [3,] 0.05635174 -0.0001650524 0.82119130 > > ##Matrix of total effects > > E=-ginv(A) > E [,1] [,2] [,3] [1,] -0.92754899 -1.19441415 -0.16040902 [2,] 0.03635079 -1.26068963 -0.03062162 matrix of total effects [3,] 0.06365752 0.08170963 -1.20674163 > > eigen(A)$values [1] 0.8708778+0.1203088i 0.8708778-0.1203088i 0.8760755+0.0000000i > eigen(E)$values [1] -1.141454+0.000000i -1.126763+0.155658i -1.126763-0.155658i >

The omnivore has a negative total effect on the plant (e13=-0.1604). The dominant eigenvalue of matrix A close to 1 (0.876) mirror the stability of the fixed point of the system. The negative real part of the eigenvalues of the TE matrix (E) reflects the stability of the solution.

> ############# Simulation with MAR(1) > > bc4<-function(N) { + + N1.t1<-coef(fit1)[1]+A[1,1]*N[1]+A[1,2]*N[2]+A[1,3]*N[3] + N2.t1<-coef(fit2)[1]+A[2,1]*N[1]+A[2,2]*N[2]+A[2,3]*N[3] + N3.t1<-coef(fit3)[1]+A[3,1]*N[1]+A[3,2]*N[2]+A[3,3]*N[3] + + c(N1.t1, N2.t1,N3.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=3) > N[1, ]<-c(2, 0.5, 0.8) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > matplot(0:t, N, type='l', col=1, ylim=c(0,4), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > lines(N3b, col='red') > title("Linear model fitting") > r<-c('Plant','Herbivore','Omnivore') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') >

Figure 8.18 is created. 130 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.16. Simulation of the dynamics of model PANT3

Figure 8.17. Dynamics of the plant consumption, in the simulation with model PANT3 © L. S. Barreto, 2017. Theoretical Ecology 131 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.18. Population dynamics with MAR(1), and model PANT3 (red line)

Now we assume that the area of discovery of the herbivore is not 1 but 8 (a1=8). Let us see what happens to the system. The dynamics of the system is evinced in figure 8.19.

Figure 8.19. Simulation of the dynamics of model PANT3, with a1=8

The fixed point is now y1= 0.51215076, y2=0.09995082, y3= 0.17363530. The consumption of the plant is exhibited in figure 8.20. 132 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.20. Dynamics of the plant consumption, in the simulation with model PANT3, with a1=8

The new community, and total effects matrices are:

> #Community matrix > A=(matrix(c( L1,L2,L3), nrow=3,ncol=3,byrow=T)) > A [,1] [,2] [,3] [1,] 1.16972321 -2.6493694 0.48446100 [2,] 0.03915076 0.7286820 -0.02593494 [3,] 0.05062359 0.2389646 0.69399462 > > ##Matrix of total effects > > E=-ginv(A) > E [,1] [,2] [,3] [1,] -0.77454315 -2.9571862 0.43017840 [2,] 0.04309751 -1.1911807 -0.07460039 [3,] 0.04165936 0.6258743 -1.44662551

Now, with a very voracious herbivore, the omnivore has a positive total effect on the plant

(e13=0.43017840). As an exercise, I suggest to the reader the comparison of the behaviours of model PANT3 with the two different vales of a1. In Barreto (2014:section 25.3) I presented a variant of model PANT3, with diet selection. In Barreto (2014:section 25.8) I presented a variant of model PANT3, with intra annual seasonality of the parameters bi. This variation can be seen as the changing of the quality of the plant for the consumers, along the year. Model PANT3, and the used analytical procedure evinced satisfactory sensitivity, and coherence. The problem of obtaining matrix E is solved with a trustful solution. © L. S. Barreto, 2017. Theoretical Ecology 133 8 Models for Predation, Omnivory, and Pantophagy

8.7 Pantophagy

Model PANT3 can be extended to four trophic levels. We consider de following situations:

y1 y2 y3 y4

Figure 8.21. Diagrammatic representation of pantophagy

The formulation of model PANT4 is:

(8.16)

(8.17)

(8.18) (8.19)

The system of ODE for model PANT4b is:

(8.20)

(8.21)

(8.22) (8.23)

In Barreto (2014:sections 25.6, 25.7) I presented variants of these two models, with diet selection. To these models we can fit MAR(1) and apply the same analytical procedure used with model PANT3. In figure 8.22, we present a graphical description of the analytical process. 134 © L. S. Barreto, 2017. Theoretical Ecology 8 Models for Predation, Omnivory, and Pantophagy

Figure 8.22. A flowchart of the proposed analytical process

8.6 References, and Related Bibliography

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. Lisbon, Portugal. Barreto, L. S., 2011. Modelling and Simulating Omnivory. Silva lusitana, 19(1):47-58. http:www.inrb.pt/inia/silva-lusitana Barreto, L. S., 2013. Ecologia Teórica. Uma outra Explanação. I. Populações Isoladas. E-book. Costa de Caparica. Barreto, L. S., 2014. Ecologia Teórica. Uma outra Explanação. II. Interações entre Populações. E-book. Costa de Ca- parica. Barreto, L. S., 2016. Ecologia Teórica. Uma outra Explanação. III. Comunidade e Ecossistema. E-book. Costa de Ca- parica. Beddington, J. R., C. A. Free e J. H. Lawton, 1976. Dynamic Complexity in Predator-prey Models Framed in Simple Difference Equations. Nature, 255:58-60. Begon, M., C. R. Towsend, and J. Harper, 2006. Ecology. From Individuals to Ecosystems. Fourth edition. Blackwell Publishing, Oxford, Reino Unido. Case, T. J., 2000. An Illustrated Guide to Theoretical Ecology. Oxford University Press, Oxford, U. K. DeAngelis, D. L.,2012. The Case for Ratio Dependence in Trophic Ecology. http://dx.doi.org/10.1016/j.tree.2012 Edelstein-Keshet, L., 2005. Mathematical Models in Biology. Society for Industrial and Applied Mathematics, Philadel- phia, E.U.A. Gillman, M. e R. Hails, 1997. An Introduction to Ecological Modelling. Putting Practice into Theory. Blackwell Science, Oxford, U. K. Hassel, M. P., 1981. Arthropod Predator-Prey Systems. Em R.M. May, compilador, Theoretical Ecology: Principles and Applications. 2nd edition. Blackwell Scientiﬁc, Oxford. Páginas 105–131. Hastings, A., 1984. Delays in Recruitment at Different Trophic Levels: Effects on Stability. Journal of Mathematical Biology, 21:35-44. Kang, Y., D. Armbruster e Y. Kuang, 2008. Dynamics of a Plant–Herbivore Model. Journal of Biological Dynamics, 2(2): 89–101. May, R.M., 1978. Host-parasitoid systems in patchy environments: A phenomenological model. Journal of Animal Ecology, 47:833–844. May, R.M., 1981. Models for two interacting populations. Em R.M. May, compilador, Theoretical Ecology: Prin- ciples and Applications. 2nd edition. Blackwell Scientiﬁc, Oxford. Páginas 78–104. © L. S. Barreto, 2017. Theoretical Ecology 135 8 Models for Predation, Omnivory, and Pantophagy

Moon, D. C., J. Moon , and A. Keagy, 2010. Direct and Indirect Interactions. Nature Education Knowledge 3(10):50. www.nature.com/scitable/knowledge/library/direct-and-indirect-interactions-15650000 Murdoch, W. W., C. J. Briggs, and R. M. Nisbet, 2003.Consumer-Resource Dynamics. Princeton University Press, Princeton. Nicholson, A.J., and V.A. Bailey, 1935. The balance of animal populations - Part I. Proceedings of the Zoological So- ciety of London, páginas 551–598. Pielou, E. C., 1969. Introduction to Mathematical Ecology. Wiley-Interscience, New York. Rockwood, L. L., 2006. Introduction to Population Ecology. Blackwell Publishing, Oxford, UK. Rosenzweig, M. L., and R. H. MacArthur, 1963. Graphical Representation and Stability Conditions of Predator–Prey Interactions. American Naturalist, 97: 209–223. Thompson, W. R., 1924. La Théorie Mathématique de lÁction des Parasites Entomophages et le Facteur du Hasard. Annls. Fac. Sci. Marseille,2 :69-89. Turchin, P., 2003. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton University Press, Princeton. Varley, G. C., G. R. Gradwell, and M. P. Hassel, 1973. Insect Population Ecology. An Analitical Approach. University of California Press, Berkeley, U.S.A.. 136 © L. S. Barreto, 2017. Theoretical Ecology 9 Systems of Herbivory

9 Systems of Herbivory

9.1 Introduction

The main purpose of this chapter is to apply models SBPRE11, and SBPRED11-de to herbivory. Turchin (2003: 112-116) developed an analysis that sustains the use of functional response type 2 in models for herbivory. We will use this functional response.

9.2 Extension of Model SBPRED11 to Herbivory

Let me start by reintroducing model SBPRED11:

(9.1)

(9.2)

To extend the model to herbivory we must reinterpret the variables, and the parameters.

The variables y1, and y2 are expressed as biomass. The values of ci have usual meaning, in the EGZ.

K is the maximum biomass of the part of the plant that is consumed by herbivore y2. It can refer to a single individual (e.g., the foliage of a tree), a spot of vegetation, or a continuous, and large area of the same type of vegetation. The choice depends on the adopted strategy for modelling. After it had been chosen, it must be consistently applied.

9.2.1 An Equation for b The expression for b must reflect both the nutritional properties of the vegetation, and the efficiency of the herbivore in extracting benefit from it. We propose the following equation (Barreto, 2011):

(9.3)

In equation (9.3) the meaning of the parameters is the following: m is an estimation of the mean of the values that b can assume; v is the rate of conversion of plant biomass into herbivore biomass; e is the benefit received by the herbivore per unit of biomass consumed (e.g., energy, protein). e/h is the profitability of the resource for the consumer. As equation (9.3) is formulated, the degradation of the quality of food does not affect its energetic or protein content, but others qualities as digestibility.

9.2.2 Simulations I will use the model of equations (9.1) to (9.3) to simulate herbivory with four different variations of b:

 Case 1. It decreases as y2 increases;  Case 2. Both b, and K changes randomly with the environment;  Case 3. Cyclic, and simultaneous variations of b, and K. I use the following Scilab script for the first case: © L. S. Barreto, 2017. Theoretical Ecology 137 9 Systems of Herbivory

//herbivory with decreasing quality //as the population of herbivores increases clear clf dx=[];x=[];x0=[];k1=[];k2=[];k3=[];k4=[];v=[]; c1=0.54;c2=c1*3;k=30;a=1;h=1;a1=1; t=0:1:249;v1=0.5;e1=1;d=1; v=[];a=1;x0(1)=5;x0(2)=2;z=[];Z=[]; for i=1:1:250 x=x0; b1=0.2*exp(log(x(1)/x(2))*v1*a1*e1/h);

dx(1)=c1*x(1)*(log(k)-log(x(1)))-a*x(1)*x(2)/(1+a*h*x(1)); dx(2)=c2*x(2)*(log(b1*x(1))-log(x(2))); k1=dx; x=x0+d/2*k1; b1=0.2*exp(log(x(1)/x(2))*v1*a1*e1/h);

dx(1)=c1*x(1)*(log(k)-log(x(1)))-a*x(1)*x(2)/(1+a*h*x(1)); dx(2)=c2*x(2)*(log(b1*x(1))-log(x(2))); k2=dx; x=x0+d/2*k2; b1=0.2*exp(log(x(1)/x(2))*v1*a1*e1/h);

dx(1)=c1*x(1)*(log(k)-log(x(1)))-a*x(1)*x(2)/(1+a*h*x(1)); dx(2)=c2*x(2)*(log(b1*x(1))-log(x(2)));

k3=dx; x=x0+d*k3; b1=0.2*exp(log(x(1)/x(2))*v1*a1*e1/h);

dx(1)=c1*x(1)*(log(k)-log(x(1)))... -a1*x(1)*x(2)/(1+a1*h*x(1)); dx(2)=c2*x(2)*(log(b1*x(1))-log(x(2)));

k4=dx; for i=1:2 x(i)=x0(i)+d*1/6*(k1(i)+2*(k2(i)+k3(i))+k4(i)); end x0=x; z=[z;x(1) x(2) ];Z=[Z;b1]; end subplot(2,2,1) plot2d(t,[z]) xtitle('Sizes of populations','Time','Biomass') legend(['Vegetation';'Herbivore']) subplot(2,2,2) plot2d(z(:,2),[Z]) xtitle('b versus herbivore','Herbivore','b') subplot(2,2,3) plot2d4(z(:,1),z(:,2)) xtitle('Phase portrait','Vegetation ',' Herbivore') disp('Sizes of populations') disp(x') disp('Last value of b') 138 © L. S. Barreto, 2017. Theoretical Ecology 9 Systems of Herbivory

disp([b1])

The script prints the graphics of figure 9.1, and writes the following:

Sizes of populations

16.511213 5.6467556

Last value of b

0.3419952

Figure 9.1. Simulation of Case 1 of herbivory,

In figure 9.1, occurs a stable node, and it is evince that the value of b decreases as the population of the herbivore increases. For Case 2, we consider three levels of quality for b, and K: low, normal, high. The probability of occurrence of the levels are, respectively, 30%, 50%, and 20%. The parameters of the model are the same as in Case 1. The script used is the following one:

//herbivory with three levels //for b, and K clear clf dx=[];x=[];x0=[];k1=[];k2=[];k3=[];k4=[];v=[]; c1=0.54;m1=3;k=30;a=1;h=1;a1=1; t=0:1:149;v1=0.5;e1=1;d=1; © L. S. Barreto, 2017. Theoretical Ecology 139 9 Systems of Herbivory v=[];a=1;x(1)=5;x(2)=2;z=[];Z=[]; for i=1:0.1:150 if i==int(i) then r=grand(1,'unf',0,0.99); end

if r<0.4 then b=0.3; elseif r>0.4 & r<0.8 then b=0.6; else b=0.75; end

dx(1)=c1*x(1)*(log(k)-log(x(1)))-a*x(1)*x(2)/(1+a*h*x(1)); dx(2)=c1*m1*x(2)*(log(b*x(1))-log(x(2))); x=x+0.1*dx; if i==int(i) then z=[z x]; end if i==int(i) then Z=[Z;b]; end end subplot(1,2,1) plot2d(t,[z']) xtitle('Sizes of populations','Time','Biomass') legend(['Vegetation';'Herbivore']) subplot(1,2,2) plot2d(t,[Z]) xtitle('Variation of b','Time','b') disp('Sizes of populations') disp(x') disp('Last value of b') disp([b])

Sizes of populations

15.352056 5.7510268

Last value of b

0.75

The graphics of this simulation are inserted in figure 9.2. 140 © L. S. Barreto, 2017. Theoretical Ecology 9 Systems of Herbivory

Figure 9.2. Simulation o Case 2. The variations of K, and b are synchronized

If the value of c2 is very low, emerges a limit cycle. If in the previous script we made c2=0,0222222 c1, instead of c2=3 c1, we obtain figure 9.3.

Figure 9.3. Limit cycle caused by a low value of c2

For Case 3, we consider 12 units of time. The time unit depends on the biology of the species or the variation of the local environment. The variations of b, and K is given by the two following equations: b=0.5*(1+sin(t*2*%pi)/12) (9.4)

K=30*(1+sin(t*2*%pi)/12) (9.5)

The script used is the following one:

clear clf © L. S. Barreto, 2017. Theoretical Ecology 141 9 Systems of Herbivory dx=[];x=[]; c1=0.54;m1=3;a=1;h=1; t=0:1:49;v1=0.5;e1=1;d=1; b=0.2;v=1; x(1)=5;x(2)=2;z=[];Z=[]; j=1:0.1:50; b=0.5*(1+sin(j*2*%pi)/12); k=30*(1+sin(j*2*%pi)/12); for i=1:0.1:50 dx(1)=c1*x(1)*(log(k(i))-log(x(1)))-a*x(1)*x(2)/(1+a*h*x(1)); dx(2)=c1*m1*x(2)*(log(b(i)*x(1))-log(x(2))); x=x+0.1*dx; if i==int(i) then z=[z x]; end if i==int(i) then Z=[Z;b(i)]; end end subplot(1,2,1) plot2d(t,[z']) xtitle('Sizes of populations','Time','Biomass') legend(['Vegetation';'Herbivore']) subplot(1,2,2) plot2d(t,Z') xtitle('Variation of b','Time','b') disp('x') disp(x')

12.470977 5.7541783

We obtain figure 9.4. 142 © L. S. Barreto, 2017. Theoretical Ecology 9 Systems of Herbivory

Figure 9.4. Simulation o Case 3

9.3 Extension of Model SBPRED11-de to Herbivory

Let us rewrite model SBPRED11-de as:

y1,t+1= y1,t exp(c1ln K-a1 ln y1,t+ m1 ln y2,t) (9.6)

y2,t+1= y2,t exp(c2 *(m2* ln y1,t)- c2 ln y2,t) (9.7)

These two equations to not include explicitly the parameters of the functional response, but they continue characterizing the way the consumer explores the resource. This assumption lead us to the following equation for m2:

(9.8)

Where m is an estimate of the average of m2. The comments about MSP11, inserted in the previous section, apply to this model. It is only a matter of the choice of the values of the parameters of the models. Let me introduce a Scilab script that originates a stable node:

clear clf c2=0.02;c1=0.9; k=20;a=1;h=0.5;a=1; t=1:1:50;v=0.1;e=1;d=1; a=1;y(1)=12;y(2)=4; B=[];z=[]; for n=1:1:50 m2=0.6+(y(1)/y(2))*v*a*e/h; ya(1)=y(1)*exp(c1*log(k)-0.190129*log(y(1))-1.606273*log(y(2))); ya(2)=y(2)*exp(c2*(m2*log(y(1)) )-c2*log(y(2))); © L. S. Barreto, 2017. Theoretical Ecology 143 9 Systems of Herbivory

y=ya; B=[B;y']; z=[z m2]; end subplot(2,2,1) plot2d(t, B) xtitle('Sizes of the populations','Time','Biomass') legend(['Vegetation';'Herbivore']) subplot(2,2,2) plot2d(B(:,2),z) xtitle('m2 versus herbivore','Herbivore','m2') subplot(2,2,3) plot2d4(B(:,1),B(:,2)) xtitle('Phase portrait','Vegetation',' Herbivore')

The script produces figure 9.5.

Figur 9.5. Discrete model MSP11-de adapted to herbivory. The simulated system evinces a stable node

If we increase m from 0.6 to 6, the solution becomes a limit cycle, as illustrated in figure 9.6. Herbivores the has a slow or very fast reaction to the vegetation growth destabilizes the system. 144 © L. S. Barreto, 2017. Theoretical Ecology 9 Systems of Herbivory

Figure 19.6. System of herbivory with an extremely reactive herbivore to population growth that gives origin to a limit cycle

Equations (9.3), and (9.8) mimic the effect of the intensity of grazing on the quality of food, but ignore the plant toxicity. If this toxicity exists, this two equation must be multiplied by the variable τ written as:

T τ =1− (9.9) T m

Where

 Tm is the maximum tolerance of the herbivore to the content of the toxin per biomass unit of vegetation  T is the concentration of the toxin in the vegetation being consumed Three situations occur:  T=0 → τ=1 only the effect of the intensity of grazing actuates

 0Tm→τ<0: for equation (9.2) the execution of the script is interrupted because negative numbers do not have logarithm; for equation (9.7) a negative exponential emerges, and the population goes to extinction. Here, I conclude the introduction of the models I established for herbivory. © L. S. Barreto, 2017. Theoretical Ecology 145 9 Systems of Herbivory

9.4 References, and Related Bibliography

Barreto, L. S., 2011. Modelling and Simulating Omnivory. Silva lusitana, 19(1):67-83. http:www.inrb.pt/inia/silva-lusitana Begon, M., C. R. Towsend, and J. Harper, 2006. Ecology. From Individuals to Ecosystems. Fourth edition. Blackwell Publishing, Oxford, UK. Edelstein-Keshet, L., 1984. Mathematical Theory of Plant-Herbivore Systems. Journal of Mathematical Biology, 24:25-58. Kang, Y., D. Armbruster, and Y. Kuang, 2008. Dynamics of a Plant–Herbivore Model. Journal of Biological Dynamics, 2(2): 89–101. Keddy, P. A., 2007. Plants and Vegetation. Origins, Processes, Consequences. Cambridge University Press,Cam- bridge, UK. Liu, R., S. A.Gourley, D. L. DeAngelis, and J. P. Bryant, 2011. Modeling the Dynamics of Woody Plant–Herbivore Interactions with Age-Dependent Toxicity. J. Math. Biol. DOI 10.1007/s00285-011-0470-0 146 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

10 Models for Amensalism, Commensalism, and Detritivory

10.1 Introduction

The emergence of total effects suggests that all interactions are equally important for the dynamics of the community. Thus, the establishment of models for communities, and ecosystems, can not iignore interactions such as amensalism, commensalism, and detritivory. In this chapter I recall the models I presented in Barreto (2005). In chapter 7, we stated that the interactions approached in this chapter do not have periodic solutions. In Barreto (2014:chapter 20) I used the Bendixson-Dulac’s negative criterion (Kot, 2001:125-127) to confirm this assertion. Here, I apply the criterion only amensalism. Also, I carry on an analysis of stability of the models, applying wxMaxima.

10.2 Amensalism

10.2.1 Assumptions I do not know any model for amensalism (0,-). The model I here propose assumes the following:  Both populations have Gompertzian dynamics;  The population that receives the negative effect has its carrying capacity diminished by the other species.

10.2.2 The Model The model for amensalism has the two following equations:

(10.1)

(10.2)

I choose the simplest formulation to mimic the effect on the carrying capacity. I call b the

coefficient of amensalism. It can be a constant or a function of y1 e y2, or other factors that affect the interaction.

10.2.3 Model Analysis

It can be easely deduced that the equilibrium populations are:

y1*=y1f e y2*=y2f-by1f

Now I introduce a script in Scilab with a function to simulate amensalism:

// Simulator for amensalism clear clf disp("g(c1,c2,k1,k2,b)"); function []=g(c1, c2, k1, k2, b) deff("yprim=g(t,y)",["yprim1=c1*y(1)*(log(k1)-log(y(1)))";... "yprim2=c2*y(2)*(log(k2-b*y(1))-log(y(2)))";... © L. S. Barreto, 2017. Theoretical Ecology 147 10 Models for Amensalism, Commensalism, and Detritivory

"yprim=[yprim1;yprim2]"]) y0=[2,5]; t0=0; t=0:100; [M]=matrix(ode(y0,t0,t,g),2,101)';

subplot(1,2,1) plot2d(t,M); xtitle("Simulation of amensalism","Time","Number or biomass"); xgrid(); subplot(1,2,2) plot2d4(M(:,1),M(:,2)) xtitle("Phase portrait","Species 1","Species 2"); endfunction

In this function the initial sizes of the population are 2, and 5. A simulation with g(0.5,0.4,20,30,0.6) is exhibited in figure 10.1.

Figure 10.1. Simulation of amensalism with the introduced function. For more details see the text

Now for amensalism, I can replicate the simulation for predation, done in R:

> ##amensalismo > library(deSolve) > > amen<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + + + c2<-parms[3] + k2<- parms[4] + a21<-parms[5] + + + 148 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

+ dn1.dt<- c1*n[1]*(log(k1)-log(n[1])) + dn2.dt<- c2*n[2]*(log(k2-a21*n[1])-log(n[2])) + + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.05, k1=20, c2=0.03, k2=30, a21=0.003) > initialn<-c(10, 10) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, amen, parms=parms) > >split.screen(figs=c(1,2)) >screen(1) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Number or biomass") > title("Amensalism") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') > > > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > N1<-dat0[,1] > N2<-dat0[,2] > > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.026274 -0.001886 0.001051 0.001556 0.010596

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.7020930 0.0038523 182.3 <2e-16 *** N1b 0.9707993 0.0004034 2406.8 <2e-16 *** N2b -0.0039981 0.0001599 -25.0 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.00404 on 296 degrees of freedom © L. S. Barreto, 2017. Theoretical Ecology 149 10 Models for Amensalism, Commensalism, and Detritivory

Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 3.133e+07 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.0047245 -0.0004889 0.0002220 0.0004074 0.0017832

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.586e-01 7.353e-04 487.7 <2e-16 *** N1b 2.885e-02 7.699e-05 374.7 <2e-16 *** N2b 9.687e-01 3.052e-05 31742.8 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.000771 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 5.685e+09 on 2 and 296 DF, p-value: < 2.2e-16

> > > > L1=coef(fit1)[-1] > L2=coef(fit2)[-1] > > > > A=(matrix(c( L1,L2), nrow=2,ncol=2,byrow=T)) > > bc4<-function(N) { + + N1.t1<-coef(fit1)[1]+A[1,1]*N[1]+A[1,2]*N[2] + N2.t1<-coef(fit2)[1]+A[2,1]*N[1]+A[2,2]*N[2] + + + c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(10,10) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,30), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > > title("Linear model fitting") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') > > 150 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

The fitting of MAR(1) to the output of the model is confirmed, and we obtain also figure 10.2.

Figure 10.2. Simulation of amensalism with the R script. The system od ODE and MAR(1) produce virtualy the same values

Now I apply the Bendixson-Dulac’s negative criterion to this model: © L. S. Barreto, 2017. Theoretical Ecology 151 10 Models for Amensalism, Commensalism, and Detritivory

The stability analysis is as follows: 152 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

10.3 Commensalism

10.3.1 Assumptions Also, for this interaction (0,+), I do not know any model. The assumptions of the model I here propose are the followings:  The two populations have Gompertzian dynamics;  We can consider two types of commensalism:

 Obligatory commensalism. The commensal (y2) receives a service (e.g., shelter) indispensable for its survival. The presence of the other species affects its rate of

growth per capita (this rate is zero in the absence of y1), but do not increase its carrying capacity that depends mostly on biotic, and abiotic factores of its habitat.

 Facultative commensalism. The commensal (y2) can persist without the interaction, ocorrer. The interaction only increases the carrying capacity. © L. S. Barreto, 2017. Theoretical Ecology 153 10 Models for Amensalism, Commensalism, and Detritivory

10.3.2 The Model Given the stated assumptions, this model comprehends the following equations::

(10.3)

The facultative commensal has the following equation:

(10.4)

The obligatory commensal has the following equation:

(10.5)

Parameter b is the coefficient of commensalism.

10.3.3 Model Analysis Facultative commensalim

As population y1 is not affected by y1, it attains its carrying capacity y1f. When this happens, y2 atains the equilibrium size y2f+by1f. Thus, the fixed point is

y1*=y1f, and y2*=y2f+by1f

The equilibrium of the commensal is symmetric of the amensal, in the previous model, and the analysis of the model is parallel and give the same results, thus we will not repeat it. We introduce the following script, in R, of a simulator for facultative commensalism:

> ##comenfac > > library(deSolve) > > comen<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + + + c2<-parms[3] + k2<- parms[4] + b<-parms[5] + + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1])) + dn2.dt<- c2*n[2]*(log(k2+ b*n[1])-log(n[2])) + + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.5, k1=20, c2=0.4, k2=30, b=0.6) > initialn<-c(2, 5) 154 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

> t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, comen, parms=parms) > > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Number or biomass") > title("Facultative comm.") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') > > > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > N1<-dat0[,1] > N2<-dat0[,2] > > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.72892 0.00386 0.00386 0.00386 0.57597

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.186175 0.054369 77.00 <2e-16 *** N1b 1.025333 0.012507 81.98 <2e-16 *** N2b -0.111826 0.005307 -21.07 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0694 on 296 degrees of freedom Multiple R-squared: 0.9969, Adjusted R-squared: 0.9969 F-statistic: 4.778e+04 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: © L. S. Barreto, 2017. Theoretical Ecology 155 10 Models for Amensalism, Commensalism, and Detritivory lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.49273 0.00238 0.00238 0.00238 0.36757

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.612243 0.035419 158.45 <2e-16 *** N1b 0.585202 0.008148 71.82 <2e-16 *** N2b 0.587651 0.003457 169.98 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04521 on 296 degrees of freedom Multiple R-squared: 0.9998, Adjusted R-squared: 0.9998 F-statistic: 7.332e+05 on 2 and 296 DF, p-value: < 2.2e-16

> > > > L1=coef(fit1)[-1] > L2=coef(fit2)[-1] > > > > A=(matrix(c( L1,L2), nrow=2,ncol=2,byrow=T)) > > bc4<-function(N) { + + N1.t1<-coef(fit1)[1]+A[1,1]*N[1]+A[1,2]*N[2] + N2.t1<-coef(fit2)[1]+A[2,1]*N[1]+A[2,2]*N[2] + + + c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(10,10) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,50), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > > title("MAR(1) fitting") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') >

We also obtain figure 10.3. The fitting of MAR(1) is good. 156 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

Figure 10.3. Simulation of facultative commensalism

Obligatory commensalism The fixed point is:

y1*=y1f e y2*=y2f

The analysis of stability is as follows: © L. S. Barreto, 2017. Theoretical Ecology 157 10 Models for Amensalism, Commensalism, and Detritivory

A simulator for this interaction is the following: 158 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

> ##comenoblig > library(deSolve) > > comen<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + + + c2<-parms[3] + k2<- parms[4] + b<-parms[5] + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1])) + dn2.dt<- c2*n[2]*b*n[1]*(log(k2)-log(n[2])) + + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.5, k1=20, c2=0.3, k2=30, b=0.4) > > initialn<-c(2, 5) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, comen, parms=parms) > > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Number or biomass") > title("Obligatory comm.") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') > > > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > N1<-dat0[,1] > N2<-dat0[,2] > > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) © L. S. Barreto, 2017. Theoretical Ecology 159 10 Models for Amensalism, Commensalism, and Detritivory

> > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.72395 0.00021 0.00021 0.00021 0.55533

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.747119 0.058136 64.45 <2e-16 *** N1b 0.587368 0.007619 77.09 <2e-16 *** N2b 0.150177 0.006084 24.68 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.06275 on 296 degrees of freedom Multiple R-squared: 0.9975, Adjusted R-squared: 0.9975 F-statistic: 5.849e+04 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -3.5531 0.0049 0.0049 0.0049 3.4732

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 8.63440 0.30697 28.128 < 2e-16 *** N1b -0.33159 0.04023 -8.242 5.54e-15 *** N2b 0.93308 0.03213 29.044 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3313 on 296 degrees of freedom Multiple R-squared: 0.9463, Adjusted R-squared: 0.9459 F-statistic: 2606 on 2 and 296 DF, p-value: < 2.2e-16

> > > > L1=coef(fit1)[-1] > L2=coef(fit2)[-1] > > > > A=(matrix(c( L1,L2), nrow=2,ncol=2,byrow=T)) > > bc4<-function(N) { + + N1.t1<-coef(fit1)[1]+A[1,1]*N[1]+A[1,2]*N[2] + N2.t1<-coef(fit2)[1]+A[2,1]*N[1]+A[2,2]*N[2] + + + c(N1.t1, N2.t1) 160 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

+ } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(10,10) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,35), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > > title("MAR(1) fitting") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') >

The simulator produces figure 10.4. The fitting of MAR(1) is good.

Figure 10.4. Simulation of obligatory commensalism © L. S. Barreto, 2017. Theoretical Ecology 161 10 Models for Amensalism, Commensalism, and Detritivory

10.4 Detritivory

10.4.1 Assumptions Roughly, detritivory can be seen as a case of obligatory commensalism. The detritus are the only resource that the active populations consumes (Begon, Towsend e Harper, 2006:chapter 11).. Before we approach the interaction, let us execute a simple simulatory exercise. We will see if it is possible to fit a EGZ to the sum of four variables that all of them follow EGZ.

> #Criate four EGZ with the > #form f(i)*r(i)êxp(-c(i)*t > > #Vectors of parameters > t=0:500; > r=c(0.2, 0.02, 0.002, 0.0002) > k=c(0.0002, 0.002, 0.02, 0.2) > f=c(5, 50, 500, 5000) > #Calculating the 4 EGZ > g1=f[1]*r[1]êxp(-k[1]*t) > > g2=f[2]*r[2]êxp(-k[2]*t) > > g3=f[3]*r[3]êxp(-k[3]*t) > > g4=f[4]*r[4]êxp(-k[4]*t); > > #Summing the time series of the data > G=g1+g2+g3+g4; > > #Fitting the EGZ > #(section 4.10) > > x<-log(G[-501]) > y<-log(G[-1]) > > model0<-lm(y~x) > summary(model0)

Call: lm(formula = y ~ x)

Residuals: Min 1Q Median 3Q Max -0.36377 -0.00207 0.00127 0.00168 0.13669

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.233136 0.012848 95.98 <2e-16 *** x 0.856651 0.001507 568.36 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02085 on 498 degrees of freedom Multiple R-squared: 0.9985, Adjusted R-squared: 0.9985 F-statistic: 3.23e+05 on 1 and 498 DF, p-value: < 2.2e-16

> 162 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

> #Plotting the logarithms of the data, > #and fitted values > plot(x,y) > cf<-coef(model0) > > a<- cf[1] > b<-cf[2] > yv<-a+b*x > lines(x,yv, , col='red')

The obtained graphic is in figure 10.5. As we obtained a good fitting, we accept the hypothesis that to the sum of EGZ variables we can fit a EGZ.

Figure 10.5. Data (circles) and the EGZ (red line) fitted to the sum of four EGZ

Now, we formulate our assumption to model detritivory:  The biomass of each population in the community follows the EGZ;  The dynamics of the total biomass of the community can be admitted to be also Gom- pertzian;  At equilibrium, the biomass of a given kind of detritus is a constant fraction of the total biomass of the community, as all populations loose biomass for the detritus.

10.4.2 The Model

Being y1 the total biomass of the community, at equilibrium, the model for detritivory is written as:

(10.6) © L. S. Barreto, 2017. Theoretical Ecology 163 10 Models for Amensalism, Commensalism, and Detritivory

The equation for the detritivore is::

(10.7)

In equation (10.7), b is a parameter that combines the fraction of the detritus consumed by the detritivore, of the total biomass of detritus, and the contribution of the detritus to the growth of the consumer.

10.4.3 Model Analysis

The fixed point is: y1*=y1f, and y2*= b y1f. The analysis of the stability is as follows: 164 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

The fixed point is a stable node. Now, I introduce a simulator for the interaction:

> ##detritiv > > comen<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + + + c2<-parms[3] + b<-parms[4] + + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1])) + dn2.dt<- c2*n[2]*(log(b*k1)-log(n[2])) + + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.5, k1=20, c2=0.3, b=0.4) > initialn<-c(3, 1) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, comen, parms=parms) > > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass") > title("Detritivory") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') > > > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > N1<-dat0[,1] > N2<-dat0[,2] > > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > © L. S. Barreto, 2017. Theoretical Ecology 165 10 Models for Amensalism, Commensalism, and Detritivory

> fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.57370 0.00374 0.00374 0.00374 0.45792

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.52273 0.05181 87.30 <2e-16 *** N1b 0.89067 0.00761 117.04 <2e-16 *** N2b -0.29248 0.01486 -19.68 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05651 on 296 degrees of freedom Multiple R-squared: 0.9974, Adjusted R-squared: 0.9974 F-statistic: 5.677e+04 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.055127 0.000361 0.000361 0.000361 0.042425

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.836319 0.004888 171.1 <2e-16 *** N1b 0.075210 0.000718 104.7 <2e-16 *** N2b 0.707389 0.001403 504.3 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.005332 on 296 degrees of freedom Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999 F-statistic: 2.148e+06 on 2 and 296 DF, p-value: < 2.2e-16

> > > > L1=coef(fit1)[-1] > L2=coef(fit2)[-1] > > > > A=(matrix(c( L1,L2), nrow=2,ncol=2,byrow=T)) > > bc4<-function(N) { + + N1.t1<-coef(fit1)[1]+A[1,1]*N[1]+A[1,2]*N[2] + N2.t1<-coef(fit2)[1]+A[2,1]*N[1]+A[2,2]*N[2] + 166 © L. S. Barreto, 2017. Theoretical Ecology 10 Models for Amensalism, Commensalism, and Detritivory

+ + c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(10,10) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,25), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > > title("MAR(1) fitting") > r<-c('y1','y2') > legend('topleft',paste(rev(r)),lty=3:1,col=1, bty='n') >

Figure 10.6 is also produced. The fitting of MAR(1) is good. © L. S. Barreto, 2017. Theoretical Ecology 167 10 Models for Amensalism, Commensalism, and Detritivory

Figure 10.6. Simulation of detritivory

In the next two chapters, we will approach models for competition.

10.5 References, and Related Bibliography

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach.Lisbon, Portugal. Livro eletrónico. Begon, M., C. R. Towsend e J. Harper, 2006. Ecology. From Individuals to Ecosystems. Fourth edition. Blackwell Publishing, Oxford, Reino Unido. Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, Cambridge, United Kingdom. Vandermeer, J. H. e D. E. Goldberg, 2003. Population Ecology. First Principles. Princeton University Press, Prince- ton. 168 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

11 Non Predictive Models for Competition

11.1 Introduction

We will dedicate more then one chapter to competition. We will approach non predictive, and predictive models, both continuous, and discrete. In this chapter we will introduce models that can only be fitted to data, but can not let us anticipate the dynamics of any given competitive interaction. We will introduce continuous model SB-BACO3, and discrete model SB-BACO4.

11.2 Modelo SB-BACO3

11.2.1. Assumptions The assumptions of model SB-BACO3 (Barreto, 1997a,b, 1999, 2001, 2003, 2004) for competition (MSB3) are the following ones::  P1. The dynamics of the two competitors is a continuous process;  P2. The competitors when isolated have a Gompertzian dynamics;  P3. Competition can be symmetric (-,-) or asymmetic (-,+);  P4. When populations interact, the effects of intraspecific, and interspecific competition are formally equal. We must convert the competitor of one specie in equivalent individuals of the other specie that is being affected. For this purpose we use a conversion factor named coefficient of interespecífic competition (coefc). We reprent coefc by

aij specie i receives the competitive effect of specie j. Individuals of specie j are transformed in equivalent individuals of specie i by multiplying its number or biomass by coefc, this is, its competitive effect (positive or negative) impinged on specie i is measured by

aijyj

 P5. The coefc can be constant or variable. We assume that they are constant.

11.2.2. The Model The equation for each population is an extension to the equation introduced in chapter 4:

y1’=c y (ln yf- ln y) (11.1)

A generic formulation of MSB3, for both asymmetric, and symmetric competition, is:

y1’= y1c1 (ln y1f-ln y1+a12 ln y2) (11.2)

y2’= y2 c2 (ln y2f+a21 ln y1- ln y2) (11.3)

Symmetric competition occurs if occurs a12<0 e a21<0. Asymmetic competition is present if one coefc is positive, and the other negative. © L. S. Barreto, 2017. Theoretical Ecology 169 11 Non Predictive Models for Competition

We will consider symmetric competition, and the coefc are constant. With these assumptions MSB3 is formulated as:

y1’= y1 c1 (ln y1f- ln y1 - a12 ln y2) (11.4)

y2’= y2 c2 (ln y2f-a21 ln y1- ln y2) (11.5)

In this model all parameters are positive.

11.2.3 Análise do Modelo The fixed point of the model of equations (11.4)-(11.5) is:

y1*=exp((a12 ln y2f-ln y1f)/(a12a21-1)) (11.6)

y2*=exp((a21 ln y1f-ln y2f)/(a12a21-1)) (11.7)

The stability analysis of the solutions of SBC3 are summarized in Table 11.1. The analogy with the Lotka-Volterra model for competition is conspicuous. In this model, the winner population do not completely exterminate the looser population, but reduces it to a very small size, comparing the winner dimension.

Table 11.1. Summary of the stability analysis of MSB3. ln=natural logarithm

Case Outcome Equivalent conditions A B C

1 Stable coexistence of the two species ln K1 ln K1/ ln K2>a12

ln K2

2 Specie 1 persists, specie 2 converges ln K1> ln K2/a21 a21> ln K2/ ln K1 1/a21< ln K1/ ln K2>a12

to zero ln K2< ln K1/a12 a12< ln K1/ ln K2

3 Specie 2 persists, specie 2 converges ln K1< ln K2/a21 a21< ln K2/ ln K1 1/a21> ln K1/ ln K2

to zero ln K2> ln K1/a12 a12> ln K1/ ln K2

4 Saddle point ln K1> ln K2/a21 a21> ln K2/ ln K1 1/a21< ln K1/ ln K2

ln K2> ln K1/a12 a12> ln K1/ ln K2

Now, I apply the Bendixson-Dulac’s negative criterion to show that periodic solutions do not occur. We use wxMaxima. 170 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

We continue the analysis, considering the asymmetric competition. © L. S. Barreto, 2017. Theoretical Ecology 171 11 Non Predictive Models for Competition

Now, we illustrate the four possible cases, described in Table 11.1, using Scilab. The fixed point is calculated applying equations (11.6), and (11.7). In the graphics, the red line corresponds to competitor 1 (y1). We use column C, of table 11.1, to characterize the stability of the fixed points. 172 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

Figure 11.1. Model SB-BACO3. Simulation of Case 1. Used parameters: c1=0.05; c2=0.03; y1f=25; y2f=18; a12=0.3; a21=0.4. ZGL = zero growth lines

The script also prints:

!Fixed point:14.475527 ! ! ! !Fixed point:6.1804309 !

1/a21

2.5

(log(k1)/log(k2))

1.1136546

a12

0.3

Stable equilibrium: (1/a21)>(log(k1)/log(k2))>a12

We present a magnification of zero growth lines (ZGL) in figure 11.2. © L. S. Barreto, 2017. Theoretical Ecology 173 11 Non Predictive Models for Competition

Figure 1 1.2. Partial magnification of the ZGL of case 1 174 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

Figure 11.3. Model SB-BACO3. Simulation of Case 2. Used parameters: c 1=0.05; c2=0.03; y1f=55; y2f=12; a12=0.2; a21=0.8

!Fixed point:65.299887 ! ! ! !Fixed point:0.4238882 !

1/a21

1.25

(log(k1)/log(k2))

1.6126695

a12

0.2

Competitor 1 wins: (1/a21)<(log(k1)/log(k2))>a12 © L. S. Barreto, 2017. Theoretical Ecology 175 11 Non Predictive Models for Competition

Figure 1 1.4. Partial magnification of the ZGL of case2

Figure 11.5. Model SB-BACO3. Simulation of Case 3. Used parameters: c 1=0.05; c2=0.03; y1f=12; y2f=55; a12=0.8; a21=0.2 176 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

!Fixed point:0.4238882 ! ! ! !Fixed point:65.299887 !

1/a21

(log(k1)/log(k2))

0.6200899

a12

0.8

Competitor 2 wins: (1/a21)>(log(k1)/log(k2))

Figure 1 1.6. Partial magnification of the ZGL of case3 © L. S. Barreto, 2017. Theoretical Ecology 177 11 Non Predictive Models for Competition

Figure 11.7. Model SB-BACO3. Simulation of Case 4. Used parameters: c 1=0.05; c2=0.03; y1f=20; y2f=60; a12=0.8; a21=1.8

!Fixed point:1.8884953 ! ! ! !Fixed point:19.104804 !

1/a21

0.5555556

(log(k1)/log(k2))

0.7316757

a12

0.8

Unstable equilibrium: (1/a21)<(log(k1)/log(k2))

178 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

Figure 1 1.8. Partial magnification of the ZGL of case4

Now, I use R to simulate the model, and fit a MAR(1) to its output. The script is as follows:

> rm(list=ls(all=TRUE)) > ##BACO3 > > baco3<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + a12<-parms[3] + + c2<-parms[4] + k2<- parms[5] + a21<-parms[6] + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1])-a12*log(n[2])) + dn2.dt<- c2*n[2]*(log(k2)-a21*log(n[1])-log(n[2])) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.05, k1=20, a12=0.02, c2=0.03, k2=25, a21=0.03) > initialn<-c(5, 5) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco3, parms=parms) > > matplot(out[,1], out[,-1], type="l", xlab="Tempo", ylab="Biomassa") > title("Competition simulated with model SB-BACO3") > > r<-c('Competidor 1','Competidor 2') > legend('bottomright',paste(rev(r)),lty=2:1,col=2:1, bty='n') > > > library(rootSolve) > y<-initialn © L. S. Barreto, 2017. Theoretical Ecology 179 11 Non Predictive Models for Competition

> ST2 <- runsteady(y=y,func=baco3,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 18.78605 22.89417 > > h<-matrix(c(out[,2],out[,3]),2991,3) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > > > > > N1<-dat0[,1] > N2<-dat0[,2] > > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > > > eqS1 <- lm(N1a ~ N1b+N2b) > eqS2 <- lm(N2a ~ N1b+N2b) > > summary(eqS1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.073618 -0.005415 0.002807 0.004309 0.030478

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.5500491 0.0053747 102.34 <2e-16 *** N1b 0.9842952 0.0007908 1244.62 <2e-16 *** N2b -0.0113358 0.0004960 -22.85 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01182 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 8.135e+06 on 2 and 296 DF, p-value: < 2.2e-16

> summary(eqS2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.0158066 -0.0012347 0.0005741 0.0009609 0.0059972

Coefficients: Estimate Std. Error t value Pr(>|t|) 180 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

(Intercept) 0.2957857 0.0010919 270.9 <2e-16 *** N1b 0.0231681 0.0001607 144.2 <2e-16 *** N2b 0.9680251 0.0001008 9606.5 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0024 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 5.165e+08 on 2 and 296 DF, p-value: < 2.2e-16

> > G=data.frame(c(coef(eqS1), coef(eqS2))) > G c.coef.eqS1...coef.eqS2.. 1 0.55004914 2 0.98429517 3 -0.01133583 4 0.29578571 5 0.02316806 6 0.96802511 > > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2] + N2.t1<-G[4,1]+G[5,1]*N[1]+G[6,1]*N[2] + + + c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(5, 5) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > windows() > matplot(0:t, N, type='l', col=2:1, ylim=c(0,23), xlab="Time", ylab="Bio- mass",) > lines(N1b, col='red') > lines(N2b, col='red') > > title("MAR(1)") > r<-c('Competitor 1','Competitor 2') > legend('topleft',paste(rev(r)),lty=2:1,col=1:2, bty='n') > >

We obtain also the graphics of figures 11.9, and 11.10. © L. S. Barreto, 2017. Theoretical Ecology 181 11 Non Predictive Models for Competition

Figure 11.9. Output of the R script. Simulation of competition with model SB-BACO3

Figure 11.10. Output of the R script. Comparison of the data generated by model SB-BACO3, and simulated with the fitted MAR(1)

We can accept that we can fit MAR(1) to the data generated by modl SB-BACO3. I leave to the reader the confirmation that model SB-BACO3 is a competitive system. Model SB-BACO3 can be easily extended to more then 2 competitors.

11.3 A generalization of model SB-BACO3

Let us extend model SB-BACO3 to three populations: 182 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

(11.8) (11.9) (11.10)

We find the per capita growth rate:

We divide these equations by ln yif, and expand them to obtain:

This system cab be written as:

(11.11)

We can express this equation more concisely as:

(11.12)

At equilibrium we find:

(11.13)

Matrix A is the community matrix. From equation (11.12) we can obtain a recursive model to project the system:

(11.14)

These results can be extended to any number of populations. © L. S. Barreto, 2017. Theoretical Ecology 183 11 Non Predictive Models for Competition

A script in R to apply equation (11.14) to three populations is the following:

> > #generalized BACO3 > rm(list=ls(all=TRUE)) > > y=c(3, 6, 8, 10) > c=c(0.2, 0.15, 0.3, 0.27) > c=matrix(c,4,1) > Y=c(y) > A=matrix(c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),4,4,byrow=T) > G=c(0,0,0,0) > > B=matrix(c(-1, 0.01, 0.03, 0.04, -0.01, -1, 0.03, 0.04,-0.015, -0.002, -1, 0.04,-0.02, -0.02, -0.03, -1),4,4,byrow=T) > for(i in seq(0,200,0.01)){ + Z=1/log(runif(1,20,50)) + A[1,]=B[1,]*Z + Z=1/log(runif(1,20,50)) + A[2,]=B[2,]*Z; + Z=1/log(runif(1,20,50)) + A[3,]=B[3,]*Z; + Z=1/log(runif(1,20,50)) + A[4,]=B[4,]*Z; + + g=0.01*(c+A%*%log(y)) + y2=y+g + #if (i==int(i)) {Y=c(Y,y2)} + Y=c(Y,y2) + G=c(G,g) + y=y2 + #y2=y+0.01*y.*(c+A*log(y)')'; + #y=y2; + } > L=matrix(Y,20002,4,byrow=T) > matplot(L,type='l',col=1:4, xlab="Time", ylab="Number",lty=4:1,lwd=3) > r<-c('Sp. 1','Sp. 2','Sp. 3','Sp. 4') > legend('topright',paste(rev(r)),lty=1:4,col=4:1,lwd=3, bty='n') > > LG=matrix(G,20002,4,byrow=T) > final=matrix(c(L[20002,],LG[20002,]),2,4,byrow=T) > final [,1] [,2] [,3] [,4] [1,] 2.1606345590 1.798655e+00 2.929482e+00 2.420938426 Final values [2,] 0.0001094819 9.482099e-05 1.870287e-06 -0.000363534 Last increment > cor(L) [,1] [,2] [,3] [,4] [1,] 1.0000000 0.9864696 0.9980110 0.9974593 [2,] 0.9864696 1.0000000 0.9797852 0.9893239 [3,] 0.9980110 0.9797852 1.0000000 0.9979693 [4,] 0.9974593 0.9893239 0.9979693 1.0000000 >

The size of the populations are highly correlated, given the parameters used. The graphical output is inserted in figure 11.11. 184 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

Figure 11.11. Simulation of competition among four species, applying equation (11.14), and the previous R script

11.4 Discrete model SB-BACO4

11.4.1 Assumptions Discrete model for competition SB-BACO4 (MSB4; Barreto, 2003) accepts all assumptions of m, but the continuity of time. Thus, when isolated the populations follow equation (4.42):

yt+1=yt exp(c ln yf - c ln yt) (11.15)

11.4.2 The Model Model MSB4 is written as:

y1,t+1=y1,t exp(a10 – a11 ln y1,t+a12 ln y2,t) (11.16)

y2,t+1=y2,t exp(a20+ a21 ln y1,t-a22 ln y2,t) (11.17)

Being a10=c1lny1f, a11=c1, a20=c2lny2f, a22=c2

11.4.3 Model Analysis The fixed point is:

(11.18)

(11.19) © L. S. Barreto, 2017. Theoretical Ecology 185 11 Non Predictive Models for Competition

The equilibrium analysis describes in Table 11.1 is applicable to MSB4. Let us simulate a system with two competitors, using MSB4, in R.

> bc4<-function(N, r=c(0.05,0.041)) { + + N1.t1<-N[1]*exp(0.292897-r[1]*log(N[1])+0.00003*log(N[2])) + N2.t1<- N[2]*exp( 0.2401753-r[2]*log(N[2])+0.00015*log(N[1])) + c(N1.t1, N2.t1) + } > > > t<-210 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(192.4, 68.9) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > matplot(0:t, N, type="l", col=c('black','red'), ylim=c(0,400)) > title("Competition simulated with model SB-BACO4") > > r<-c('Competitor 1','Competitor 2') > legend('bottomright',paste(rev(r)),lty=2:1,col=2:1, bty='n') > >

The graphic of figure 11.12 is produced.

Figure 11.12. Simulation of a competitive system with two species, applying MSB4

Now, we approach the question of the coherence of the two models introduced in this chapter. For this purpose, we simulate a system first wit MSB3, after we fit to the simulated data MSB4, Finally, we compare the outputs of the continuous, and discrete models. Let us introduce the script, and its output:

> rm(list=ls(all=TRUE)) > ##BACO3 & BACO4 > > rm(list=ls(all=TRUE)) 186 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

> > baco3<-function(times,y,parms) { + n<-y + + c1<-parms[1] + k1<- parms[2] + a12<-parms[3] + + c2<-parms[4] + k2<- parms[5] + a21<-parms[6] + + dn1.dt<- n[1]*(c1*log(k1)-c1*log(n[1])-a12*log(n[2])) + dn2.dt<- n[2]*(c2*log(k2)-a21*log(n[1])-c2*log(n[2])) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.05, k1=40, a12=0.02, c2=0.04, k2=45, a21=0.03) > initialn<-c(10, 10) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco3, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > h<-matrix(c(out[,2],out[,3]),2991,2) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > > > yp<-out[,2] > y1<-yp[-1] > x1<-yp[-300] > yq<-out[,3] > y2<-yq[-1] > x2<-yq[-300] > > ##Competitor 1 > > init<-c(a12=0.003) > fit1<-nls(y1~x1*exp(0.19033331-0.05*log(x1)+a12*log(x2)), start=init) > summary(fit1)

Formula: y1 ~ x1 * exp(0.19033331 - 0.05 * log(x1) + a12 * log(x2))

Parameters: Estimate Std. Error t value Pr(>|t|) a12 -2.480e-02 2.331e-05 -1064 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.041 on 2989 degrees of freedom

Number of iterations to convergence: 3 Achieved convergence tolerance: 2.583e-08

> yp<-predict(fit1,list(x1,x2)) © L. S. Barreto, 2017. Theoretical Ecology 187 11 Non Predictive Models for Competition

> > zz<-c(y1,yp) > out<-matrix(zz,ncol=2) > > > screen(1) > matplot(out, type="l", xlab="Time", ylab="y1") > title("Models MSB3, and MSB4 for competitor 1") > > ##Competidor 2 > init<-c(a21=0.002) > fit2<-nls(y2~x2*exp(0.1522665+a21*log(x1)-0.04*log(x2)), start=init) > summary(fit2)

Formula: y2 ~ x2 * exp(0.1522665 + a21 * log(x1) - 0.04 * log(x2))

Parameters: Estimate Std. Error t value Pr(>|t|) a21 -2.863e-02 2.578e-05 -1111 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02217 on 2989 degrees of freedom

Number of iterations to convergence: 3 Achieved convergence tolerance: 4.356e-07

> yq<-predict(fit2,list(x1,x2)) > > zz<-c(y2,yq) > out<-matrix(zz,ncol=2) > > screen(2) > matplot(out, type="l", xlab="Time", ylab="y2") > title("Models MSB3, and MSB4 for competitor 2") > >

The fittings are good, and the values generated by the two models are virtually coincident, as exhibited in figure 11.13. We simulated a symmetric competition system (-,-), and the fitted coefc of model MSB4, coherently, are both negative (-2.480e-02, -2.863e-02).

188 © L. S. Barreto, 2017. Theoretical Ecology 11 Non Predictive Models for Competition

Figure 11.13. In each graphic, for each competitor, we inserted the output obtained with model SB-BACO3 (continuous red line), and SB-BACO4 (dashed line). The values are coincident

11.5 References, and Related Bibliography

Barreto, L. S., 1997a. A Quasi Lotka-Volterra Model for Tree Competition. Departamento de Engenharia Florestal, Instituto Superior de Agronomia. Barreto, L. S., 1997b. Coexistence and Competitive Ability of Tree Species. Elaborations on Grime's Theory. Silva Lusitana, 5(1):79-93. Barreto, L. S., 1999. A Tentative Typification of the Patterns of Interaction with Models BACO2 and BACO3. Silva Lusitana, 7(1):117-125. Barreto, L. S., 2001. O Modelo BACO3 para a Competição entre Plantas. Research Paper SB-02/01. Departamento de Engenharia Florestal, Instituto Superior de Agronomia. Barreto, L. S., 2003. SB-BACO 4. A Gompertzian Discrete Model for Tree Competition. Silva Lusitana 11(1):77-89. Barreto, L. S., 2004. Tree Competition: Concepts, Models, and Patterns. Research Paper SB-01/04. Departamento de Engenharia Florestal, Instituto Superior de Agronomia. Barreto, L. S., 2008. The Reconciliation of the r-K, and C-S-R Models for Life-History Strategies. Silva Lusitana, 16(1):97-103. Barreto, L. S, 2009. Growth, Regeneration, and Survival Indices for Tree Species. Silva Lusitana, 17(1):83-95. Barreto, L. S., 2011. From Trees to Forests. A Unified Theory. Livro eletrónico. Costa de Caparica, Portugal. Leslie, P. H. e J. C. Gower, 1958. The Properties of a Stochastic Model for Two Competing Species. Biometrika, 45: 316–330.

12 Predictive Models for Competition SB-BACO2, and SB-BACO6

12.1 Introduction

Predictive models SB-BACO2 (MSB2), and SB-BACO6 (MSB6) are in the line of thought that led us to the clarification of growth, regeneration, and survival indices of species (Barreto, 2006), and of their strategies of life (Barreto, 2008). All these three elaborations have a common root: the relative variation rate of the species (RVR; equations (4.30), and (4.31)). Empirical evidence that sustains this statement ( Grime’s hypothesis) can be found in Grime (1977, 1979), and Radosevich, and Roush (1990). It is coherent that the competitive ability of a specie is an attribute strategically relevant for its life in the community, and the upper mentioned indices must reflect its life-history. Remember the quotation of Sibly, Hone, Clutton-Brock (2002:1149) presented in section 4.8. In this chapter I also include model SB-BACO5 given its plasticity and high degree of generalization.

12.2 Model SB-BACO2

12.2.1 Assumptions If the growth of a specie in competition is different from its growth when in isolated, because its RVR of the biomass or its mortality rate is changed by the presence of the other specie. Then, the RVR in competition (cRVR) is equal to the RVR of the specie, plus or minus a fraction f of RVR. This is, occurs cRVR=RVR + f RVR (12.1)

From the definition of RVR, in a given interval of time, the variation of the population is

∆y= y cRVR (12.2)

We start by introducing our assumptions about the competitive interaction, and after we focus on the estimation of f. Remember equation (4.31):

RVR=-c ln (y0/yf) exp(-ct) (12.3)

The assumptions of MSB2 are:  Intraspecific, and interspecific competitions are similar phenomena;  By large, the most frequent type of interspecific competition is the asymmetric kind. For empirical evidence in vegetation see Keddy (2007:chapter 5), and particularly for forests see Stadt et al. (2007);  The coefficients of competition, as the RVR, change with age;  The two main factors that control interespecific competition are the RVR of the species, and their relative density (proportion of the individuals or biomass of each specie); 190 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

 Thus, the process of competition is sensitive to the initial sizes of the populations (deterministic butterfly effect);  The coevolution of the species causes the emergence of very similar RVR, or/and changing competition dominance [(+,-) → (-,+)] ;

 Given species i, and j, in competition, and their RVR, RVRi, and RVRj, if RVRi> RVRj specie i is dominant, and receives a positive effect from specie j. Specie j is dominated, and receives a negative impact from specie i. The interaction [i, j] is represented by (+.-).

12.2.2 The Model

In equation (12.1) the expression for f of specie yi (number of individuals or biomass) is:

y j RVR j f i= ln ⁡( ) (12.4) yi + y j RVRi

proportion ratio of the RVR

thus the continuous model MSB2 has the two following ODE:

' y2 RVR2 y1= y1 RVR1(1+ l n ) (12.5) y1+ y2 ( RVR1 )

' y1 RVR1 y2= y2 RVR2(1+ l n ) (12.6) y1+ y2 ( RVR2 )

having RVR the expression of equation (12.3). Let me introduce a numerical illustration, reproduced from Barreto (2011b). We consider a situation of self-thinning in a mixed forest. In this context, the RVR in action are the relative mortality rates (rmr) of the two tree species. Consider a self-thinned mixed even-aged stand (SMES) of Pinus pinaster (Ppi; r strategist; species a), and Quercus robur (Qro; K strategist; species b). The characteristic parameters of Ppi are c=0.0500, R-2=6.0191, R0,67=0.549742; Qro: c=0.0410, R-2=125.9635, R0,67= 0.1999492. Being t0=10 years. The mixed stand at age 40 has 800 trees of species a, and 500 trees of species b. First, I calculate the relative mortality rates as in pure stand , using the characteristic parameters of the species: Species a: rmra= -0.05* log(6.019)* exp(-0.05*30)= - 0.0200251 Species b: rmrb= -0.041*log(125.9635) *exp(-0.041*30)= - 0.0579545 As the rmr calculate the trees to be subtracted to the actual density, they have negative values, being the self-thinning larger in the common oak. Let us use the following notation: m rmrat is the rmr in the SMES of specie a, at age t; m rmrbt is the rmr in the SMES of specie b, at age t; From equations (12.5), and (12.6) we can write: © L. S. Barreto, 2017. Theoretical Ecology 191 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

(12.7)

(12.8)

We apply these two equations. Let us see the dynamics in the SMES. We have fra40=800/(800+500)=0.6153846, and frb40=500/(800+500)=0.3846154. Replacing the calculated values in eq. (12.7), it is obtained: Species a: mrmra=- 0.0282097 The absolute value of the rmr is larger than in pure stand, thus the self-thinning is more intense. For Qro it is estimated: Species b: mrmrb=- 0.0200550 Now, for the oak, the absolute value of the rmr is smaller than in pure stand, thus the self- thinning is less intense. Let us assume that the total biomasses of the two populations were the same, and equal to 100 Mg. This is, fra40=frb40=0.5. How did competition affect their growths, ignoring the biomasses of the respective mean trees? The relative growth rates in pure stands are: Species a: RVRa0,67= -0.05* log(0.549742)* exp(-0.05*(40-10))= 0.0066750 Species b: RVRb0,67= -0.041*log(0.1999492;) *exp(-0.041*(40-10))= 0.0192905 For the same populations in the SMES, replacing the values in equations homologous to eq. (14.6) it is obtained: Species a: mRVRa0,6740= 0.0031331 The relative growth rate in mixed stand is smaller than in pure stand. This is consistent with a more intense self-thinning in mixed stand. For Qro it is found: Species b: mRVRb0,6740= 0.0295265 The relative growth rate in mixed stand is greater than in pure stand. This is consistent with a less intense self-thinning in mixed stand. The species with fast growth require high mortality rate to liberate the necessary resources, and space for the fast growth of the survival trees. It is verified rmrb / rmra= RVRb0,67 / RVRa0,67= 2.894, thus the oak is the dominant specie. In figure 12.1, I display the values of this ratio from 10 to 100 years. Qro is always the dominant species (ratio>1) and the dominance increase with age. These examples are consistent with the life-history strategies of the two species. 192 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.1. Ratio of the RVR of common oak, and maritime pine (Qro/Ppi)

The following R script uses MSB2 to simulate the number of trees in a SMES of Ppi+Qro, fit a MAR(1), to the data generated, and compares the final densities of the two populations when isolated, and in the mixed stand:

> rm(list=ls(all=TRUE)) > ##BACO2 > rm(list=ls(all=TRUE)) > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + cop1<-c1*log(r1) + cop2<-c2*log(r2) + + dn1.dt<-n[1]*cop1*exp(c1*times)*(1+n[2]/ (n[1]+n[2])*log((cop2*exp(c2*times))/(cop1*exp(c1*times)))) + dn2.dt<- n[2]*cop2*exp(c2*times)*(1+n[1]/ (n[1]+n[2])*log((cop1*exp(c1*times))/(cop2*exp(c2*times)))) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=-0.05, r1=6.019, c2=-0.041, r2=125.9635) > initialn<-c(5000, 5000) > t.s<- seq(1,90, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > © L. S. Barreto, 2017. Theoretical Ecology 193 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Number", ylim=c(30,5000)) > title("Competition BACO2: Ppi+Qro") > r<-c('Ppi','Qro') > legend('topright',paste(rev(r)),lty=2:1,col=1, bty='n') > > > > h<-matrix(c(out[,2],out[,3]),891,2) > > > g<-seq(1,891,10) > dat0<-matrix(c(h[g,]),90,2) > > > N1<-dat0[,1] > N2<-dat0[,2] > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > n<-m[1]-1 > > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -4.8873 -1.9149 0.3171 2.2635 3.0194

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -43.79800 2.02991 -21.576 <2e-16 *** N1b -0.22159 0.02309 -9.596 3e-15 *** N2b 1.11949 0.02325 48.141 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.405 on 86 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 5.394e+06 on 2 and 86 DF, p-value: < 2.2e-16 Good fitting

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -4.3958 -1.8600 0.2378 2.1073 3.0419 194 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -31.71482 1.87777 -16.89 <2e-16 *** N1b -1.04896 0.02136 -49.11 <2e-16 *** N2b 1.94927 0.02151 90.62 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.225 on 86 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 6.257e+06 on 2 and 86 DF, p-value: < 2.2e-16 Good fitting

> > > > G=data.frame(c(coef(fit1), coef(fit2))) > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2] + N2.t1<-G[4,1]+G[5,1]*N[1]+G[6,1]*N[2] + + + c(N1.t1, N2.t1) + } > > > t<-90 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(5000, 5000) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > windows() > split.screen(figs=c(1,2)) [1] 3 4 Warning message: In par(new = TRUE) : calling par(new=TRUE) with no plot > screen(1) > plot(0:t, N[,1], type='l', col=2:1, ylim=c(0,5080), xlab="Time", ylab="Number",) > title("MAR(1), Ppi") > lines(N1b, col='red') > > screen(2) > plot(0:t, N[,2], type='l', col=2:1, ylim=c(0,5080), xlab="Time", ylab="Number",) > title("MAR(1), Qro") > lines(N2b, col='red') > > > > pip=5000*6.019^(exp(-0.05*90)-1) > print('Ppi: density pure, and mixed') [1] "Ppi: density pure, and mixed" > c(pip,out[891,2]) 1 847.433 394.799 Ppi is dominated. It has a larger population when in pure stand > © L. S. Barreto, 2017. Theoretical Ecology 195 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> qr=5000*125.9635^(exp(-0.041*90)-1) > print('Qro: density pure, and mixed') [1] "Qro: density pure, and mixed" > c(qr,out[891,3]) 2 44.78913 473.79094 Qro is dominant. It has a larger population when in mixed stand >

We obtain also the graphics in figures 12.2, and 12.3.

Figure 12.2. Simulation of the density of a SMES of maritime pine, and common oak, with MSB2. The two populations start with 5000 trees, but the final density of the oak is larger then the density of maritime pine

Figure 12.3. In each graphic, for each competitor, we inserted the output obtained with model SB-BACO2 (continuous red line), and MAR(1) (dashed line). The values are almost coincident 196 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

It is clear now, that one way to detect the competitive hierarchy between species i, and j, is to estimate the ratio of the RVR of any variable:

TRV i RTRV = (12.9) TRV j

If RRVR>1 specie i is dominant

If RRVR<1 specie j is dominant

If RRVR=1 both species have the same competitive capacity

We represent competitive dominance by ‘>’. Specie i>Specie j implies RRVR>1 We consider three Mediterranean pines that coexist in natural forests. They are Pinus pinaster, P. halepensis, and P. pinea. They show the following shifts of dominance (see figure 12.4): Till age 25 years: P. pinea> P. halepensis > P. pinaster From 25 to 30 years: P. pinea > P. pinaster > P. halepensis From 30 to 38 years: P. pinaster > P. pinea > P. halepensis After 38 years: P. pinaster > P. halepensis > P. pinea In figure 15.3, I illustrate de dynamics of the absolute value of the rmr of these three Mediterranean pines.

Figure 12.4. Absolute value of the relative mortality rates of three Mediterranean pines

Besides shifts of dominance, similitude of RVR can occur among species that had coevolved, In North-America, there are natural mixed stands of the three following species: Picea sitchensis (Psi), Pseudotsuga manziesii (Pme) e Thsuga hetrophylla (The). We expect that their rmr have very close values. This antecipation in confirmed in figure 12.5. Simultaneously, the SMES of these species evince the following shifts of dominance: Till age 14 years: Psi>Pme>The From 14 to 32 years: Pme>Psi>The From 32 to 34 years: Pme>The>Psi Ages greater than 34 years: The>Pme>Psi © L. S. Barreto, 2017. Theoretical Ecology 197 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.5. Absolute value of the relative mortality rates of Picea sitchensis (Psi), Pseudotsuga manziesii (Pme), and Thsuga hetrophylla (The)

We argue, in chapter 4, that in nature populations follow the EGZ. Does the populations simulated by MSB2 follow the Gompertz equation? Let us clarify this issue with a simulation in R . The competitors are the maritime pine, and the common oak, characterized by their total biomasses. Here is the script:

> rm(list=ls(all=TRUE)) > > ##BACO2 > # Total biomass of each population > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + cop1<-c1*log(r1) + cop2<-c2*log(r2) + + + + + + dn1.dt<-n[1]*cop1*exp(c1*times)*(1+n[2]/ (n[1]+n[2])*log((cop2*exp(c2*times))/(cop1*exp(c1*times)))) + dn2.dt<- n[2]*cop2*exp(c2*times)*(1+n[1]/ (n[1]+n[2])*log((cop1*exp(c1*times))/(cop2*exp(c2*times)))) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=-0.05, r1=0.5497, c2=-0.041, r2=0.1995) > initialn<-c(30, 30) 198 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> t.s<- seq(1,60, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass") > title("Competition BACO2: Pb+Qr") > r<-c('Ppi','Qro') > legend('topright',paste(rev(r)),lty=2:1,col=1, bty='n') > > #======> > y1<-out[,2] > t<-seq(1,60, by=0.01) > > fit<-nls(y1~a*b^exp(-c*(t-10)), start=c(a=200, b=10, c=0.04)) > fit Nonlinear regression model model: y1 ~ a * b^exp(-c * (t - 10)) data: parent.frame() a b c 69.7918 0.5759 0.0473 residual sum-of-squares: 0.03853

Number of iterations to convergence: 8 Achieved convergence tolerance: 5.581e-08 > summary(fit)

Formula: y1 ~ a * b^exp(-c * (t - 10))

Parameters: Estimate Std. Error t value Pr(>|t|) a 6.979e+01 1.764e-04 395673 <2e-16 *** b 5.759e-01 1.685e-06 341688 <2e-16 *** c 4.730e-02 4.958e-07 95390 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.002556 on 5898 degrees of freedom

Number of iterations to convergence: 8 Achieved convergence tolerance: 5.581e-08

>#Graphic to compare the two time series > av<-seq(1,60,0.01) > bv<-predict(fit,list(t=av)) > z1<-cbind(bv,y1) > windows() > matplot(z1,type="l", main='Ppi',xlab="Time", ylab="Biomass") > > #======> > y2<-out[,3] > > fit2<-nls(y2~a*b^exp(-c*(t-10)), start=c(a=250, b=8, c=0.04)) > fit2

Nonlinear regression model model: y2 ~ a * b^exp(-c * (t - 10)) data: parent.frame() © L. S. Barreto, 2017. Theoretical Ecology 199 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

a b c 63.69612 0.62130 0.05138 residual sum-of-squares: 5.586

Number of iterations to convergence: 6 Achieved convergence tolerance: 1.309e-06 > summary(fit2)

Formula: y2 ~ a * b^exp(-c * (t - 10))

Parameters: Estimate Std. Error t value Pr(>|t|) a 6.370e+01 1.782e-03 35737 <2e-16 *** b 6.213e-01 2.120e-05 29304 <2e-16 *** c 5.138e-02 7.108e-06 7228 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03077 on 5898 degrees of freedom

Number of iterations to convergence: 6 Achieved convergence tolerance: 1.309e-06

>#Graphic to compare the two time series > windows() > bv<-predict(fit2,list(t=av)) > z2<-cbind(bv,y2) > #plot(bv) > matplot(z2,type="l", main='Qro',xlab="Time", ylab="Biomass") > > > > #======> > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 72.63497 65.41441 Fixed point > >

The graphic of the simulation is inserted in figure 12.6. The graphics to compare the data generated by MSB2, and the respective GPZ are displayed in figures 12.7, and 12.8. We conclude that the biomass of each competitor follows the EGZ. Maritime pine is a pioneer species that has an initial explosive growth of biomass, but a longevity much smaller (100-120 years) then the one of common oak (500 years). The trees of oak are much smaller then the trees of the pine. Contrary to the other models already presented, MSB2 is not autonomous, as the time variable t is present in the second member of its ODE. A consequence of this characteristic is variables coefficients of competition, that preclude the possibility of a formal stability analysis. We will appreciate the capacity of the model to origin correct predictions of real competitive systems. 200 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.6. Simulation of the biomasses of maritime pine, and common oak, in competition, by MSB2

Figure 12.7. Biomass of maritime pine, simulated with MSB2, and with a fitted EGZ. The two time series are virtually coincident

The effects of the proportions of the initial sizes of the dominant, and the dominator competitors can be easily deduced. The greater is the proportion of the dominated species, the more is the interaction beneficial to the dominant specie, and the less depressive is the interaction to the dominated specie. Simulations with MSB2 are sensitive to the initial values, this is, they evince deterministic chaos or the butterfly effect, as it will be illustrated in chapter 13. © L. S. Barreto, 2017. Theoretical Ecology 201 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.8. Biomass of common oak, simulated with MSB2, and with a fitted EGZ. The two time series are virtually coincident

12.2.3 Model Evaluation Populations of sessile organisms, easy to count, measure, and where intense competion occurs are the most appropriate to study competition. Forests satisfy these requirements, but unfortunately time series of mixed forests are scarce, because in the past forest management was concentrate in pure, and even-aged forests (isolated cohorts of tree populations). In this CD I include my e-book Árvores e Arvoredos. Geometria e Dinâmica, and its English expanded version, From Trees to Forests. A Unified Theory. In their chapters 15, I present a more detailed, and deep evaluation of MSB2. I suggest the reading of these texts, as a complement to this chapter. I already verified:  MSB2 reproduces the dynamics of biomass in mixed forests of Fagus sylvatica+Picea abies, and of Larix decídua+Picea abies, measured in Germany (Barreto, 2013);  MSB2 predicts the over yielding that occurs in mixed Mediterranean forests (Barreto, 2012:101), empirically verified by Vila et all. (2007).  MSB2 detected RVR with very close values, and/or shifts of competitive dominance in natural forests, both European and North American.  A controversial issue is the relation between local productivity, and local intensity of competition. MSB2 predicts that the less is the local productivity the more intense is competition, measured by the absolute value of the coefficients of competition.  An area of intense research, and dispute is the composition of communities. MSB2 let us formulate the two following predictions: a) When the species that colonize an empty space did not coevolved together, and have very distinct competitive abilities, it is expected that the competitive interaction is relevant for the final composition of the community. The experimental study carried on by Ejrnæs, Bruun, and Graae (2006) sustain this conclusion. 202 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

b) When the species had coevolved, and are well adapted to the physical environment, the colonization process is neutral (Hubbell, 2001).  MSB2 successfully simulated laboratory experiments of mixtures of species of Paramecium (see chapter 13).

12.2.4 A Simulator for MSB2 We can conjecture MSB3 with variable coefficients of competition. From equations (11.2), and (11.3) we can obtain the following expressions for yhe coefficients of competition:

(12.10)

(12.11)

Now, we use MSB2, and equations (12.10), and (12.11) to elaborate a simulator, of general application, that also provides the coefficients of competition. The simulator is written with Scilab. To illustrate its utilization we simulate the competition of a mixture of common oak (specie 1), and maritime pine (specie 2). The variable simulated is density. Initially, both populations have 3000 trees. The script is:

//msb2CC //Simulador para o MSB2 clear disp('[]=g(c1,c2,r1,r2,p01,p02)') function []=g(c1, c2, r1, r2, p01, p02) e1=p01;e2=p02; k1=p01/r1;k2=p02/r2;P=[];C=[]; w=[];x=[10:0.01:89]; cop1=-c1*log(r1); cop2=-c2*log(r2); p1=p01;p2=p02;y=1; for q=0:0.01:79 g=p1+p2; ep1=exp(-c1*q);ep2=exp(-c2*q); rm1=cop1*ep1;rm2=cop2*ep2; f1=log(rm2/rm1);f2=log(rm1/rm2);g1=p2/g;g2=p1/g; d1=p1*(0.01*rm1*(1+g1*f1)); d2=p2*(0.01*rm2*(1+g2*f2)); p1=p1*(1+0.01*rm1*(1+g1*f1)); p2=p2*(1+0.01*rm2*(1+g2*f2)); a1=(c1*p1*log(p1)-c1*log(k1)*p1+d1)/(p1*log(p2)); a2=(c2*p2*log(p2)-c2*log(k2)*p2+d2)/(p2*log(p1)); C=[C; a1 a2]; P=[P;p1 p2] end clf subplot(1,2,1) plot2d(x,P) b=get("current_axes"); m=b.title; m.font_size=3; m.font_style=5; xtitle('Density','Age','Nº of trees') legend(['Specie 1';'Specie 2']) © L. S. Barreto, 2017. Theoretical Ecology 203 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

subplot(1,2,2) plot2d(x,C) b=get("current_axes"); m=b.title; m.font_size=3; m.font_style=5; xtitle('Coefficients of competition','Age','Coef. comp.') disp('Final densities, Sp.1, Sp.2') LS=string(int(P(7901,:))); disp(LS) disp('Final densities of pure stands') legend(['a12';'a21']) PP=string(int([e1/r1 e2/r2])); disp(PP) endfunction

We entered with g(0.041,0.05,125.9635,6.019,3000,3000). The simulator provides the graphics in figure 12.9. The written output is:

-->g(0.041,0.05,125.6935,6.019,3000,3000)

Final densities, Sp.1, Sp.2

!256 214 !

Final densities of pure stands

!23 498 ! 204 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.9. Graphical output of the general simulator for MSB2

The following is verified:  The oak (specie 1) as the dominant competitor, has the larger final density (256>214);  The final density of the oak is larger in mixed stand then in pure (256>23), because as the dominant competitor he transfers to the pine a fraction of its mortality;  The maritime pine (species 2) as the dominated competitor, has a final population smaller on mixed stand then in pure (214<256);  The final density of the pine is smaller in mixed stand then in pure (214<498), because as the dominated competitor receives part of the mortality the oak would have;

 The values of the coefficient of competition of the effect of the pine on the oak a12, coherently, are positive, and they decrease as the proportion of pine in the mixed stand decreases also;

 The values of the coefficient of competition of the effect of the oak on the pine a21, coherently, decrease, change from positive to negative, and when negative their absolute

value increases, reflecting the increasing of both RTRM oak/pine from 2.0 to 5.4, from age 10 to 120 years (figure 12.1) , and the proportion of oaks in the mixture;  The interaction changes from symmetric, and cooperative (+,+) to asymmetric (+,-)

12.3 Model SB-BACO5

Model SB-BACO5 (MSB5) is a generic model for competition, explicitly formulated. The model is written as :

yi,t=AB (12.12) where: © L. S. Barreto, 2017. Theoretical Ecology 205 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

 A is the Gompertzian dynamics (density or biomass), for instance, given by equation (4.11)  B represents the effects of the competitors, and has the form:

(12.13)

If • There are two competitors; • The coefficients of competition are constant, and negative; • Equation (4.11) is used, The the analysis inserted in Table 11.1 is applicable. Forests of oak, and ash (Fraxinus excelsior) occupie large areas, in Europe. Thus, these two species had coevolved during a long period of time. I present a script in R, that simulates this mixed stand with MSB2 (figure 12.10), and after fits MSB5 to the generated data. As it can be seen in figures 12.11, and 12.12, the values obtained with the two models are coincident. Here is the script:

> ##BACO2 ->BC5 > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + cop1<-c1*log(r1) + cop2<-c2*log(r2) + + + + + + dn1.dt<-n[1]*cop1*exp(c1*times)*(1-n[2]/ (n[1]+n[2])*log((cop2*exp(c2*times))/(cop1*exp(c1*times)))) + dn2.dt<- n[2]*cop2*exp(c2*times)*(1-n[1]/ (n[1]+n[2])*log((cop1*exp(c1*times))/(cop2*exp(c2*times)))) + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=-0.038, r1=0.225022, c2=-0.041, r2=0.199492) 206 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> initialn<-c(78.76, 68.9) > t.s<- seq(1,201, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass") > title("Competition SB-BACO2: ash+oak") > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 331.1791 368.6611 > > > # Fitting MSB5 to ash > t<-seq(1,201,0.01) > x1<-78.76*0.225022^(exp(-0.038*t)-1) > > y1<-out[,2] > > > y2<-out[,3] > > > x1g<-log(y1) > x2g<-log(y2) > > > fit1<-nls(y1~x1*exp(a*x2g), start=list(a=0.002)) > summary(fit1)

Formula: y1 ~ x1 * exp(a * x2g)

Parameters: Estimate Std. Error t value Pr(>|t|) a -1.893e-02 5.323e-06 -3556 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.206 on 20000 degrees of freedom

Number of iterations to convergence: 3 Achieved convergence tolerance: 3.477e-07

> © L. S. Barreto, 2017. Theoretical Ecology 207 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> > windows() > split.screen(c(1,2)) [1] 3 4 Warning message: In par(new = TRUE) : calling par(new=TRUE) with no plot > > yp<-predict(fit1,list(x1,x2g)) > > > zz<-c(y1,yp) > out<-matrix(zz,ncol=2) > > screen(1) > > matplot(out, type="l", xlab="Years/100", ylab="Biomass") > title("Ash") > screen(2) > > plot(y1-yp, type="l", xlab="Years/100", ylab="Residuals") > title("Residuals") > # Fitting MSB5 to oak > > x2<-350*0.199^exp(-0.041*t) > fit2<-nls(y2~x2*exp(a*x1g), start=list(a=0.002)) > summary(fit2)

Formula: y2 ~ x2 * exp(a * x1g)

Parameters: Estimate Std. Error t value Pr(>|t|) a -2.726e-03 5.062e-06 -538.5 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.244 on 20000 degrees of freedom

Number of iterations to convergence: 3 Achieved convergence tolerance: 6.813e-10

> > > windows() > split.screen(c(1,2)) [1] 3 4 Warning message: In par(new = TRUE) : calling par(new=TRUE) with no plot 208 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> > yq<-predict(fit2,list(x1,x2)) > > zz1<-c(y2,yq) > out1<-matrix(zz1,ncol=2) > > screen(1) > matplot(out1, type="l", xlab="Years/100", ylab="Biomass") > title("Oak") > screen(2) > plot(y2-yq, type="l", xlab="Years/100", ylab="Residuals" ) > title("Residuals") >

Figure 12.10. Simulation of the biomass of a mixed forest of ash (black line), and oak with model SB-BACO2 © L. S. Barreto, 2017. Theoretical Ecology 209 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.11. Ash population. Simulated data with MSB2, and fitted model MSB5. The two time series are virtually coincident 210 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Figure 12.12. Oak population. Simulated data with MSB2, and fitted model MSB5. The two time series are virtually coincident

12.4 Model SB-BACO6

12.4.1 Assumptions Model SB-BACO6 (MSB6) is focused in the competition of two plants for a limiting nutrient. Now the model contains not only the variables of the population biomass, but also the weight of the assimilable limiting nutrient k. Then, we have three state variables, y1, y2, and k. It is accepted that in terrestrial ecosystems, excluding C, the cycles of the other elements, can be seen as closed (e. g., CHAPIN III, MATSON, and MOONEY, 2002:222). In these ecosystems, the most common situation is the one where the most strongly limiting nutrient imposes the quantitative expression of the cycles of the other essential nutrients (e. g., CHAPIN III, MATSON, and MOONEY, 2002:220). This pattern abides Liebig´s Law of the Minimum. This situation is the starting point for MSB6. The basic assumptions of MSB6 are the following ones: 1 The dynamics of the assimilable nutrient can be modelled as a Gompertz equation. This assumption is coherent with section 10.4.1; 2 The dynamics of the biomass of both plants have the same Gompertzian pattern of growth. 3 The appropriation of the usable nutrient is proportional to the competitive ability of each species; this is, to its RVR; 4 The amount of nutrient appropriated by each plant species defines its carrying capacity.

12.4.2 The Model MSB6 consists in an algorithm that mimics the dynamics of the system nutrient+competitor 1+competitor 2. The algorithm comprehends the following steps:

Step 1. At age t, for the competitors, calculate RVR1 e RVR2 (equaçã0 (12.3))

Step 2. Find the sum tot of the two RVR (tot=RVR1+RVR2)

Step 3. Find the fraction of each RVR in the total (fi=RVRi/tot)

Step 4. Estimate the appropriation of the usable nutrient by plant i represented by ki (ki=k*fi) © L. S. Barreto, 2017. Theoretical Ecology 211 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Step 5. Find the carrying capacity of each plant biomass as yif=ki*efi, being efi the efficiency of the plant using the nutrient. It represents the units of biomass produced by the uptake of one unit of nutrient. This parameter can be modelled as function of age and/or k. Step 6. Find the accretion of biomass of each species using equation (4.10)

dy(i)=ci * y(i) *(ln yif – ln y(i))

Biomass of specie i Step 7. Find the variation of the quantity of assimilable nutrient (recycling less plant uptake) using the equation:

Nutrient absortion by the species

cn=Rate of recycling of the nutrient. It reflects the process of mineralization of the detritus. It can be modelled as a function of environmental parameters, such as precipitation, temperature, and redox potential. K= Maximum value k can attain in the closed system.

Step 8. Actualize the values of yi, and k, keep the values. Step 9. Return to step 1. The simulator, in Scilab, for MSB6, describes the dynamics of the biomass of oak (specie 1), and maritime pine (specie 2). The limiting nutrient is phosphorous. Initial populations are 10 years old, and the standing biomass of each population is 50 units of biomass per unit of area (50 ub/ua). As I do not have specific values of the efficiency of the oak, and the pine using phosphorous, I use the average value for hardwood (1859), and for conifers (1519), obtained from Cole, and Rapp (1981:366). The script of the simulator is:

//**************************************** // Competition with explicit partition * // of limiting nutrient (Liebig’ law) * // (c) L. S. Barreto, 2014 * // Algorithm SB-BACO6 * //**************************************** clear clf //ua=unit of are K=0.39;//maximum assimilable nutrient k=0.06;//available assimilable nutrient //Recycling rate of the nutrient cn=0.035; //Parameters of competitor 1 c1=0.041;r1=0.199; ef1=1859;//Efficiency in nutrient use of competitor 1 //Parameters of competitor 2 c2=0.05;r2=0.55; ef2=1519;// Efficiency in nutrient use of competitor 2 //Initial biomasses; ub=units of biomass y=[50 50];y0=y; //Auxiliary vectors to keep calculated values kv=[]; Gq=[];Gp=[]; y1=[y(1)];y2=[y(2)]; //Loop of 99 years for i=0:0.01:99 212 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

//Steps 1 to 3 trv1=-c1*log(r1)*exp(-c1*i); trv2=-c2*log(r2)*exp(-c2*i); tot=trv1+trv2; f1=trv1/tot;f2=trv2/tot; //Step 4 k1=k*f1;k2=k*f2; Gq=[Gq k1]; Gp=[Gp k2];

//Step 5 y1f=k1*ef1;y2f=k2*ef2; //Step 6 dy(1)=y(1)*c1*(log(y1f)-log(y(1))); dy(2)=y(2)*c2*(log(y2f)-log(y(2))); //Step 7 dk=cn*k*(log(K)-log(k))-sum([dy(1)/ef1 dy(2)/ef2]); //Step 8 y=[y]+0.01*[dy(1) dy(2)];k=k+0.01*dk; //Step 9 y1=[y1 y(1)]; y2=[y2 y(2)]; kv=[kv k]; end B=[y1;y2]'; M=[Gq;Gp]'; time=10:0.01:109; subplot(2,2,1) plot2d(time,M) b=get("current_axes"); m=b.title; m.font_size=3; m.font_style=5; xtitle('Nutrient uptake by each specie','Age, years','Weight/ua') ti=10:0.01:109.01; legend(['Specie 1';'Specie 2'],2) subplot(2,2,2) plot2d(ti,B) b=get("current_axes"); m=b.title; m.font_size=3; m.font_style=5; xtitle('Sizes of populations','Age, years','ub/ua') legend(['Specie 1';'Specie 2'],2) subplot(2,2,3) plot2d(time,kv) b=get("current_axes"); m=b.title; m.font_size=3; m.font_style=5; xtitle('Assimilable nutrient','Age, years','Weight/ua')

t=[0:1:80]'; rmr1=(-c1*log(r1)*exp(-c1*t)); rmr2=(-c2*log(r2)*exp(-c2*t)); [k]=rmr2';[l]=rmr1'; [M]=[l./k]'; tt=10:1:90; subplot(2,2,4) plot2d(tt,[M]) b=get("current_axes"); m=b.title; m.font_size=3; © L. S. Barreto, 2017. Theoretical Ecology 213 12 Predictive Models for Competition SB-BACO2, and SB-BACO6 m.font_style=5; xtitle("Ratio of the RVR (Specie 1/Specie 2)","Age, years","Ratio") xgrid(2) disp(' ') printf(' Final population of competitor 1=%f',y(1)); disp(' ') printf(' Final population of competitor 2=%f',y(2)); Misto=y(1)+y(2); Spuros=y0(1)*r1^(exp(-c1*99)-1)+y0(2)*r2^(exp(-c2*99)-1); R=Misto/Spuros; disp(' ') printf(' Ratio«Mixed total/Pure total»=%f',R); if R>1 then disp(' It occurs overyielding'); end

The written output is:

Final population of competitor 1=370.345706

Final population of competitor 2=71.368883

Ratio«Mixed total/Pure total»=1.319047 It occurs overyielding

The overyielding is empirically corroborated in Vila et all. (2007). The graphics in figure 12.13 are also produced by the script.

Figure 12.13. Graphical output of the simulator for MSB6. For more details see the text 214 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

As a general comment, we state that this simulation is consistent with the one presented in subsection 12.2.4, executed with MSB2, using the same species. Now, let us concentrate in figure 12.13. In the upper line, we verify: • The left graphic shows that the uptake of phosphorous, by the species, is consistent with the competitive hierarchy, where the oak is the dominant species (oak>pine); • The right graphic also mirrors the competitive hierarchy, as the biomass of the oak is much more larger the biomass of the pine (both species started with 50 ub/ua). These two graphics mirror the ratio of the bottom right graphic of the figure. In the bottom line we have the dynamics of the assimilable nutrient in the soil, k, and the already known graphic of the ratio of the RVR of the species (figure 12.1).

12.5 Competition, and Total Effects

I suggest that the reader recall section 8.6. On this section, I will approach, with MSB3, a system of three competitors with the following competitive hierarchy: Competitor 2 > competitor 1 > competitor 3. First, we simulate the system, and find the fixed, with all three populations initially equal to 10. Here is the script:

> ##BACO33 > > baco3<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + a12<-parms[3] + a13<-parms[4] + + c2<-parms[5] + k2<- parms[6] + a21<-parms[7] + a23<-parms[8] + + c3<-parms[9] + k3<- parms[10] + a31<-parms[11] + a32<-parms[12] + + + dn1.dt<- c1*n[1]*(log(k1)-log(n[1])-a12*log(n[2])-a13*log(n[3])) + dn2.dt<- c2*n[2]*(log(k2)-a21*log(n[1])-log(n[2])-a23*log(n[3])) + dn3.dt<- c3*n[3]*(log(k3)-a31*log(n[1])-a32*log(n[2])-log(n[3])) + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > parms<-c(c1=0.05, k1=40, a12=0.02, a13=0.005, c2=0.04, k2=45, a21=0.03, a23=0.002, c3=0.033, k3=31, a31=0.0004, a32=0.005) > initialn<-c(10, 10, 10) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco3, parms=parms) > © L. S. Barreto, 2017. Theoretical Ecology 215 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> matplot(out[,1], out[,-1], type="l", xlab="Time", col=c('blue','red','green'),ylab="Number", lw=2) > title("Competition, SB-BACO3") > > > # Fixed point > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco3,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 36.52406 40.12081 30.38924 Fixed point >

The fixed point mirrors the competitive hierarchy: y2 > y1 > y3, as rhe graphic in figure 12.14.

Figure 12.14. Simulation of a system of three competitors, with MSB3. For more details see the text

The MAR(1) is obtained as follows:

> h<-matrix(c(out[,2],out[,3], out[,4]),2991,3) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > > library(MASS) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > > ###### MAR(1) > > m<-c(dim(dat0)) 216 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.052853 -0.004249 -0.001715 0.005672 0.023429

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.522869 0.016761 31.20 <2e-16 *** N1b 1.067626 0.001982 538.67 <2e-16 *** N2b -0.158062 0.003971 -39.80 <2e-16 *** N3b 0.110349 0.003496 31.56 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.008565 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 3.785e+07 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.0276178 -0.0023061 -0.0008984 0.0030190 0.0118940

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.485596 0.008655 56.11 <2e-16 *** N1b 0.084191 0.001023 82.26 <2e-16 *** N2b 0.860288 0.002051 419.53 <2e-16 *** N3b 0.067370 0.001805 37.32 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.004423 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 2.253e+08 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b)

Residuals: © L. S. Barreto, 2017. Theoretical Ecology 217 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Min 1Q Median 3Q Max -0.0111082 -0.0007453 -0.0002992 0.0009958 0.0046012

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.5084679 0.0031498 161.43 <2e-16 *** N1b 0.0302058 0.0003725 81.10 <2e-16 *** N2b -0.0313911 0.0007463 -42.06 <2e-16 *** N3b 0.9884352 0.0006570 1504.38 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.00161 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 8.512e+08 on 3 and 295 DF, p-value: < 2.2e-16

We obtained a good fitting of MAR(1). Let us fin the matrix of total effects:

> > ############# Matrix of total effects > > G=data.frame(c(coef(fit1), coef(fit2), coef(fit3))) > > A=matrix(c( G[2,1] , G[3,1], G[4,1], G[6,1], G[7,1], G[8,1], G[10,1], G[11,1], G[12,1]), nrow=3, byrow=T) > E=-ginv(A) > round(E, 3) [,1] [,2] [,3] [1,] -0.927 -0.166 0.115 Matrix of total effects [2,] 0.088 -1.144 0.068 [3,] 0.031 -0.031 -1.013 > eigen(A)$values [1] 0.9758433+0.0000000i 0.9702529+0.0339493i 0.9702529-0.0339493i > eigen(E)$values [1] -1.029399+0.036019i -1.029399-0.036019i -1.024755+0.000000i >

We focus on the matrix of total effects (E). As the elements of the main diagonal are negative we assume that intraspecific competition prevails. The effect of competitor 1 on the two others is in column 1. It benefits the two others competitor, mainly the dominant competitor (0.088 > 0.031). As competitor 1 deviates from competitor 3 part of the negative impact of competitor 2, it has a positive impact on competitor 3. Competitor 2, as the dominant one, has negative effects on the two others (column 2). Competitor 3 benefits the two other competitors. As the weakest competitor of the system, he gives more benefit to competitor 1 (0.068) then he receives from it (0.031). A real eigenvalue of matrix A close to 1, indicates a final series of unchangeable values of the state variables. The eigenvalues of matrix E with negative real parts indicates a stable fixed point. We now present a graphical comparison of the time series generated by MSB3, and MAR(1), in figure 12.15. The script is: 218 © L. S. Barreto, 2017. Theoretical Ecology 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

> ############# Simulation with MAR(1) > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2]+G[4,1]*N[3] + N2.t1<-G[5,1]+G[6,1]*N[1]+G[7,1]*N[2]+G[8,1]*N[3] + N3.t1<-G[9,1]+G[10,1]*N[1]+G[11,1]*N[2]+G[12,1]*N[3] + + c(N1.t1, N2.t1,N3.t1) + } > > # Graphical comparison > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=3) > N[1, ]<-c(2, 0.5, 0.8) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > windows() > matplot(0:t, N, type='l', col=1, ylim=c(0,50), xlab="Time", ylab="Number",) > lines(N1b, col='red') > lines(N2b, col='red') > lines(N3b, col='red') > title("Comparison MSB3, and MAR(1)") > >

Figure 12.15. Comparison of the simulations with MSB3, and MAR(1)

12.6 References, and Related Bibliography

Barreto, L. S., 2004a. Pinhais Bravos. Ecologia e Gestão. “e-book”. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. © L. S. Barreto, 2017. Theoretical Ecology 219 12 Predictive Models for Competition SB-BACO2, and SB-BACO6

Barreto, L. S., 2004b. SB-KHRONOSKHABA. A Gompertzian Model for the Seasonal Growth of Forest Trees and Stands An Application to Pinus pinaster Ait. Research Paper SB-05/04. Departamento the Engenharia Florestal, In- stituto Superior de Agronomia, Lisboa. Barreto, L. S., 2005a. Theoretical Ecology. A Unified Approach. “e-book”. Edição do autor, Costa de Capar- ica. Included in the CD. Barreto, L. S., 2005b. Gause’s Competition Experiments with Paramecium sps. Revisited. Research Paper SB-01/05. Departamento the Engenharia Florestal, Instituto Superior de Agronomia, Lisboa. Versão revista sub- metida à revista Silva Lusitana. Barreto, L. S., 2010a. Árvores e Arvoredos. Geometria e Dinâmica. Livro eletrónico. Edição do autor. Costa de Ca- parica. In the CD. Barreto, L. S., 2010b. Simulação do Carbono Retido no Pinhal Bravo e da sua Acreção. Silva Lusitana, 18(1):47-58. Barreto, L.S., 2011a. Modelling and Simulating Omnivory. Silva Lusitana, 19(1):67-83. Barreto, L. S., 2011b. From Trees to Forests. A Unified Theory. E-book. Costa de Caparica. Included in the CD. Barreto, L. S., 2012. The Global Yield and Allometry of Self-Thinned Mixed Forests. A Theoretical and Simulative In- quiry. Silva Lusitana, 20(1/2):83-103. Barreto, L. S., 2013. The Total Biomass of Self-Thinned Mixed Forests. A Theoretical and Simulative Inquiry. Silva Lusitana, 21(2):179-203. Chapin III, F. S., P. A. Matson e H. A. Mooney, 2002. Principles of Terrestrial Ecosystem Ecology. Springer, New York. Cole, D. W., and M. Rapp, 1981. Elemental Cycling in Forest Ecosystems. Em D. E. Reichle, (Editor), Dynamic Prop- erties of Forest Ecosystems. International Biological Programme 23. Cambridge University Press, pages 341-409. Ejrnæs, R., H. H. Bruun, and B. J. Graae, 2006. Community Assembly in Experimental Grasslands: Suitable Environment or Timely Arrival?? Ecology 87:1225–1233. Fryxell, J. M., and P. Lunberg, 1997. Individual Behaviour and Community Dynamics. Chapman & Hall, London. Grime, J. P. 1977. Evidence for the Existence of Three Primary Strategies in Plants and its Relevance to Ecological and Evolutionary Ecology. Am. Nat. 111:1169-1194. Grime, J. P. 1979. Plant Strategies and Vegetation Processes. Academic Press, London. Hubbell, S P, 2001. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton. Keddy, P. A., 2007. Plants and Vegetation. Origins, Processes, Consequences. Cambridge University Press, Cambridge. Radosevich, S. R., and M. L. Roush, 990. The Role of Competition in Agriculture. Em J. B. Grace, and D. Tilman, (Ed- itors), Perspectives on Plant Competition. Academic Press, San Diego, California. Pages 341-363. Ricker, W.E. 1975. Computation and interpretation of biological statistics of fish populations. Bulletin of the Fish- eries Research Board of Canada. 191: 382 pp. Sibly, R. M., D. Barker, M. C. Denham, J. Hone,and M. Pagel, 2005. On the regulation of populations of mammals, birds, ﬁsh, and insects. Science, 309:607–610. Stadt, K. J., C. Huston, K. D. Coates, Z. Feng, M. R. T. Dale, and V. J. Lieffers, 2007. Evaluation of competition and light estimation indices for predicting diameter growth in mature boreal mixed forests. Ann. For. Sci. 64:477–490. Vandermeer, J. H., 1969. The Competitive Structure of Communities. An Experimental Approach with Protozoa. Ecology 50:362-371. Vila, M., J. Vayreda, L. Comas, J. Ibáñez, T. Mata, and B. Obón, 2007. Species richness and wood production: a positive association in Mediterranean forests. Ecology Letters, (2007) 10: 241–250. Waring, R. H. e Schlesinger,W. H., 1985. Forest Ecosystems. Concepts and Management. Academic Press, Orlando. 220 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

13.1 Introduction

In the previous chapter I used tree species to illustrate the application of MSB2, and MSB6. In this chapter I will use unicellular ciliates of the genus Paramecium. This chapter is supported by Barreto (2005a: section 11.4). Paramecia can be found in marine ecosystems, freshwater, and brackish. The species that are readily cultivated, and easily induced to conjugate, and divide, it has been used as teaching material en biology. The classical experiments of Gause (1934), about competition, with P. aurelia e P. caudatum, are included in almost all ecology textbooks. This study is a fecund contribution that contains precocious predictions to the dynamics of this interaction (Hastings, 2004). This chapter develops as follows:  I fit the Gompertz equation (equation (4.11)) to data from the literature, about the growth of populations of Paramecium aurelia (Pa), P. bursaria (Pb) e P. caudatum (Pc). These species coexist in aquatic environments, and had experimented a process of coevolution.  The information previously obtained is used establish the competitive hierarchy of the three species.  MSB2 is used two simulate competitive systems with two, and three species.  The results of the simulations are compared with the empirical evidence available.

13.2 The Characteristic Parameters of the Species

I used information from Hastings (2004), Hutchinson (1978), Roughgarden (1998:110 e 227), and Vandermeer (1969) to fit equation (4.11). The fitted values are displayed in table 13.1.

Table 13.1. Fitted values of c, and R-3 for Paramecium aurelia, P. bursaria, P. cadatum

Parameters P. aurelia P. bursaria P. caudatum c 0,297 0,289 0,253

R-3 0,0033 0,0097 0,0090

The information displayed in table 13.1 is used to execute the simulations presented in this chapter.

The size of the populations are referred to the number of individuals in 0.5 cm3. The time unit is the day

In figure 13.1, I exhibit the growth of the isolated populations of the three species, with the initial sizes of 2 individuals/0.5 cm3. © L. S. Barreto, 2017. Theoretical Ecology 221 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figura 13.1. Simulations of the growth of three isolated populations of the species of Paramecium

13.3 The Relative Variation Rates

Now, we calculate the RVR of the species, and compare them. As the species coexist in mature, we expect that these rates are very close, and the eventual occurrence of shifts of dominance. The equation used is (4.31), here recalled: rgr= -c ln R exp(-c t) (13.1)

For this purpose, we use the following R script:

> #ratios > t=1:40 > A=-0.297019*log(0.0033233)*exp(-0.297019*t) > # > C=-0.253*log(0.0090)*exp(-0.253*t)#Pc > B=-0.289*log(0.0097)*exp(-0.289*t)#Pb > rab=A/B > rac=A/C > rbc=B/C > > ab=matrix(c(A,B), nrow=40, ncol=2, byrow=FALSE) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(t,ab, type='l', lw=2) > title('RVR of Pa, and Pb') > screen(2) > plot(t,rab, type='l') 222 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

> title('Ratio of RGR: Pa/Pb') > abline(h=1, col='red') > > ####################### > windows() > > ac=matrix(c(A,C), nrow=40, ncol=2, byrow=FALSE) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(t,ac, type='l', lw=2) > title('RGR of Pa, and Pc') > screen(2) > plot(t,rac, type='l') > title('Ratio of RVR: Pa/Pc') > abline(h=1, col='red') > > ############################ > > > bc=matrix(c(B,C), nrow=40, ncol=2, byrow=FALSE) > windows() > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(t,bc, type='l', lw=2) > title('RGR of Pb, and Pc') > screen(2) > plot(t,rbc, type='l') > title('Ratio of VGR: Pb/Pc') > abline(h=1, col='red')

In figures 13.2 to 13.4, we present the RVR of the three possible pairs of mixtures with the three species. Also we display the ratio of the relative growth rates of the two species of each pair, as we did in chapter 11 (for instance, figure 12.1). After, we put in the same graphic the three RVR, From these four figures, we can conclude the following: • The three species have very close RVR, as their ratios are very close to 1; • Shifts of dominance are present because the ratios are smaller, and greater then 1. These two verifications are coherent with the coexistence of the species in nature, and the coevolution they experienced. © L. S. Barreto, 2017. Theoretical Ecology 223 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figure 13.2. RVR of Pa, and Pb, and their ratio. Pa has a small dominance over Pb (ratio>1) in the first 28 days. After the dominance is inverted [Pa>Pb → Pb>Pa]. The ratio has a small range of vatiation

Figure 13.3. RVR of Pa e Pc, and their ratio. Pa has a small dominance over Pb (ratio>1) in the first 9 days. After the dominance is inverted [Pa>Pc → Pc>Pa] 224 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figure 13.4. RVR of Pb e Pc, and their ratio. Pb has a very small dominance over Pb (ratio>1) in the first 4 days. After the dominance is inverted [Pb>Pc → Pc>Pb]

Figure 13.5. RVR of Pa (black), Pb (red), and Pc (green)

13.4 Simulations with model SB-BACO2

In this section, we simulate mixtures of two, and three species, and we will interpret the obtained results. All simulations have initial sizes of 2 individuals/0.5 cm3. In figures 13.6 to 13.9, we display simulations of the mixtures Pa+Pc, Pa+Pb, Pb+Pc, and Pa+Pb+Pc . The figures include the dynamics of the sizes of the population, and of the coefficients of competition (CC). Let us display the R script for the first simulation, and its output: © L. S. Barreto, 2017. Theoretical Ecology 225 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

First simulation: Pa+Pc

> ##BACO2 > ##Pa+Pb > > rm(list=ls(all=TRUE)) > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + + + rm1=-c1*log(r1)*exp(-c1*times) + rm2=-c2*log(r2)*exp(-c2*times) + + + g=n[1]+n[2]; + g1=n[1]/g;g2=n[2]/g; + f12=log(rm2/rm1); + f21=log(rm1/rm2); + + + dn1.dt<- n[1]*rm1*(1+g2*f12) + dn2.dt<- n[2]*rm2*(1+g1*f21) + + return(list(c(dn1.dt,dn2.dt))) + } > > > parms<-c(c1=0.2970191, r1=0.0033233,c2=0.289, r2=0.0097) > initialn<-c(2,2) > t.s<- seq(0,38, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", xlab="Tempo", ylab="N") > title("Competition MSB2: Pa+Pb") > > #Fixed point > library(rootSolve) Warning message: package ‘rootSolve’ was built under R version 3.3.2 > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye 226 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

[1] 331.9604 333.3443 > > # Coefficients of competition > c1=parms[1] > r1=parms[2] > > c2=parms[3] > r2=parms[4] > k1=2/r1 > k2=2/r2 > > h<-matrix(c(out[,2],out[,3]),nrow=3801,ncol=2, byrow=F) > > > N1<-h[,1] > N2<-h[,2] > > N1a<-N1[-1] > N1b<-N1[-3801] > N2a<-N2[-1] > N2b<-N2[-3801] > d1=N1a-N1b > d2=N2a-N2b > > p1=N1a > p2=N2a > a1=c() > a2=c() > a1=(c1*p1*log(p1)-c1*log(k1)*p1+d1)/(p1*log(p2)) > a2=(c2*p2*log(p2)-c2*log(k2)*p2+d2)/(p2*log(p1)) > > > > cc=cbind(a1,a2) > t=1:3800 > screen(2) > matplot(t,cc, type='l', lw=2) > title('CC Pa, and Pb') > > > > cat("Pa: Pure= ", k1, "Mixed ",ye[1],"\n") Pa: Pure= 601.8115 Mixed 331.9604 > > cat("Pb: Pure= ", k2, "Mixed ",ye[2],"\n") Pb: Pure= 206.1856 Mixed 333.3443 > > cat("CC: a12= ", cc[3800,1], "a21=",cc[3800,2],"\n") CC: a12= -0.03042156 a21= 0.0239138 > © L. S. Barreto, 2017. Theoretical Ecology 227 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figure 13.6. Simulation of the mixture Pa+Pb. The scale of the time in the graphic of CC is day/100

The analysis of this simulation is as follows:  The sizes of the two populations are almost coincident, what mirrors their coevolution;  The CC are not constant; there is an initial phase with strong variation followed by a phase with CC almost stationary;  Comparing the isolated population the size of Pa is reduced from 601 to 331. Pa is the dominated species;

 Consistently, the values of the CC a12 (effect of Pb on Pa) are always negative;  Comparing the isolated population, the size of Pb increases from 206 o 333. Pb is the dominant species;

 Consistently, the values of the CC a21 (effect of Pa on Pb) are negative till age 8, and after

they become positive (Pa benefits Pb). When negative, it is also verified |a21|<|a12|, smaller negative impact of Pa on Pbthen of Pb on Pa.

Second simulation: Pa+Pc

> rm(list=ls(all=TRUE)) > ##BACO2 > ##Pa+Pc > > rm(list=ls(all=TRUE)) > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] 228 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

+ r2<- parms[4] + + + + + rm1=-c1*log(r1)*exp(-c1*times) + rm2=-c2*log(r2)*exp(-c2*times) + + + g=n[1]+n[2]; + g1=n[1]/g;g2=n[2]/g; + f12=log(rm2/rm1); + f21=log(rm1/rm2); + + + dn1.dt<- n[1]*rm1*(1+g2*f12) + dn2.dt<- n[2]*rm2*(1+g1*f21) + + return(list(c(dn1.dt,dn2.dt))) + } > > > parms<-c(c1=0.2970191, r1=0.0033233,c2=0.2532272, r2=0.0090313) > initialn<-c(2,2) > t.s<- seq(0,38, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", xlab="Dias", ylab="Number/0,5 c.c.") > title("Competition MSB2: Pa+Pc") > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 335.1878 337.2999 > > > # Coefficients of competition > c1=parms[1] > r1=parms[2] > > c2=parms[3] > r2=parms[4] > k1=2/r1 > k2=2/r2 > > h<-matrix(c(out[,2],out[,3]),nrow=3801,ncol=2, byrow=F) > > > N1<-h[,1] > N2<-h[,2] > > N1a<-N1[-1] > N1b<-N1[-3801] © L. S. Barreto, 2017. Theoretical Ecology 229 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

> N2a<-N2[-1] > N2b<-N2[-3801] > d1=N1a-N1b > d2=N2a-N2b > > p1=N1a > p2=N2a > a1=c() > a2=c() > a1=(c1*p1*log(p1)-c1*log(k1)*p1+d1)/(p1*log(p2)) > a2=(c2*p2*log(p2)-c2*log(k2)*p2+d2)/(p2*log(p1)) > > > > cc=cbind(a1,a2) > t=1:3800 > screen(2) > matplot(t,cc, type='l', lw=2) > title('CC Pa, and Pc') > > cat("Pa: Pure= ", k1, "Mixed ",ye[1],"\n") Pa: Pure= 601.8115 Mixed 335.1878 > cat("Pc: Pure= ", k2, "Mixed ",ye[2],"\n") Pc: Pure= 221.4521 Mixed 337.2999 > cat("CC: a12= ", cc[3800,1], "a21=",cc[3800,2],"\n") CC: a12= -0.02986866 a21= 0.0183218 > > >

Figure 13.7. Simulation of the mixture Pa+Pc. The scale of the time in the graphic of CC is day/100

The analysis of this simulation is as follows:  The sizes of the two populations are almost coincident, what mirrors their coevolution;  The CC are not constant; there is an initial phase with strong variation followed by a phase with CC almost stationary; 230 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

 Comparing the isolated population the size of Pa is reduced from 601 to 335. Pa is the dominated species;

 Consistently, the values of the CC a12 (effect of Pc on Pa) are always negative;  Comparing the isolated population, the size of Pc increases from 221 o 337. Pc is the dominant species;

 Consistently, the values of the CC a21 (effect of Pa on Pc) are negative till age 10, and after

they become positive (Pa benefits Pc). When negative, it is also verified |a21|<|a12|, smaller negative impact of Pa on Pc then of Pc on Pa.

Third simulation: Pb+Pc

> ##BACO2 > ##Pb+Pc > > rm(list=ls(all=TRUE)) > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + + + rm1=-c1*log(r1)*exp(-c1*times) + rm2=-c2*log(r2)*exp(-c2*times) + + + g=n[1]+n[2]; + g1=n[1]/g;g2=n[2]/g; + f12=log(rm2/rm1); + f21=log(rm1/rm2); + + + dn1.dt<- n[1]*rm1*(1+g2*f12) + dn2.dt<- n[2]*rm2*(1+g1*f21) + + return(list(c(dn1.dt,dn2.dt))) + } > > > parms<-c(c1=0.253, r1=0.0090,c2=0.289, r2=0.0097) > initialn<-c(2,2) > t.s<- seq(0,38, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) © L. S. Barreto, 2017. Theoretical Ecology 231 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

> matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Number/0,5 c.c.") > title("Competition MSB2: Pb+Pc") > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 209.3667 209.8236 > > > # Coefficients of competition > c1=parms[1] > r1=parms[2] > > c2=parms[3] > r2=parms[4] > k1=2/r1 > k2=2/r2 > > h<-matrix(c(out[,2],out[,3]),nrow=3801,ncol=2, byrow=F) > > > N1<-h[,1] > N2<-h[,2] > > N1a<-N1[-1] > N1b<-N1[-3801] > N2a<-N2[-1] > N2b<-N2[-3801] > d1=N1a-N1b > d2=N2a-N2b > > p1=N1a > p2=N2a > a1=c() > a2=c() > a1=(c1*p1*log(p1)-c1*log(k1)*p1+d1)/(p1*log(p2)) > a2=(c2*p2*log(p2)-c2*log(k2)*p2+d2)/(p2*log(p1)) > > > > cc=cbind(a1,a2) > t=1:3800 > screen(2) > matplot(t,cc, type='l', lw=2) > title('CC Pb, and Pc') > > cat("Pb: Pure= ", k1, "Mixed ",ye[1],"\n") Pb: Pure= 222.2222 Mixed 209.3667 > cat("Pc: Pure= ", k2, "Mixed ",ye[2],"\n") Pc: Pure= 206.1856 Mixed 209.8236 > cat("CC: a12= ", cc[3800,1], "a21=",cc[3800,2],"\n") CC: a12= -0.002824004 a21= 0.0009393544 > 232 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figure 13.8. Simulation of the mixture Pb+Pc. The scale of the time in the graphic of CC is day/100

The analysis of this simulation is as follows:  The sizes of the two populations are almost coincident, what mirrors their coevolution;  The CC are not constant; there is an initial phase with strong variation followed by a phase with CC almost stationary; the two time series of CC are virtully coincident;  Comparing the isolated population the size of Pb is reduced from 222 to 209. Pb is the dominated species;

 Consistently, the values of the CC a12 (effect of Pc on Pb) are always negative;  Comparing the isolated population, the size of Pc increases from 206 to 209. Pc is the dominant species;

 Consistently, the values of the CC a21 (effect of Pb on Pc) are negative till age 10, and after they become positive, but very small (Pb slightly benefits Pc). When negative, it is also

verified |a21|<|a12|, smaller negative impact of Pb on Pc then of Pc on Pb.

The three simulations of two species we presented gave place to the emergence of the following competitive hierarchy: Pc>Pb>Pa. The dominance that prevails in this the hierarchy is very weak, particularly the one of Pc over Pb.

Fourth simulation: Pa+Pb+Pc

> ##BC2_3_mAr > ##Pa+Pb+Pc > > rm(list=ls(all=TRUE)) > > baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] © L. S. Barreto, 2017. Theoretical Ecology 233 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

+ + + c2<-parms[3] + r2<- parms[4] + + c3<-parms[5] + r3<-parms[6] + + + rm1=-c1*log(r1)*exp(-c1*times) + rm2=-c2*log(r2)*exp(-c2*times) + rm3=-c3*log(r3)*exp(-c3*times) + + g=n[1]+n[2]+n[3]; + g1=n[1]/g;g2=n[2]/g;g3=n[3]/g; + f12=log(rm2/rm1);f13=log(rm3/rm1); + f21=log(rm1/rm2);f23=log(rm3/rm2); + f31=log(rm1/rm3);f32=log(rm2/rm2); + + dn1.dt<- n[1]*rm1*(1+g2*f12+g3*f13) + dn2.dt<- n[2]*rm2*(1+g1*f21+g3*f23) + dn3.dt<- n[3]*rm3*(1+g1*f31+g2*f32) + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > > parms<-c(c1=0.2970191, r1=0.0033233,c2=0.2532272, r2=0.0090313, c3=0.289, r3=0.0097) > initialn<-c(2,2,2) > t.s<- seq(0,38, by=0.01) > > library(deSolve) > out<- ode(y=initialn, times=t.s, baco2, parms=parms) > oute=out > matplot(out[,1], out[,-1], type="l", xlab="t", ylab="Number/0,5 c.c.") > title("Competition BACO2: Pa+Pb+Pc") > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=baco2,parms=parms,times=c(0,1000)) > ye<-ST2$y > ye [1] 270.9821 279.4320 281.1184 > > ########################## > > h<-matrix(c(out[,2],out[,3], out[,4]),3801,3) > > g<-seq(1,3801,100) > dat0<-matrix(c(h[g,]),39,3) > m<-c(dim(dat0)) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] 234 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

> N3a<-N3[-1] > N3b<-N3[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -5.5634 -0.9375 -0.1232 1.1123 3.7233

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 10.0398 1.0144 9.897 1.52e-11 *** N1b 4.3908 0.7725 5.684 2.22e-06 *** N2b -6.8942 0.9401 -7.334 1.70e-08 *** N3b 3.5531 0.2260 15.724 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.718 on 34 degrees of freedom Multiple R-squared: 0.9996, Adjusted R-squared: 0.9996 F-statistic: 2.836e+04 on 3 and 34 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -5.7757 -0.9892 -0.1281 1.1694 3.8738

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 10.3217 1.0587 9.749 2.23e-11 *** N1b 3.7454 0.8062 4.646 4.92e-05 *** N2b -6.4366 0.9812 -6.560 1.63e-07 *** N3b 3.7496 0.2358 15.899 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.793 on 34 degrees of freedom Multiple R-squared: 0.9996, Adjusted R-squared: 0.9996 F-statistic: 2.77e+04 on 3 and 34 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -6.5949 -1.1371 -0.1508 1.3414 4.4577

Coefficients: © L. S. Barreto, 2017. Theoretical Ecology 235 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Estimate Std. Error t value Pr(>|t|) (Intercept) 11.4362 1.2175 9.393 5.65e-11 *** N1b 4.4793 0.9271 4.831 2.84e-05 *** N2b -8.2166 1.1283 -7.282 1.97e-08 *** N3b 4.8142 0.2712 17.751 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.062 on 34 degrees of freedom Multiple R-squared: 0.9994, Adjusted R-squared: 0.9994 F-statistic: 2.057e+04 on 3 and 34 DF, p-value: < 2.2e-16

> > G=data.frame(c(coef(fit1), coef(fit2), coef(fit3))) > > library(MASS) > A=matrix(c( G[1,1] , G[2,1], G[3,1], G[4,1], G[5,1], G[6,1], G[7,1], G[8,1], G[9,1]), nrow=3, byrow=T) > A [,1] [,2] [,3] Community matrix [1,] 10.039791 4.390812 -6.894239 [2,] 3.553106 10.321675 3.745390 [3,] -6.436566 3.749608 11.436219 > E=-ginv(A) > eigen(A)$values [1] 17.4324736 13.4711142 0.8940971 > eigen(E)$values [1] -1.11844673 -0.07423291 -0.05736421 > round(E,4) [,1] [,2] [,3] [1,] -0.4953 0.3623 -0.4172 Matrix of total effects [2,] 0.3083 -0.3355 0.2958 [3,] -0.3799 0.3139 -0.4192 > > > cat("Pa Pure= ", 2/0.0033233, "Mixed ",ye[1],"\n") Pa Pure= 601.8115 Mixed 270.9821 > > cat("Pb: Pure= ", 2/0.0090313, "Mixed ",ye[2],"\n") Pb: Pure= 221.4521 Mixed 279.432 > > cat("Pc: Pure= ", 2/0.0097, "Mixed ",ye[3],"\n") Pc: Pure= 206.1856 Mixed 281.1184 236 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figura 13.10. Simulation of the mixture Pa+Pb+Pc. The scale of the time in the graphic of CC is day/100

The graphic, and written output of the simulation with Pa+Pb+Pc, let us conclude: ● It is possible to good fitting of MAR(1) to the data generated by the simulation; ● The matrix of total effects it is not intuitively predictable. During the process of competition emerges a complex network of direct, and indirect effects mirrored by this matrix. Referring solely to the process of competition, coherently, we verify the following:  The three populations, at equilibrium, have very close sizes;  The competitive hierarchy previously detected prevails;  As Pa is simulataneously receinving the dominance of both Pb, and Pc, its size (270) is smaller then in the mixtures with two species (331, 335);  As Pc is simulataneously receiving the benefits from both Pa, and Pb, its size (281) is greater then in the mixtures with two species (209, 337);  As Pb benefits from the presence of Pa, its size (279) is greater then in the misture Pc+Pb (209).

13.5 The Butterfly Effect

In subsection 12.2.2, we state that the greater is the proportion of the dominated species, the more is the interaction beneficial to the dominant specie, and the less depressive is the interaction to the dominated specie. We also declared that, in this chapter, we will illustrate deterministic chaos, or deterministic butterfly effect. In the present situation, this is understood as a high sensitivity of the dynamics of the interaction to small changes in the initial values of the populations of the competitors. © L. S. Barreto, 2017. Theoretical Ecology 237 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

The following script in R simulates the butterfly effect, applying MSB2 the mixture Pa+Pc. Figure 13.11 is obtained. In this figure, it can be seen that a small difference between the initial values of two simulations is amplified by the divergence of the respective trajectories.

> ##butterPaPc > ini<-matrix(c(30,30,30,35,35,30,30,45,45,30,30,50,50,30),7,2,byrow=T) > colnames(ini)=c('Pa','Pc') > ini Pa Pc [1,] 30 30 Initial populations used [2,] 30 35 [3,] 35 30 [4,] 30 45 [5,] 45 30 [6,] 30 50 [7,] 50 30 > #criating a graphic with a with line to receive the other lines > x<-c(30:2800);y<-c(30:2800) > plot(x,y,main='Butterfly effect',type='l',col=0,xlab='Pa',ylab='Pc') > grid(nx = NULL, ny = NULL, col = "lightgreen", lty = "dotted", + lwd = par("lwd")) > cr=c(1,2,3,4,5,6,11) > for (i in 1:7){ + + baco2<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + r1<- parms[2] + + + c2<-parms[3] + r2<- parms[4] + + + cop1<-c1*log(r1) + cop2<-c2*log(r2) + + dn1.dt<-n[1]*cop1*exp(c1*times)*(1+n[2]/ (n[1]+n[2])*log((cop2*exp(c2*times))/(cop1*exp(c1*times)))) + dn2.dt<- n[2]*cop2*exp(c2*times)*(1+n[1]/ (n[1]+n[2])*log((cop1*exp(c1*times))/(cop2*exp(c2*times)))) + return(list(c(dn1.dt,dn2.dt))) + } + + + parms<-c(c1=-0.297, r1=0.0033, c2=-0.253, r2=0.0090) + initialn<-ini[i,] + t.s<- seq(1,40, by=1) + + library(deSolve) + out<- ode(y=initialn, times=t.s, baco2, parms=parms) + + lines(out[,2],out[,3], col=cr[i],lw=2) + + } 238 © L. S. Barreto, 2017. Theoretical Ecology 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

Figura 13.11. Illustration of the deterministic butterfly effect with seven mixtures Pa+Pc. For more details see the text

13.6 Evaluation of the Simulations

Although the results presented in this chapter revealed high level of internal coherence, it is necessary to confront them with the available empirical evidence. It is not possible for me to made cultures of the simulated mixtures in the lab, but this can be done by somme readers. The empirical evidence that I had access is the following:  In nature, populations that coevolved, when in mixtures, have very close sizes. For instance, this is verified in European forests with Fagus sylvaticand, Larix decidua, and Picea abies. This also happens in North American forests with Picea sitchensis, Pseudotsuga menziesii, and Tsuga heteropylla (Barreto, 2010: figures 15.1 e 15.5);  The variation of the CC, with two phases, was already detected by Gause, in 1934. In the first phase, the values of the RVR are high, the appropriation of the resources in intense, and the same happens with the mutual interference among the competitors, giving rise to CC with high absolute values. When the values of the RVR became low, and asymptotic the intensity of competition becomes residual, and this is reflected in the CC;  In Gause’s experiments with Pa+Pc, the former specie eliminates Pc, because this specie is sensitive to metabolic products that are produced. If these products are eliminated, Pc becomes the dominant species;  In mixtures with Pa+Pb, the populations coexists in a stable equilibrium (Hutchinson, 1978:122)  Vandermeer (1969) carried on experiments with mixtures of the three species, and Blephanaria. In this mixture, P. bursaria e Blephanaria became extinct about day 32, and Pa, and Pc attained very close sizes, having each population about 200 individuals;  The same author detected asymmetric competition in the mixture Pa+Pc. Now, we move to the analysis of mutualism. © L. S. Barreto, 2017. Theoretical Ecology 239 13 Virtual Laboratory: Simulations of Experiments with Paramecium sps.

13.7 References, and Related Bibliography

Barreto, L. S., 2005a. Theoretical Ecology. A Unified Approach. E-book. Costa de Caparica. Include in the CD. Barreto, L. S., 2005b. Gause’s Competition Experiments with Paramecium sps. Revisited. Research Paper SB-01/05. Departamento the Engenharia Florestal, Instituto Superior de Agronomia, Lisboa. Revised version submitted to Silva Lusitana. Barreto, L. S., 2011. Árvores e Arvoredos. Geometria e Dinâmica. Edição do autor, Costa de Caparica. Include in the CD. Gause, G. F. 1934. The Struggle for Existence. Williams and Wilkins, Baltimore. In 2003, Dover Publications published this oook. Available in the intenet. Hastings, A., 2004. Old wine in a new bottle. Trends Ecol. Evol., 19(2):64-65. Hutchinson, G. E., 1978. An Introduction to Population Ecology. Yale University Press. Roughgarden, J., 1998. Primer of Ecological Theory. Prentice Hall. Vandermeer, J. H., 1969. The Competitive Structure of Communities. An Experimental Approach with Protozoa. Ecology 50:362-371. 240 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

14 Modelling Mutualism

14.1 Introduction

Mutualism occurs when the interaction of two species is beneficial for both; in same cases, the interaction may be indispensable for their survival. Mutualism is accepted as a very important and common species relation, in nature. In this chapter we introduce models for two basic mutualistic relations that a specie can have with another specie: facultative mutualism (FM), and obligatory mutualism (OM). Using these two basic models, first we introduce mutualism between two species. The following systems are modelled: FM+ FM, OM +OM, FM+ OM. We move to systems with three mutualists. The basic structure comprehends two species that do not interact between them, but shares a mutualist. It can be seen as two plant species that have a common pollinator, and do not interact with each other. The interactions that the ´pollinator’ has with the two plants contemplates the three combinations already introduced: FM+ FM, OM +OM , FM+ OM. The presentation of the two basic models is more easily understood when they are integrated in the systems FM+ FM, OM +OM. We will apply this methodology. Two recommended readings about this interaction are Boucher (1985), and Bascompte, and Jordano (2014).

14.2 A Gompertzian Model for Facultative Mutualism (FM+FM)

14.2.1 Assumptions The basic assumptions of this model are the following ones:  When isolated, the mutualists have Gompertzian growth;  The growth of each mutualist is accelerated, and its carrying capacity increased by the presence of the other.

14.2.2 The Model We model this interaction with the following system of ODE:

(14.1)

(14.2)

We call a12, and a21 the coefficients of facultative mutualism (cfm).

In the equation of mutualist i, when mutualist j is absent (yj=0), the expression exp(-aij yj) is equal to 1 and occurs ci yi (ln yif – ln yi), that is, it reduces to the EDO of the EGZ. The number two has no biological meaning, and it is used to obtain the desired behaviour. This number is equal to the number of mutualists in the system. The equation for FM can be generalized for a system of n species. For instance, for species 1 is written:

Although we already know that the fixed point of this system is a stable node, we will present its analysis.

14.2.3 Model Analysis It is not possible to obtain an explicit solution for the system of equations (14.1), and (14.2). To clarify the stability of their solutions, we show that the system is cooperative, recurring to Maxima:

In figure 14.1, we illustrate the effects of the cfm. The greater they are, the faster the species grow, and the higher is their carrying capacities. It is also evident that the product a12 a21=15>1 does not destabilize the system. 242 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

Figure 14.1. Two simulations of facultative mutualism with equations (14.1) e (14.2). The parameters are: c 1=0.05, c2=0.04, y1f=40 y2f=45. Both species have the same initial value: 10 individuals. The effect of the value of the coefficients of facultative mutualism is conspicuous; the greater they are, the greater are the carrying capacities of the populations

To finalize this section, we present a simulation with the system of equations (14.1), and (14.2), using R:

> rm(list=ls(all=TRUE)) > ##mutfac > > > library(deSolve) > library(rootSolve) > > > mutfac<-function(t=0,y, parms=NULL,...) { + + c1<-0.05 + k1<- 40 + a12<-0.005 + + c2<-0.04 + k2<- 45 + a21<--0.003 + + + dn1<- c1*y[1]*(log(k1*(2-exp(-a12*y[2])))-log(y[1])) + dn2<- c2*y[2]*(log(k2*(2-exp(-a21*y[1])))-log(y[2])) + + return(list(c(dn1,dn2))) + } > initialn<-c(10, 10) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, mutfac, parms=parms) > > split.screen(figs=c(1,2)) © L. S. Barreto, 2017. Theoretical Ecology 243 14 Modelling Mutualism

[1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass", ylim=c(10,51)) > title("Facultative mutualism")

> h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > > ######################### > > > y<-initialn > ST2 <- runsteady(y=y,func=mutfac,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 46.95356 38.19311 Fixed point > ############################## > > > > N1<-dat0[,1] > N2<-dat0[,2] > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > n<-m[1]-1 > > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.153728 -0.010731 0.006870 0.008563 0.068377

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.577916 0.026121 22.12 <2e-16 *** N1b 0.801868 0.005887 136.20 <2e-16 *** N2b 0.228220 0.007860 29.04 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0265 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.356e+07 on 2 and 296 DF, p-value: < 2.2e-16 244 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.118876 -0.009061 0.005812 0.007279 0.055261

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.548488 0.021267 25.79 <2e-16 *** N1b -0.135187 0.004793 -28.20 <2e-16 *** N2b 1.151641 0.006399 179.96 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02157 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.144e+07 on 2 and 296 DF, p-value: < 2.2e-16

> > > G=data.frame(c(coef(fit1), coef(fit2))) > G c.coef.fit1...coef.fit2.. 1 0.5779156 2 0.8018677 3 0.2282203 4 0.5484878 5 -0.1351865 6 1.1516410 > > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2] + N2.t1<-G[4,1]+G[5,1]*N[1]+G[6,1]*N[2] + + + c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(5, 5) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=2:1, ylim=c(10,51), xlab="Time", ylab="Biomass",) > lines(N1b, col='green') > lines(N2b, col='red') > > title("MAR(1)") > r<-c('Mutualist 1','Mutualist 2') > legend('topleft',paste(rev(r)),lty=2:1,col=1:2, bty='n') > © L. S. Barreto, 2017. Theoretical Ecology 245 14 Modelling Mutualism

We obtain also the graphic of the figure 14.2. The fitting of MAR(1) to the data generated by the system of ODE is good.

Figure 14.2. Simulation of facultative mutualism with equations (14.1), and (14.2). The fixed point is [ 46.953, 38.193].Comparison of the simulations of the same model (red), and fitted MAR(1)

14.3 A Gompertzian Model for Obligatory Mutualism (OM+OM)

14.3.1Assumptions The main assumptions of this model are:  The populations have Gompertzian dynamics;  The presence of each species is indispensable for the growth of the other;

 The size of population yi when specie yj is absent is zero; when yj is present the

equilibrium of yi is yif.

14.3.2 The Model The presented assumptions give rise to the following model:

(14.4)

(14.5)

The constant z controls how fast the specie becomes extinct in the absence of the other mutualist. We assume z=10-10, in all simulations made with the model. Specie i, in the absence of specie j,is modelled by::

y’1=-cz(ln yif-ln yi)<0 (14.6) and decreases monotonically . 246 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

Suppose a situation where the two mutualists coexist, but for any reason y2 becomes extinct. Then, y1 slowly declines to extinction. This occurrence is simulated by the short script in R:

> #modecline > R=c() > y1=19 #initial value > > s=seq(0,10, by=0.1) > for (t in s) { + d1=0.5*(-1)*y1*(log(20)-log(y1)) + y1=y1+0.1*d1 + R=c(R,y1) + } > > plot(s,R, xlab='Time', ylab='y1', type='l') > grid(nx = NULL, ny = NULL, col = "lightgreen", lty = "dotted", + lwd = par("lwd"), equilogs = TRUE) > >

The script produces the graphic of figure 14.3, where y1, an aged structured population, goes to extinction.

Figure 14.3. Simulation of the extinction of an obligatory mutualist, when the other is absent

For a system where specie y1 interacts with n species, its equation is:

(14.7)

Now, we move to the model analysis.

14.3.3 Model Analysis As it is not possible to obtain an explicit solution to the model, we show that it is a cooperative system: © L. S. Barreto, 2017. Theoretical Ecology 247 14 Modelling Mutualism

To conclude this section, we present a simulation with the system of equations (14.4), and (14.5), using R:

> rm(list=ls(all=TRUE)) > ##mutobrig > rm(list=ls(all=TRUE)) > > #the one with smaller vi act as dominant > > mutobri<-function(times,y,parms) { + n<-y + + + c1<-parms[1] + k1<- parms[2] + v1<-parms[3] + + c2<-parms[4] + k2<- parms[5] + v2<-parms[6] + 248 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

+ dn1.dt<- c1*(v1*n[2]-10^(-10))*n[1]*(log(k1)-log(n[1])) + dn2.dt<- c2*(v2*n[1]-10^(-10))*n[2]*(log(k2)-log(n[2])); + + + return(list(c(dn1.dt,dn2.dt))) + } > > parms<-c(c1=0.05, k1=20, v1=0.5, c2=0.04, k2=30, v2=0.2) > initialn<-c(10, 10) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, mutobri, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass", ylim=c(10,31)) > title("Obligatory mutualism") > > > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > > ######################### > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=mutobri,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 20 30 Fixed point > ############################## > > > > N1<-dat0[,1] > N2<-dat0[,2] > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > n<-m[1]-1 > > > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Residuals: Min 1Q Median 3Q Max -0.256702 0.002814 0.002817 0.002817 0.187268

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.748630 0.046056 81.39 <2e-16 *** N1b 0.846597 0.003710 228.22 <2e-16 *** N2b -0.022780 0.001228 -18.55 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02956 on 296 degrees of freedom Multiple R-squared: 0.9984, Adjusted R-squared: 0.9984 F-statistic: 9.152e+04 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.0195223 -0.0009185 -0.0009185 -0.0009173 0.0272931

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.5236527 0.0078985 66.3 <2e-16 *** N1b 0.1743465 0.0006362 274.1 <2e-16 *** N2b 0.8663446 0.0002106 4112.9 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.005069 on 296 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 3.847e+07 on 2 and 296 DF, p-value: < 2.2e-16

> ################# > > > ############################# > G=data.frame(c(coef(fit1), coef(fit2))) > G c.coef.fit1...coef.fit2.. 1 3.74862961 2 0.84659715 3 -0.02277964 4 0.52365270 5 0.17434646 6 0.86634455 > > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2] + N2.t1<-G[4,1]+G[5,1]*N[1]+G[6,1]*N[2] + + 250 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

+ c(N1.t1, N2.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) > N[1, ]<-c(5, 5) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=2:1, ylim=c(0,31), xlab="Time", ylab="Biomass",) > lines(N1b, col='green') > lines(N2b, col='red') > > title("MAR(1)") > r<-c('Mutualist 1','Mutualist 2') > legend('bottomright',paste(rev(r)),lty=2:1,col=1:2, bty='n') >

We obtain also the graphic of the figure 14.4. The fitting of MAR(1) to the data generated by the system of ODE is good.

Figure 14.4. Simulation of obligatory mutualism with equations (14.4), and (14.5). The fixed point is [20,30 ]. Comparison of the simulations of the same model, and fitted MAR(1)

14.4 A Gompertziano Model for Facultative, and Obligatory Mutualism

14.4.1 Assumptions The assumptions of this model associate the assumptions of the two previous models. Thus:  Both mutualists have a Gompertzian dynamics;

 The growth of mutualist y1 is accelerated, and its carrying capacity increased in the

 The presence of muyualist y1 is indispensable for the growth of mutualist y2;

 Species y2 in the absence of y1 extinguishes, and in its presence it attains y2f.

14.4.2 The Model The model associates equations (14.1), and (14.5):

(14.8)

(14.9)

14.4.3 Model Analysis As it is not possible to obtain an explicit solution to the model, we show that it is a cooperative system:

We close this section, presenting a simulation with the system of equations (14.8), and (14.9), using R:

> ##mutfacobl > > > library(deSolve) > library(rootSolve) > > > mutfac<-function(t=0,y, parms=NULL,...) { + + c1=0.5 + c2=0.3 + k1=20 252 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

+ k2=30 + a12=0.003 + a21=0.02 + + dn1<- c1*y[1]*(log(k1*(2-exp(-a12*y[2])))-log(y[1])) + dn2<- c2*(a21*y[1]-10^(-10))*y[2]*(log(k2)-log(y[2])); + + + return(list(c(dn1,dn2))) + } > > initialn<-c(2, 5) > t.s<- seq(1,300, by=0.1) > > > out<- ode(y=initialn, times=t.s, mutfac) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", xlab="Time", ylab="Biomass", ylim=c(0,31)) > title("Facultative & obligatory mutualism") > > h<-matrix(c(out[,2],out[,3]),2991,2) > > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,2) > > > ######################### > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=mutfac,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 21.72138 30.00000 > ##############################

> N1<-dat0[,1] > N2<-dat0[,2] > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > n<-m[1]-1 > > fit1<-lm(N1a ~ N1b+N2b) > fit2=lm(N2a ~ N1b+N2b) > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b)

Min 1Q Median 3Q Max -1.07207 0.00535 0.00539 0.00539 0.70772

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.483896 0.069581 64.441 < 2e-16 *** N1b 0.805874 0.004925 163.645 < 2e-16 *** N2b -0.009087 0.002054 -4.423 1.37e-05 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09655 on 296 degrees of freedom Multiple R-squared: 0.9956, Adjusted R-squared: 0.9956 F-statistic: 3.365e+04 on 2 and 296 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b)

Residuals: Min 1Q Median 3Q Max -0.258720 -0.008417 -0.008417 -0.008336 0.216591

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.178668 0.035530 5.029 8.57e-07 *** N1b 0.113093 0.002515 44.975 < 2e-16 *** N2b 0.912441 0.001049 869.848 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0493 on 296 degrees of freedom Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999 F-statistic: 1.076e+06 on 2 and 296 DF, p-value: < 2.2e-16 > > ############################# > > G=data.frame(c(coef(fit1), coef(fit2))) > G c.coef.fit1...coef.fit2.. 1 4.483895718 2 0.805874112 3 -0.009086788 4 0.178668263 5 0.113092790 6 0.912440598 > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2] + N2.t1<-G[4,1]+G[5,1]*N[1]+G[6,1]*N[2] + + + c(N1.t1, N2.t1) + } > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=2) 254 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

> N[1, ]<-c(5, 5) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=2:1, ylim=c(10,51), xlab="Time", ylab="Biomass",) > lines(N1b, col='green') > lines(N2b, col='red') > > title("MAR(1)") > r<-c('Mutualist 1','Mutualist 2') > legend('topleft',paste(rev(r)),lty=2:1,col=1:2, bty='n')

The script produces also figure 14.5. The fitting of MAR(1) to the data generated by the system of ODE is good.

Figure 14.5. Simulation of facultative, and obligatory mutualism with equations (14.8), and (14.9). The fixed point is [ 21.721, 30.000]. Comparison of the simulations of the same model (red), and fitted MAR(1)

14.5 First Gompertzian Model for a System with Three Mutualists

14.5.1 Context In nature, as there are food webs, there are mutualist networks. In figure 14.6, we exhibit a mutualist network. It is called a bipartite network because it has two disjunct sets of nodes or vertices, and the edges only connect nodes of different sets. The network in figure 14.6 is a pollination network obtained in Parque Nacional do Catimbau, Brasil, in a steppe formation called Caatinga (Bezerra, Machado, e Mello, 2009; http://www.nceas.ucsb.edu/interactionweb). The pollinators are the nodes in the upper row, and the plants (gathered in 13 taxonomic groups) are in the inferior set of nodes. The figure was obtained from package bipartite, of R (Butts, 2013). © L. S. Barreto, 2017. Theoretical Ecology 255 14 Modelling Mutualism

Pollinators

Links with width proportional to intensity

Plants

Figure 14.6. Mutualist network of Brazilian Caatinga. Figure obtained from R. For more details see the text

14.5.2 Assumptions

We consider a system of three mutualists, y1, y2 e y3, where y1 has a reciprocal interaction of facultative mutualism with f y2, and reciprocal interaction of obligatory mutualism with y3. The assumptions related to these types of mutualism, already presented, prevail. A network of the system is presented in figure 14.7.

Figure 14.7. Network of the first system with three mutualists. Facultativo=Facultative; Obrigatório= obligatory

14.5.3 The Model The model for this system is the following:

y’1=c1 (a13y3-z) y1 (log(k1 (2-exp(-a12*y2)))-log(y1)) (14.10) obligatory facultative

y’2=c2y2 (log(k2 (2-exp(-a21y1)))-log(y2)) Facultative (14.11) 256 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

y’3= c3 (a31y1-z) y3 (log(k3)-log(y3)) Obligatory (14.12)

The mutualist y1 in its ODE contains both obligatory, and facultative mutualism.

The equation for mutualist y2 only has facultative mutualism with y1.

The equation for mutualista y3 only exhibits the obligatory interaction with y1.

14.5.4 Model Analysis It is not possible to establish an explicit solution for the model, thus we will show that it is a cooperative system. To fulfil this desideratum, we mus show that the following equations are satisfied:

∂f ∂ f ∂ f ∂ f 1 ≥ 0, 1 ≥ 0 , 2 ≥ 0 , 3 ≥ 0 (14.13) ∂ y2 ∂ y3 ∂ y1 ∂ y1

These four equations are satisfied under the following conditions:  All parameters are positive  All variables are positive

 z

 ln(k3)-ln(y3)>0

 ln(k1*(2-exp(-a12*y2)))-ln(y1)>0

 2-exp(-a12*y2)>0 (-10)  a13*y3-10 >0

 2-exp(-a21*y1)>0 Now, we use wxMaxima to show that the model is a cooperative system. © L. S. Barreto, 2017. Theoretical Ecology 257 14 Modelling Mutualism

Using R, we present a simulation of the model, and use the output to carry on some analysis. 258 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

> rm(list=ls(all=TRUE)) > ##mutualismo3 > #y1 fac (->y2) e oblig(->y3) > rm(list=ls(all=TRUE)) > > ############ Parameters > > c1=0.5;c2=0.3;c3=0.25 > k1=20;k2=30;k3=25 > a12=0.0003;a21=0.0002 > a13=0.06;a31=0.05 > > #*************** Model, and solution > mutual3<-function(times,y,parms) { + n<-y + + dn1.dt=c1*(a13*n[3]-10^(-10))*n[1]*(log(k1*(2-exp(-a12*n[2])))- log(n[1])) + dn2.dt=c2*n[2]*(log(k2*(2-exp(-a21*n[1])))-log(n[2])) + dn3.dt=c3*(a31*n[1]-10^(-10))*n[3]*(log(k3)-log(n[3])) + + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > initialn<-c(2,5,4) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, mutual3, parms=parms) > > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", col=1:3, xlab="Time", ylab="Biomass",ylim=c(0,35)) > title("Mutualism") > r<-c('Fac&oblig','Facultative','Obligatory') > legend('bottomright',paste(rev(r)),lty=3:1,col=3:1, bty='n') > #****************** > > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=mutual3,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 20.17991 30.12084 25.00000 Fixed point > > #************************* > > h<-matrix(c(out[,2],out[,3], out[,4]),2991,3) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > > library(MASS) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] © L. S. Barreto, 2017. Theoretical Ecology 259 14 Modelling Mutualism

> > ###### MAR(1) > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.223843 0.004085 0.004085 0.004085 0.268284

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.335128 0.044727 7.493 7.9e-13 *** N1b 0.921974 0.007603 121.272 < 2e-16 *** N2b 0.113899 0.004214 27.030 < 2e-16 *** N3b -0.087816 0.003216 -27.306 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04637 on 295 degrees of freedom Multiple R-squared: 0.9996, Adjusted R-squared: 0.9996 F-statistic: 2.556e+05 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.4585 -0.0017 -0.0017 -0.0017 0.3064

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.868770 0.043541 88.853 < 2e-16 *** N1b -0.140218 0.007401 -18.945 < 2e-16 *** N2b 0.957465 0.004102 233.406 < 2e-16 *** N3b 0.009748 0.003131 3.114 0.00203 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04514 on 295 degrees of freedom Multiple R-squared: 0.9996, Adjusted R-squared: 0.9996 F-statistic: 2.381e+05 on 3 and 295 DF, p-value: < 2.2e-16

Call: lm(formula = N3a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.045172 0.000783 0.000783 0.000783 0.057669

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.6149010 0.0087012 70.67 <2e-16 *** N1b 0.2723574 0.0014790 184.15 <2e-16 *** N2b -0.0085991 0.0008198 -10.49 <2e-16 *** N3b 0.7658873 0.0006256 1224.17 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009021 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.439e+07 on 3 and 295 DF, p-value: < 2.2e-16

> > G=data.frame(c(coef(fit1), coef(fit2), coef(fit3))) > > ############# Matrix of total effects > > A=matrix(c( G[2,1] , G[3,1], G[4,1], G[6,1], G[7,1], G[8,1], G[10,1], G[11,1], G[12,1]), nrow=3, byrow=T) > A [,1] [,2] [,3] [1,] 0.9219745 0.113898700 -0.087815570 [2,] -0.1402175 0.957465080 0.009747808 [3,] 0.2723574 -0.008599139 0.765887281 > E=-ginv(A) > E [,1] [,2] [,3] Matrix of total effects [1,] -1.0307736 0.12154403 -0.119734015 [2,] -0.1546675 -1.02606747 -0.004674713 [3,] 0.3648171 -0.05474266 -1.263149005 > > ############# Simulation with MAR(1) > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2]+G[4,1]*N[3] + N2.t1<-G[5,1]+G[6,1]*N[1]+G[7,1]*N[2]+G[8,1]*N[3] + N3.t1<-G[9,1]+G[10,1]*N[1]+G[11,1]*N[2]+G[12,1]*N[3] + + c(N1.t1, N2.t1,N3.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=3) > N[1, ]<-c(2, 0.5, 0.8) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,35), xlab="Time", ylab="Biomass",) © L. S. Barreto, 2017. Theoretical Ecology 261 14 Modelling Mutualism

> lines(N1b, col='red') > lines(N2b, col='red') > lines(N3b, col='red') > title("MAR(1)") > r<-c('Fac&oblig','Facultative','Obligatory') > legend('bottomright',paste(rev(r)),lty=3:1,col=1, bty='n') > > #******************************************* > >

The script produces also figure 14.8. The fitting of MAR(1) to the data generated by the system of ODE is good. I suggest to the reader to comment on the matrix of total effects E.

Figure 14.8. Simulation of the model of equations (14.10), (14.11), and (14.12). The fixed point is [20.180, 30.121, 25.000]. Comparison of the simulations of the same model (red), and fitted MAR(1)

14.6 Second Gompertzian Model for a System with Three Mutualists

14.6.1 Assumptions

Let us consider a system with three mutualists, y1, y2 e y3, where y1 has a facultative mutualistic interaction with the other two species. The network of the model is similar to the one in figure 14.7, corresponding now the two edges to facultative interactions.

14.6.2 The Model The model of this system is written as:

(14.14)

(14.16)

14.6.3 Model Analysis It is not possible to establish an explicit solution for the model, thus we will show that it is a cooperative system. The system is cooperative if the following is verified:  All parameters, and variables are positive;

 3-exp(-a13*y3)-exp(-a12*y2) >0 )  log(k1*(2-e xp(-a12*y2 ))-log(y1)>0 (-10)  a13*y3-10 >0

 2-exp(-a21*y1)>0

 2-exp(-a31*y1)>0 Now, we usw wxMaxima to proof that the system is cooperative. © L. S. Barreto, 2017. Theoretical Ecology 263 14 Modelling Mutualism

Again, using R, we present a simulation of the model, and use the output to accomplish some analysis.

> rm(list=ls(all=TRUE)) > ##mutualismo3 > #3 facultative mutualists > rm(list=ls(all=TRUE)) > > ############ Parameters > 264 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

> c1=0.5;c2=0.3;c3=0.35 > > k1=20;k2=18;k3=25 > > a12=0.002;a13=0.005; > a21=0.004; > a31=0.003; > > #*************** Model, and solution > mutual3<-function(times,y,parms) { + n<-y + + dn1.dt=c1*n[1]*(log(k1*(3-exp(-a12*n[2])-exp(-a13*n[3])))-log(n[1])) + dn2.dt=c2*n[2]*(log(k2*(2-exp(-a21*n[1])))-log(n[2])) + dn3.dt=c3*n[3]*(log(k3*(2-exp(-a31*n[1])))-log(n[3])) + + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > initialn<-c(2,2,2) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, mutual3, parms=parms) > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > matplot(out[,1], out[,-1], type="l", col=1:3, xlab="Time", ylab="Biomass",ylim=c(0,35)) > title("Mutualism") > r<-c('y1','y2','y3') > legend('bottomright',paste(rev(r)),lty=3:1,col=3:1, bty='n') > #****************** > > #Fixed point > library(rootSolve) > y<-initialn > ST2 <- runsteady(y=y,func=mutual3,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 23.26700 19.59963 26.68552 > > #************************* > > h<-matrix(c(out[,2],out[,3], out[,4]),2991,3) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > > library(MASS) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > > ###### Modelo linear > > m<-c(dim(dat0)) > N1a<-N1[-1] © L. S. Barreto, 2017. Theoretical Ecology 265 14 Modelling Mutualism

> N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.39104 -0.00305 -0.00305 -0.00305 0.32055

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.18631 0.05310 60.00 <2e-16 *** N1b 1.35869 0.01597 85.06 <2e-16 *** N2b 0.64224 0.03504 18.33 <2e-16 *** N3b -0.90374 0.03675 -24.59 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04288 on 295 degrees of freedom Multiple R-squared: 0.9993, Adjusted R-squared: 0.9993 F-statistic: 1.397e+05 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.098442 -0.000634 -0.000634 -0.000634 0.087814

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.792556 0.012883 139.15 <2e-16 *** N1b 0.283406 0.003875 73.13 <2e-16 *** N2b 0.839600 0.008501 98.76 <2e-16 *** N3b -0.196442 0.008915 -22.03 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0104 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 2.625e+06 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b)

-0.183922 -0.001366 -0.001366 -0.001366 0.157088

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.294222 0.024540 93.49 <2e-16 *** N1b 0.580137 0.007382 78.58 <2e-16 *** N2b 0.268079 0.016194 16.55 <2e-16 *** N3b 0.211364 0.016983 12.45 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01981 on 295 degrees of freedom Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999 F-statistic: 1.277e+06 on 3 and 295 DF, p-value: < 2.2e-16

> > G=data.frame(c(coef(fit1), coef(fit2), coef(fit3))) > > ############# Simulation with MAR(1) > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2]+G[4,1]*N[3] + N2.t1<-G[5,1]+G[6,1]*N[1]+G[7,1]*N[2]+G[8,1]*N[3] + N3.t1<-G[9,1]+G[10,1]*N[1]+G[11,1]*N[2]+G[12,1]*N[3] + + c(N1.t1, N2.t1,N3.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=3) > N[1, ]<-c(2, 2, 2) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,35), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > lines(N3b, col='red') > title("MAR(1)") > r<-c('y1','y2','y3') > legend('bottomright',paste(rev(r)),lty=3:1,col=1, bty='n') > > > ############# Matrix of total effects > > > A=matrix(c( G[2,1] , G[3,1], G[4,1], G[6,1], G[7,1], G[8,1], G[10,1], G[11,1], G[12,1]), nrow=3, byrow=T) > A [,1] [,2] [,3] [1,] 1.3586943 0.6422391 -0.9037356 [2,] 0.2834058 0.8396004 -0.1964418 [3,] 0.5801373 0.2680788 0.2113640 > E=-ginv(A) > E [,1] [,2] [,3] © L. S. Barreto, 2017. Theoretical Ecology 267 14 Modelling Mutualism

[1,] -0.4019356 0.66025125 -1.10493046 [2,] 0.3036745 -1.41732184 -0.01882926 [3,] 0.7180463 -0.01458386 -1.67455566 >

From the script we obtain figure 14.9. The fitting of MAR(1) to the data generated by the system of ODE is good. I suggest to the reader to comment on the matrix of total effects E, and compare it with the same matrix of the previous model

Figure 14.9. Simulation of the model of equations (14.14), (14.15), and (14.16). The fixed point is [23.267, 19.600, 26.685]. Comparison of the simulations of the same model (red), and fitted MAR(1)

14.7 Third Gompertzian Model for a System with Three Mutualists

14.7.1 Assumptions

Let us consider a system with three mutualists, y1, y2 e y3, where y1 has an obligatory mutualistic interaction with the other two species. The network of the model is similar to the one in figure 14.7, corresponding now the two edges to obligatory interactions.

14.7.2 The Model The model of this system is written as:

(14.17)

(14.18)

(14.19) 14.7.3 Model Analysis The analysis of the model is similar to the two previous ones. 268 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

The system is positive if all parameters, and variables are positive, and also log(k 1)- log(y1)>0, log(k2)-log(y2)>0, log(k3)-log(y3)>0. The verification in wxMaxima is as follows:

We present a simulation of the model, and use the output to accomplish some analysis, using R.

> ##mutualismo3 > #y1 oblig (->y2), oblig(->y3) > rm(list=ls(all=TRUE)) > > ############ Parameters > > c1=0.5;c2=0.4;c3=0.35 > k1=15;k2=20;k3=25 > a12=0.0003;a21=0.02 > a13=0.06;a31=0.05 > > #*************** Model, and solution © L. S. Barreto, 2017. Theoretical Ecology 269 14 Modelling Mutualism

> mutual3<-function(times,y,parms) { + n<-y + + dn1.dt=c1*(a12*n[2]+a13*n[3]-10^(-10))*n[1]*(log(k1)-log(n[1])) + dn2.dt=c2*(a21*n[1]-10^(-10))*n[2]*(log(k2)-log(n[2])) + dn3.dt=c3*(a31*n[1]-10^(-10))*n[3]*(log(k3)-log(n[3])) + + return(list(c(dn1.dt,dn2.dt,dn3.dt))) + } > > initialn<-c(7,7,7) > t.s<- seq(1,300, by=0.1) > > library(deSolve) > out<- ode(y=initialn, times=t.s, mutual3, parms=parms) > > > split.screen(figs=c(1,2)) [1] 1 2 > screen(1) > > matplot(out[,1], out[,-1], type="l", col=1:3, lw=2, + xlab="Time", ylab="Biomass",ylim=c(0,26)) > title("Mutualism") > r<-c('y1','y2','y3') > legend('bottomright',paste(rev(r)),lty=3:1,col=3:1, bty='n') > #****************** > > #Fixed point > library(rootSolve) Warning message: package ‘rootSolve’ was built under R version 3.3.2 > y<-initialn > ST2 <- runsteady(y=y,func=mutual3,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye [1] 15 20 25 > > #************************* > > h<-matrix(c(out[,2],out[,3], out[,4]),2991,3) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,3) > > library(MASS) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > > ###### MAR(1) > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] 270 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

> > fit1<-lm(N1a ~ N1b+N2b+N3b) > fit2=lm(N2a ~ N1b+N2b+N3b) > fit3=lm(N3a ~ N1b+N2b+N3b) > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.179129 -0.001973 -0.001973 -0.001936 0.148164

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.760297 0.050483 34.87 <2e-16 *** N1b 1.014961 0.007632 132.99 <2e-16 *** N2b 0.034737 0.002689 12.92 <2e-16 *** N3b -0.107099 0.004551 -23.53 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02237 on 295 degrees of freedom Multiple R-squared: 0.9987, Adjusted R-squared: 0.9987 F-statistic: 7.419e+04 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.0127487 -0.0002425 -0.0002425 -0.0002366 0.0105722

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.5714773 0.0042506 134.45 <2e-16 *** N1b 0.0706118 0.0006426 109.88 <2e-16 *** N2b 0.8925732 0.0002264 3942.90 <2e-16 *** N3b 0.0207249 0.0003832 54.09 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.001884 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.013e+08 on 3 and 295 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b)

Residuals: Min 1Q Median 3Q Max -0.045366 -0.000639 -0.000639 -0.000628 0.043299

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0935820 0.0156873 5.965 6.97e-09 *** © L. S. Barreto, 2017. Theoretical Ecology 271 14 Modelling Mutualism

N1b 0.4149492 0.0023716 174.964 < 2e-16 *** N2b 0.0110461 0.0008355 13.221 < 2e-16 *** N3b 0.7384759 0.0014142 522.178 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.006953 on 295 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 7.768e+06 on 3 and 295 DF, p-value: < 2.2e-16

> > G=data.frame(c(coef(fit1), coef(fit2), coef(fit3))) > > ############# Simulation with MAR(1) > > bc4<-function(N) { + + N1.t1<-G[1,1]+G[2,1]*N[1]+G[3,1]*N[2]+G[4,1]*N[3] + N2.t1<-G[5,1]+G[6,1]*N[1]+G[7,1]*N[2]+G[8,1]*N[3] + N3.t1<-G[9,1]+G[10,1]*N[1]+G[11,1]*N[2]+G[12,1]*N[3] + + c(N1.t1, N2.t1,N3.t1) + } > > > t<-300 > > N<-matrix(NA,nrow=t+1, ncol=3) > N[1, ]<-c(2, 0.5, 0.8) > for (i in 1:t) N[i+1, ]<-bc4(N[i, ]) > > screen(2) > matplot(0:t, N, type='l', col=1, ylim=c(0,26), xlab="Time", ylab="Biomass",) > lines(N1b, col='red') > lines(N2b, col='red') > lines(N3b, col='red') > title("MAR(1)") > r<-c('y1','y2','y3') > legend('bottomright',paste(rev(r)),lty=3:1,col=1, bty='n') > > #******************************************* > ############# Matrix of total effects > > A=matrix(c( G[2,1] , G[3,1], G[4,1], G[6,1], G[7,1], G[8,1], G[10,1], G[11,1], G[12,1]), nrow=3, byrow=T) > A [,1] [,2] [,3] [1,] 1.01496128 0.03473718 -0.10709950 [2,] 0.07061184 0.89257324 0.02072491 [3,] 0.41494917 0.01104610 0.73847591 > E=-ginv(A) > E [,1] [,2] [,3] [1,] -0.93219329 0.037965405 -0.13625937 [2,] 0.06160536 -1.123254489 0.04045799 [3,] 0.52287735 -0.004531131 -1.27818142 > > 272 © L. S. Barreto, 2017. Theoretical Ecology 14 Modelling Mutualism

The script creates also figure 14.10. The fitting of MAR(1) to the data generated by the system of ODE is good. I suggest to the reader to comment on the matrix of total effects E, and compare it with the same matrices of the previous models. What is the sensitivity of the matrix E to variations of the parameters? And initial values?

Figure 14.10. Simulation of the model of equations (14.17), (14.18), and (14.19). The fixed point is [15, 20, 25]. Comparison of the simulations of the same model (red), and fitted MAR(1)

The three models, we presented in sections 14.5 to 14.7, let foresee the capability of our basic models to be combined in models with much more variables in each set of species, applicable to larger mutualistic networks, such as the one of figure 14.6.

14.8 References, and Related Bibliography

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. “e-book”. Costa de Caparica. Include in this CD. Bascompte, J. e P. Jordano, 2014. Mutualistic Networks. Princeton University Press, Princeton. Bezerra, E.L.S., I.C.S. Machado e M.A.R. Mello, 2009. Pollination networks of oil-ﬂowers: atiny world within the smallest of all worlds. Journal of Animal Ecology, 78:1096–1101. Boucher, D. H., 1985. The Biology of Mutualism. Ecology and Evolution. Oxford University Press, New York. Brauer, F. e C. Castillo-Chávez, 2001. Mathematical Models in Population Biology and Epidemiology. Springer, New York. Butts, Carter T., 2013. sna: Tools for Social Network Analysis. R package version 2.3-1. http://CRAN.R-project.org/package=sna Craig A. J. F. K. , B. T. Bonnevie e P. E. Hulley, 2011. Honeyguide puzzles. Afring News 40:16-18 . http://safring.adu.org.za/afring_news_current.php Dean, A. M., 1982. A Simple Model of Mutualism. The American Naturalist, 121:409-317. Friedmann, H., 1955. The Honeyguides. United States National Museum Bulletin, 208: 1-292. Holland, J. N. e D. L. DeAngelis, 2010. Consumer-Resource Approach to the Density-dependent Population Dynam- ics of Mutualism. Ecology , 91:1286-1295 . © L. S. Barreto, 2017. Theoretical Ecology 273 14 Modelling Mutualism

May, R.M., 1981. Models for two interacting populations. Em R.M. May, compilador, Theoretical Ecology: Prin- ciples and Applications. 2nd edition. Blackwell Scientiﬁc, Oxford. Páginas 78–104. 274 © L. S. Barreto, 2017. Theoretical Ecology PART III

PART III

Community, and Ecosystem

In this third part of the book, I will integrate the models for interactions in a model of a community. An example of such a model will be constructed, and used to illustrate some attributes of the community. After, the model of the community will be extended to a model for the ecosystem. The main tenets we sustain are the followings: ➢ Communities, and ecosystems are linear stochastic dynamic systems that can be modelled by MAR(1). ➢ Analysis of the community, and the ecosystem that only consider the trophic interaction (the food web) give incomplete, and incorrect information about these two biosystems. ➢ The matrix of total positive effects is a tool that generates very elucidative information about the community, and the ecosystem. ➢ Probably, the holistic approach is the most fecund analytic strategy for communities, and ecosystems. This Part of the book culminates the integrated modelling effort announced in section 1.1. Here, the hierarchy of models that start at organism level attains its peak, the models for community, and ecosystem, without violating the principle that the model for the biosystem of level n+1 is an expansion of the model for the system of level n, without formal or conceptual discontinuities. To the ontological hierarchy of biosystems corresponds a conceptual hierarchy of linked mathematical models. © L. S. Barreto, 2017. Theoretical Ecology 275 15 Conceptual, and Mathematical Models for the Community

15 Conceptual, and Mathematical Models for the Community

15.1 Introduction

In this chapter, and in the next one, we will use a modelling, and simulative approach (M&S). In the last decades this methodology of inquiry has been intensively used to study the structure, dynamics, and evolution of communities. Several factors contributed for this expansion of M&S:  The complexity of the community, and the impossibility of the execution of projects for its study with very long duration;  The urgency in the implementation of sustainable resources management;  The need for immediate biodiversity conservation;  The emergence of the modern computers with their well known capabilities. This situation originated a variety of models for the community that are classified in figure 15.1.

Figure 15.1. A tentative classification of the models used in the study of the community. This figure benefits from the reading of Brännström et al. (2012)

The models of our interest are non evolutionary dynamic models of the community. Virtually, they started with the pioneer work of Lotka, and Volterra. Already, the models of Part II belong to this type of models. With very rare exceptions, the models that have been used to study the community reveal some shortcomings, such as, the only interaction considered is the food web, very small number 276 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community of species, and the use of small modules of patterns of webs. The model we introduce ahead avoids these inconvenients. A review, and analysis of the use of non evolutionary dynamic models in community ecology can be found in Brose, and Dunne (2009). See also Brännström et al. (2012). Both texts have an ample bibliography. Reflections about the use of models as tools for scientific research can be found in Frigg (2012) e Godfrey-Smith (2009).

15.2 The Nature of Communities, and Ecosystems

Given the hierarchical integration of the proposed theory, and the results of Part II, we deduce that the community can also be modelled by MAR(1). This conducted us to the following statement about the nature of the community:

Communities can be conceptualized as linear stochastic dynamic systems (LSDS) that can be modelled by MAR(1).

In this chapter will sustain this conjecture, and we will explore its admitted correctness. There is an abundant literature about LSDS, and they have an ample application in several scientific areas (e.g., Petris, Petrone, and Campagnoli, 2009). In the next chapter, we will sustain that ecosystems are also LSDS that can be modelled by MAR(1). These assertions legitimate the application of the models of state space analysis to the study of communities, and ecosystems. The all structure of our attempt to achieve a unified theory that embraces from organisms to ecosystems, is graphically sketched in figure 15.2. © L. S. Barreto, 2017. Theoretical Ecology 277 15 Conceptual, and Mathematical Models for the Community

Chapters 2 & 3

Gompertzian model of Space Subsection 4.4.2 organisms

Gompertzian model of Subsection 4.4.1 populations Metapopulation

Part II Gompertzian Models of models of community Metacommunity interactions

Environmental Models of Landscape factors ecosystem ecology

Figure 15.2. Flowchart of the hierarchy of models corresponding to the ontological hierarchy of biosystems

15.3 A Model of a Community

In this section we introduce a model of a community that integrates the results of the previous parts, particularly Part II, as illustrated in figure 15.2. The model evinces the following attributes: ● More than one type of interaction are included in the model; ● It originate time series of the variables that permit the fitting of MAR(1), and the establishment of the matrix of total effects; ● It can receive detailed modelling procedures without theoretical discontinuities. The network of the community we model is exhibited in figure 15.3. The interactions present in the community are: • Herbivory; • Competition; • Facultative mutualism; • Obligatory mutualism; • Predation; • Omnivory. 278 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

Consumer resource Competition Obligatory mutualism Facultative mutualism

Figure 15.3. Simplified network of the modelled community. For details see the text

From figure 15.3, we can infer the following:

• Variable y2 refers to the vegetation;

• Variable y10 is a top predator (can also be a parasite);

• Specie y8 is omnivore and reveals intra-trophic predation, as he consumes y9;

• There is an interaction of obligatory mutualism between species y3 e y4;

• There is an interaction of facultative mutualism between species y6 e y7; The size of the populations are measured using an arbitrary unit of biomass B. There are more competitive interactions then those represented in figure 15.3. We sup- pressed them to avoid an almost illegible network. The complementary information about the prevailing competition is as follows: • As the species have a Gompertzian dynamics they are submitted to intraspecific competition; • The interspecific competition between herbivores is asymmetric; • The species of each mutualistic interaction do not compete with each other;

• Specie y3 benefits from the presence of species y5, y6, y7;

• Specie y4 benefits from the presence of species y5, y6, y7;

• Specie y5 benefits from the presence of species y6, y7 but is depressed by species y3, y4;

• Specie y6 is depressed by y3, y4, y5;

• Specie y7 is depressed by y3, y4, y5. The constructed community do not pretend to represent any particular community, and the choice of the interactions only serves to our explanatory purposes. The same is applicable to the parametrisation of the model. One more clarification about units of the state variables, and of flows: © L. S. Barreto, 2017. Theoretical Ecology 279 15 Conceptual, and Mathematical Models for the Community

• The values of the state variables yi are expressed in the adopted units (energy, biomass) per area or volume unit (e.g., kg hectare-1); • The flows between the compartments corresponding to the state variable refer to units of biomass or energy per unit of area or volume per unit of time (e.g., kg hectare-1 year-1); • We assume that biomass is expressed in a hypothetical unit B. The script of the simulator, in R, is inserted bellow. For commodity, In the script, we assume variables y1 to y9 (y1, y2, … y9), instead of y2 to y10. Here is the simulation:

> rm(list=ls(all=TRUE)) > library(deSolve) > library(MASS) > > modul<-function(times,y,parms) { + n<-y + + r<-c(0.05, 0.069, 0.061, 0.057, 0.055, 0.05, 0.03, 0.035, 0.02) + b<-c(20, 0.2, 0.21, 0.28, 0.29, 0.27) + b7<-c(b[1], b[2], b[3])/3 + b8<-c(b[4], b[5], b[6])/2.5 + bm7<-mean(b7)/5 + bm8<-mean(b8)/4 + b9<-c(bm7, bm8) + + + #Plant consumption by herbivores, and omnivore + ex<-sum( y[1]*y[2]/(70+0.2*y[1]), y[3]*y[1]/(90+0.3*y[1]), y[4]*y[1]/ (90+0.3*y[1]), y[5]*y[1]/(80+0.3*y[1]), y[6]*y[1]/(75+0.3*y[1]), y[7]*y[1]/ (83+0.3*y[1])) + + #herbivores:competition, and predation + #initial/final<-initials/(bi*resouce) + + a1n<-c(0, 0, 0.02, 0.03, 0.04) + y1n<-log(c(y[2], y[3], y[4], y[5], y[6])) + a2n<-c(0, 0, 0.01, 0.02, 0.03) + y2n<-log(c(y[2], y[3], y[4], y[5], y[6])) + a3n<-c(-0.04, -0.03, 0, 0.01, 0.02) + y3n<-log(c(y[2], y[3], y[4], y[5], y[6])) + a4n<-c(-0.04, -0.03, -0.02, 0, 0) + y4n<-log(c(y[2], y[3], y[4], y[5], y[6])) + a5n<-c(-0.04, -0.03, -0.02, 0, 0) + y5n<-log(c(y[2], y[3], y[4], y[5], y[6])) + + #Competition effects + comp2<-sum(a1n*y1n) + comp3<-sum(a2n*y2n) + comp4<-sum(a3n*y3n) + comp5<-sum(a4n*y4n) + comp6<-sum(a5n*y5n) + + #consumptions of predators y[7], y[8], and y[9] + prey7<-y[7]*c(y[2]/(50+0.2*y[2]), y[3]/(60+0.3*y[3])) + prey8<-y[8]*c(y[4]/(50+0.21*y[4]), y[5]/(60+0.3*y[5]), y[6]/ (70+0.3*y[6])) + prey9<-c(y[9]*y[7]/(30+0.2*y[7]), y[9]*y[8]/(40+0.3*y[8])) 280 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

+ #carrying capacity of omnivore y[7] + k7<-sum(b7*c(y[1], y[2], y[3])); + #carrying capacity of predator y[8] + k8<-sum(b8*c(y[4], y[5], y[6])) + # carrying capacity of top predator + k9<-sum(b9*c(y[7], y[8])) + + #ODE of each specie + dy1.dt<-y[1]*r[1]*(log(6000)-log(y[1]))-ex + dy2.dt<-y[2]*r[2]*(4*y[3]-10^(-10))*(log(b[2]*y[1])-log(y[2])+comp2)- prey7[1] + dy3.dt<-y[3]*r[3]*(3*y[2]-10^(-10))*(log(b[3]*y[1])-log(y[3])+comp3)- prey7[2] + dy4.dt<-y[4]*r[4]*(log(b[4]*y[1])-log(y[4])+comp4)-prey8[1] + dy5.dt<-y[5]*r[5]*(log(b[5]*y[1]*(2-exp(-0.003*y[6])))-log(y[5]) +comp5)-prey8[2] + dy6.dt<-y[6]*r[6]*(log(b[6]*y[1]*(2-exp(-0.03*y[6])))-log(y[6])+comp6)- prey8[3] + dy7.dt<-y[7]*r[7]*(log(k7)-log(y[7]))-prey9[1] + dy8.dt<-y[8]*r[8]*(log(k8)-log(y[8]))-prey9[2] + dy9.dt<-y[9]*r[9]*(log(k9)-log(y[9])) + + return(list(c(dy1.dt,dy2.dt,dy3.dt, dy4.dt, dy5.dt, dy6.dt,dy7.dt,dy8.dt, dy9.dt ))) + } > > initials<-c(300, 18, 18, 18, 18, 18, 6, 6, 2) > t.s<- seq(1, 300, by=0.1) > > > out<- ode(y=initials, times=t.s, modul) > > split.screen(figs=c(2,2)) [1] 1 2 3 4 > > screen(1) > matplot(out[,2], type='l',xlab="Time", ylab="B", main='Plant') > > ys<-cbind(out[,3],out[,4],out[,5],out[,6],out[,7]) > screen(2) > #windows() > matplot(ys, type='l',xlab="Time", ylab="B", main='Herbivores') > > ys2<-cbind(out[,8],out[,9]) > screen(3) > #windows() > matplot(ys2, type='l',xlab="Time", ylab="B", main='Omnivore, and predador') > > screen(4) > #windows() > matplot(out[,10], type='l',xlab="Time", ylab="B", main='Top predator') > > ##################### > res<-out[2991,] > res<-res[-1] > #Fixed points > round(res,4) 1 2 3 4 5 6 7 8 9 16.7093 3.3321 3.2536 4.1940 4.1334 4.2708 6.0172 0.2004 2.7353 The dynamics of the populations are in figure 15.4. © L. S. Barreto, 2017. Theoretical Ecology 281 15 Conceptual, and Mathematical Models for the Community

Figure 15.4. Dynamics of the populations in the simulated community. See the fixed point in the script output

Now, we fit MAR(1), and for the sake of completeness display all results:

>h<-matrix(c(out[,2],out[,3], out[,4],out[,5], out[,6],out[,7], out[,8],out[,9],out[,10]),2991,9) > > g<-seq(1,2991,10) > dat0<-matrix(c(h[g,]),300,9) > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > N4=dat0[,4] > N5<-dat0[,5] > N6<-dat0[,6] > N7<-dat0[,7] > N8=dat0[,8] > N9<-dat0[,9] > > > ###################################### > ###### MAR(1) > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] 282 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

> N4a<-N4[-1] > N4b<-N4[-m[1]] > N5a<-N5[-1] > N5b<-N5[-m[1]] > N6a<-N6[-1] > N6b<-N6[-m[1]] > N7a<-N7[-1] > N7b<-N7[-m[1]] > N8a<-N8[-1] > N8b<-N8[-m[1]] > N9a<-N9[-1] > N9b<-N9[-m[1]] > > > ########## > fit1<-lm(N1a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit2=lm(N2a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit3=lm(N3a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit4=lm(N4a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit5<-lm(N5a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit6=lm(N6a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit7=lm(N7a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit8=lm(N8a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > fit9=lm(N9a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b) > > ########### > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.47682 -0.09047 0.04547 0.06102 0.93877

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 33.285683 0.790416 42.112 < 2e-16 *** N1b 0.347295 0.002134 162.736 < 2e-16 *** N2b 4.504671 0.631570 7.132 7.94e-12 *** N3b -4.387654 0.636905 -6.889 3.51e-11 *** N4b -28.261729 0.580452 -48.689 < 2e-16 *** N5b 43.004949 0.896550 47.967 < 2e-16 *** N6b -13.508025 0.514507 -26.254 < 2e-16 *** N7b -1.190559 0.061595 -19.329 < 2e-16 *** N8b -4.654604 0.153758 -30.272 < 2e-16 *** N9b -6.068349 0.170763 -35.537 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1415 on 289 degrees of freedom Multiple R-squared: 0.9995, Adjusted R-squared: 0.9995 F-statistic: 6.631e+04 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + © L. S. Barreto, 2017. Theoretical Ecology 283 15 Conceptual, and Mathematical Models for the Community

N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.211271 -0.019249 0.008466 0.011719 0.312458

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.7391130 0.2859784 16.572 < 2e-16 *** N1b 0.1051880 0.0007721 136.230 < 2e-16 *** N2b -1.7681232 0.2285066 -7.738 1.70e-13 *** N3b 1.8547938 0.2304371 8.049 2.19e-14 *** N4b -4.9312007 0.2100120 -23.481 < 2e-16 *** N5b 6.0653462 0.3243783 18.698 < 2e-16 *** N6b -0.7860411 0.1861524 -4.223 3.24e-05 *** N7b -0.2540763 0.0222856 -11.401 < 2e-16 *** N8b -1.4109540 0.0556309 -25.363 < 2e-16 *** N9b -0.9274045 0.0617832 -15.011 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05118 on 289 degrees of freedom Multiple R-squared: 0.9993, Adjusted R-squared: 0.9993 F-statistic: 4.491e+04 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.216901 -0.019055 0.008409 0.011658 0.308571

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.5402723 0.2907099 15.618 < 2e-16 *** N1b 0.1061969 0.0007849 135.298 < 2e-16 *** N2b -2.6791457 0.2322873 -11.534 < 2e-16 *** N3b 2.7795916 0.2342497 11.866 < 2e-16 *** N4b -4.8138005 0.2134866 -22.548 < 2e-16 *** N5b 5.7471902 0.3297451 17.429 < 2e-16 *** N6b -0.5899609 0.1892323 -3.118 0.00201 ** N7b -0.2505816 0.0226543 -11.061 < 2e-16 *** N8b -1.4338246 0.0565513 -25.354 < 2e-16 *** N9b -0.8911924 0.0628054 -14.190 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05203 on 289 degrees of freedom Multiple R-squared: 0.9993, Adjusted R-squared: 0.9993 F-statistic: 4.501e+04 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit4)

Call: lm(formula = N4a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b) 284 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

Residuals: Min 1Q Median 3Q Max -0.0164918 -0.0013572 -0.0009958 0.0019750 0.0218602

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.682e-01 1.910e-02 24.507 < 2e-16 *** N1b 4.357e-03 5.158e-05 84.464 < 2e-16 *** N2b -2.526e-01 1.526e-02 -16.549 < 2e-16 *** N3b 2.631e-01 1.539e-02 17.092 < 2e-16 *** N4b 1.331e+00 1.403e-02 94.872 < 2e-16 *** N5b -4.219e-01 2.167e-02 -19.472 < 2e-16 *** N6b -8.939e-02 1.244e-02 -7.188 5.63e-12 *** N7b 3.308e-03 1.489e-03 2.222 0.027 * N8b 2.701e-02 3.716e-03 7.268 3.42e-12 *** N9b 5.778e-02 4.127e-03 14.001 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.003419 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 5.391e+06 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit5)

Call: lm(formula = N5a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.0162568 -0.0010908 -0.0007812 0.0015428 0.0222110

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.6631307 0.0174431 38.017 <2e-16 *** N1b 0.0040833 0.0000471 86.702 <2e-16 *** N2b -0.2127669 0.0139376 -15.266 <2e-16 *** N3b 0.2226302 0.0140553 15.840 <2e-16 *** N4b 0.2370081 0.0128095 18.502 <2e-16 *** N5b 0.9201616 0.0197852 46.507 <2e-16 *** N6b -0.3337500 0.0113542 -29.394 <2e-16 *** N7b -0.0019020 0.0013593 -1.399 0.163 N8b 0.0614487 0.0033932 18.110 <2e-16 *** N9b 0.0053095 0.0037684 1.409 0.160 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.003122 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 7.051e+06 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit6)

Call: lm(formula = N6a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max © L. S. Barreto, 2017. Theoretical Ecology 285 15 Conceptual, and Mathematical Models for the Community

-0.0135001 -0.0006499 -0.0004244 0.0008760 0.0192342

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 9.412e-01 1.348e-02 69.830 <2e-16 *** N1b 3.319e-03 3.639e-05 91.210 <2e-16 *** N2b -1.396e-01 1.077e-02 -12.962 <2e-16 *** N3b 1.473e-01 1.086e-02 13.567 <2e-16 *** N4b -1.310e-03 9.898e-03 -0.132 0.895 N5b 5.829e-01 1.529e-02 38.126 <2e-16 *** N6b 2.593e-01 8.773e-03 29.557 <2e-16 *** N7b -1.087e-02 1.050e-03 -10.351 <2e-16 *** N8b 9.652e-02 2.622e-03 36.816 <2e-16 *** N9b -7.485e-02 2.912e-03 -25.706 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.002412 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.357e+07 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit7)

Call: lm(formula = N7a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.006753 -0.002296 0.001388 0.001810 0.009814

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.375e+00 1.740e-02 79.00 <2e-16 *** N1b 5.010e-04 4.699e-05 10.66 <2e-16 *** N2b 1.445e-01 1.391e-02 10.39 <2e-16 *** N3b -1.441e-01 1.402e-02 -10.28 <2e-16 *** N4b -7.763e-01 1.278e-02 -60.74 <2e-16 *** N5b 1.340e+00 1.974e-02 67.88 <2e-16 *** N6b -5.251e-01 1.133e-02 -46.35 <2e-16 *** N7b 9.388e-01 1.356e-03 692.20 <2e-16 *** N8b -8.241e-02 3.385e-03 -24.34 <2e-16 *** N9b -3.850e-01 3.760e-03 -102.39 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.003115 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 9.616e+05 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit8)

Call: lm(formula = N8a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -0.0009373 -0.0003087 0.0001540 0.0002201 0.0012814 286 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.984e-02 2.252e-03 26.571 < 2e-16 *** N1b 1.722e-05 6.081e-06 2.831 0.00496 ** N2b 3.749e-02 1.800e-03 20.830 < 2e-16 *** N3b -3.771e-02 1.815e-03 -20.781 < 2e-16 *** N4b -1.298e-01 1.654e-03 -78.456 < 2e-16 *** N5b 2.106e-01 2.555e-03 82.438 < 2e-16 *** N6b -6.482e-02 1.466e-03 -44.215 < 2e-16 *** N7b -1.771e-03 1.755e-04 -10.090 < 2e-16 *** N8b 9.072e-01 4.381e-04 2070.715 < 2e-16 *** N9b -3.022e-02 4.866e-04 -62.107 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0004031 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.546e+08 on 9 and 289 DF, p-value: < 2.2e-16

> summary(fit9)

Call: lm(formula = N9a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b)

Residuals: Min 1Q Median 3Q Max -1.522e-04 -1.455e-05 -8.548e-06 3.130e-05 1.689e-04

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -6.555e-04 3.159e-04 -2.075 0.038910 * N1b -2.297e-05 8.531e-07 -26.932 < 2e-16 *** N2b 6.017e-03 2.525e-04 23.834 < 2e-16 *** N3b -6.122e-03 2.546e-04 -24.047 < 2e-16 *** N4b 6.451e-03 2.320e-04 27.803 < 2e-16 *** N5b -1.302e-03 3.584e-04 -3.633 0.000332 *** N6b -5.287e-03 2.057e-04 -25.707 < 2e-16 *** N7b 9.004e-03 2.462e-05 365.684 < 2e-16 *** N8b 2.162e-03 6.146e-05 35.179 < 2e-16 *** N9b 9.807e-01 6.826e-05 14367.549 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.655e-05 on 289 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 2.583e+08 on 9 and 289 DF, p-value: < 2.2e-16

The good fittings we obtained sustain the modelling of communities as MAR (1). Now we obtain the community matrix A, and the MTE E.

> > c1=coef(fit1);c1=c1[-1] > c2=coef(fit2);c2=c2[-1] > c3=coef(fit3);c3=c3[-1] > c4=coef(fit4);c4=c4[-1] > c5=coef(fit5);c5=c5[-1] > c6=coef(fit6);c6=c6[-1] © L. S. Barreto, 2017. Theoretical Ecology 287 15 Conceptual, and Mathematical Models for the Community

> c7=coef(fit7);c7=c7[-1] > c8=coef(fit8);c8=c8[-1] > c9=coef(fit9);c9=c9[-1] > > #Community matrix, and MTE > > A=matrix(c( c1 , c2, c3, c4, c5, c6, c7, c8, c9), nrow=9, byrow=T) > round(A,4) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] 0.3473 4.5047 -4.3877 -28.2617 43.0049 -13.5080 -1.1906 -4.6546 -6.0683 [2,] 0.1052 -1.7681 1.8548 -4.9312 6.0653 -0.7860 -0.2541 -1.4110 -0.9274 [3,] 0.1062 -2.6791 2.7796 -4.8138 5.7472 -0.5900 -0.2506 -1.4338 -0.8912 [4,] 0.0044 -0.2526 0.2631 1.3310 -0.4219 -0.0894 0.0033 0.0270 0.0578 [5,] 0.0041 -0.2128 0.2226 0.2370 0.9202 -0.3338 -0.0019 0.0614 0.0053 [6,] 0.0033 -0.1396 0.1473 -0.0013 0.5829 0.2593 -0.0109 0.0965 -0.0749 [7,] 0.0005 0.1445 -0.1441 -0.7763 1.3400 -0.5251 0.9388 -0.0824 -0.3850 [8,] 0.0000 0.0375 -0.0377 -0.1298 0.2106 -0.0648 -0.0018 0.9072 -0.0302 [9,] 0.0000 0.0060 -0.0061 0.0065 -0.0013 -0.0053 0.0090 0.0022 0.9807 > E=-ginv(A) > > round(E, 4)

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] -21.3866 297.4126 -235.0742 -302.3208 519.3373 -208.7233 -9.0490 -23.3246 -69.9047 [2,] 38.2913 -584.8570 455.8358 450.7934 -831.3146 373.6071 11.9692 11.2515 109.5962 [3,] 37.7355 -575.2662 448.1302 444.4908 -819.7801 368.4591 11.8035 11.0802 108.0700 [4,] -0.1898 2.7408 -2.0573 -2.7823 3.8917 -2.3032 -0.0521 0.0979 -0.5027 [5,] -0.1836 2.6975 -2.0396 -1.8825 3.2660 -2.5666 -0.0580 0.1340 -0.5597 [6,] -0.1428 2.1597 -1.6495 -1.7800 4.2964 -3.6728 -0.0588 0.1681 -0.5567 [7,] -0.0652 1.1954 -0.9725 -1.6169 2.8246 -1.1397 -1.0803 -0.1310 -0.5919 [8,] -0.0079 0.1684 -0.1418 -0.2336 0.3709 -0.1143 -0.0037 -1.1114 -0.0513 [9,] 0.0010 -0.0100 0.0064 0.0234 -0.0297 0.0056 0.0099 0.0037 -1.0140

The total effects exercised by each specie on itself, and the others are estimated as:

> Einf=c(sum(E[,1]),sum(E[,2]), sum(E[,3]), sum(E[,4]), sum(E[,5]), + sum(E[,6]), sum(E[,7]),sum(E[,8]),sum(E[,9])) > round(Einf, 4) [1] 54.0518 -853.7587 662.0376 584.6916 -1117.1375 523.5519 13.4807 -1.8316 144.4852

The total effects received by each specie are:

> Erec=c(sum(E[1,]),sum(E[2,]), sum(E[3,]), sum(E[4,]), sum(E[5,]), + sum(E[6,]), sum(E[7,]),sum(E[8,]),sum(E[9,])) > round(Erec, 4) [1] -53.0333 35.1727 34.7229 -1.1571 -1.1923 -1.2363 -1.5775 -1.1247 -1.0036

Before we introduce any comments, we display a graphic with the columns of matriz E in figure 15.5. They are the total effects exercised by each specie on itself, and the others:

> windows() > nu=1:1:9 > matplot(nu,E,type='l',lw=2) 288 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

Figure 15.5. Graphic of the columns of the MTE E. In the graphic, each line represents the values of the column correspondent to the respective variable displayed in the left side. They are the total effects exercised by each specie on itself, and the others

For a more conspicuous representation, in figure 15.6, we introduce a chromatic representation of matrix E, where negative values are represented by red squares, and positive values by green squares.

Figure 15.6. A chromatic representation of the MTE E, where negative values are represented by red squares, and positive values by green squares

With the parameters we used, for any initial values of the variables, the system converges to the fixed point:

y2*=3,3321, y3*=3,2536, y4*=4,1940, y5*=4,1334, y6*=4,2708

y7*=6,0172, y8*=0,2004

y9*=2,7353 We used an artificial model adequate to our explanatory purposes. This is not a text book, thus, in this context, the analysis of the results we obtained is irrelevant. The model can be improved with the following procedures: • Introducing other types of interactions, and more species; • Using structured populations; • Including cannibalism; • Including diet selection; • Being associated to static models; • Including allometric relations in various biosystems; • Being included in meta-community models; • Including evolutive aspects. Let us move to the ecosystem ecology.

15.4 References, and Related Bibliography

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. E-book. Lisboa Brännström, A., J. Johansson, N. Loeuille, N. Kristensen, T. A. Troost, R. H. R Lambers, and U. Dieckmann, 2012. Modelling the ecology and evolution of communities: a review of past achievements, current efforts, and future promises. Evolutionary Ecology Research, 14: 601–625. Brose, U. and J. A. Dunne, 2009. Modelling the Dynamics of Complex Food Webs. Em H.A. Verhoef and P.J. Morin, (Editors), Community Ecology. Processes, Models, and Applications, pages 37-44. Oxford University Press. Cattin, M.F., L.F. Bersier, C. Banasek-Richter, R. Baltensperger, and J.P. Gabriel, 2004. Phylogenetic constraints and adaptation explain food-web structure. Nature, 427: 835–839. Cohen, J.E., and C.M. Newman, 1985. A stochastic theory of community food webs: I. Models and aggregated data. Proc. R. Soc. Lond. B, 224: 421–448. Dunne, J.A. 2006. The network structure of food webs. In M. Pascual e J.A. Dunne, (Editors), Ecological Net- works: Linking Structure to Dynamics in Food Webs, pages. 27–86. Oxford University Press, Oxford. Frigg, R., 2012. Models and Theories. Acumen Publishing. Godfrey-Smith, P., 2009. Models and Fictions in Science. Philosophical Studies, 143:101-116. Goudard A., and M. Loreau, 2008. Non-trophic interactions, biodiversity and ecosystem functioning: an interaction web model. The American Naturalist, 171:91–106. Hudson, L. N., and D. C. Reuman, 2013. A Cure for the Plague of Parameters: Constraining Models of Complex Population Dynamics with Allometries. Proc R Soc B 280:20131901. http://dx.doi.org/10.1098/rspb.2013.1901 Hudson, L., D. Reuman, and R. Emerson, 2014. Cheddar: analysis and visualization of ecological communities. R package version 0.1-626. Hudson, L. N., R. Emerson, G. B. Jenkins, K. Layer, M. E. Ledger, D. E. Pichler, M. S. A. Thompson, E. J. O'Gorman, G. Woodward, and D. C. Reuman, 2013. Cheddar: analysis and visualisation of ecological communities in R. Meth- ods in Ecology and Evolution, 4:1 99--104. (DOI 10.1111/2041-210X.12005) Loeuille, N., and M. Loreau, 2009. Emergence of complex food web structure in community evolution mod - els. In H.A. Verhoef and P.J. Morin, (Editors), Community Ecology. Processes, Models, and Applications, pages 163–178. Oxford University Press. Martinez, N. D., R.J. Williams, and J. A. Dunne, 2006. Diversity, complexity, and persistence in large model ecosystems. In M. Pascual , and J.A. Dunne, (Editors), Ecological Networks: Linking Structure to Dynamics in Food Webs, pages. 163–185. Oxford University Press, Oxford. Mayhew, P. J., 2006. Discovering Evolutionary Ecology. Bringing together ecology and evolution. Oxford University Press, Oxford. Pascual, M., and J.A. Dunne, (Editors), 2006. Ecological Networks: Linking Structure to Dynamics in Food Web., Oxford University Press, Oxford. 290 © L. S. Barreto, 2017. Theoretical Ecology 15 Conceptual, and Mathematical Models for the Community

Petris, G., S. Petrone, and P. Campagnoli, 2009. Dynamic Linear Models in R. Springer-Verlag. Roughgarden, J., 2009. Is there a general theory of community ecology? Biology and Philosophy, 24(4):521–529. Vellend, M., 2010. Conceptual Synthesis in Community Ecology. The Quarterly Review of Biology, 85(2):183-206. Williams, R. J. e N. D. Martinez, 2000. Simple rules yield complex food webs. Nature, 404:180–183. © L. S. Barreto, 2017. Theoretical Ecology 291 16 Modelling, and Analysis of the Ecosystem

16 Modelling, and Analysis of the Ecosystem

16.1 Introduction

In this chapter we extend the application of the methodology M&S to the ecosystem. We will accomplish the following: • We introduce our model of ecosystem, that is an expansion of the model for the community; • We will show that ecosystems are also linear stochastic dynamic systems (LSDS) that can be modelled by MAR(1); • We will use the analytical procedures of the ecological network analysis (ENA; Fath (2012), Fath and Patten, (1999), Ulanowicz (2004, 2011)) to scrutinize the importance of the interactions used in the model; • We will propose a procedure for the identification of keystone species, and controlling factors of the ecosystem.

16.2 The Ecosystem to Be Modelled

To adapt the community model to the ecosystem we consider the following: • The community maintains the same structure, with different parameters; • The fall of dead organic matter to the soil; • The existence of a nutrient that acts as a limiting factor; • The cycle of the nutrient is closed. In terrestrial ecosystems this can be accepted with any reluctance. For instance, see Chapin III, Matson, and Mooney (2002:220-222). A graphical representation of the ecosystem is exhibited in figure 16.1. 292 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Consumer resource Plant nutrient Competition Obligatory mutualism Facultative mutualism Nutrient return

Figure 16.1. Graphical representation of the ecosystem to be modelled, and simulated

Using the package igraph (Csárdi e Nepusz, 2006; Kolaczyk e Csárdi, 2014), in R, we can obtain a network of the flow of materials in the ecosystem, exhibited in figure 16.2. © L. S. Barreto, 2017. Theoretical Ecology 293 16 Modelling, and Analysis of the Ecosystem

Figure 16.2. Network of the flow of materials (nutrient, and biomass) in the constructed ecosystem. The arrows (edges) point from the source to the receiver. The edge between the nutrient (y1), and the plant (y2) is bidirectional because the plant uptakes, and returns the nutrient. All variables (nodes) return nutrient to y1

16.3 Modelling, and Simulating the Proposed Ecosystem

In the model of the ecosystem, the following is assumed:

• The variables y2 to y10 are he same as in the model for the community, but they loose biomass to the soil; • All species release 1% of their biomasses to the soil, in unit of time; • For each unit of nutrient assimilated by the plant, it produces 1000 units of plant biomass;

• The carrying capacity of the plant is 1000 y1; • The contents of the nutrient in the biomasses are: plant: 0,001; herbivores: 0.01;

omnivore, and predator y9:0,1; top predator:0,3; • The ODE of the nutrient is equal to the contents of the fallen litter less the plant uptake;

• The plant uptake is given by y2*y1/(50+0.4*y1). As already said, now, we use a different parametrization that can be read in the script of the model. Let us simulate the ecosystem:

> #semomissões_2 > #y(1) nutrient > #y(2) plant > #y(3 a 7) herbivores > #y(3,4) obligatory mutualism > #y(6,7) facultative mutualism 294 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

> #the mutualists do not compete with each other > #y(8-->2 a 4) predation and omnivory > #y(9-->4 a 7) predation > #y(10-->8,9) top predator > > rm(list=ls(all=TRUE)) > > library(deSolve) Warning message: package ‘deSolve’ was built under R version 3.0.3 > library(MASS) Warning message: package ‘MASS’ was built under R version 3.0.3 > library(rootSolve) Warning message: package ‘rootSolve’ was built under R version 3.0.3 > library(vegan) Loading required package: permute Loading required package: lattice This is vegan 2.2-0 Warning messages: 1: package ‘vegan’ was built under R version 3.0.3 2: package ‘permute’ was built under R version 3.0.3 3: package ‘lattice’ was built under R version 3.0.3 > > modul<-function(times,y,parms) { + n<-y + + r<-c(0.05, 0.09, 0.071, 0.057, 0.055, 0.07, 0.06, 0.035, 0.02) + b<-c(20, 0.2, 0.21, 0.28, 0.29, 0.27) + b7<-c(b[1], b[2], b[3])/3 + b8<-c(b[4], b[5], b[6])/2.5 + bm7<-mean(b7)/5 + bm8<-mean(b8)/4 + b9<-c(bm7, bm8) + + + #plant:total consumption of herbivores, and omnivore + #herbivores & omnivore + ex<-sum( y[2]*y[3]/(70+0.2*y[2]), y[4]*y[2]/(90+0.3*y[2]), y[5]*y[2]/ (90+0.3*y[2]), y[6]*y[2]/(80+0.3*y[2]), y[7]*y[2]/(75+0.3*y[2]), y[8]*y[2]/ (83+0.3*y[2])) + + #competition, and herbivores consumptions + #inicial/final<-iniciais/(bi*resouce) + + a1n<-c(0, 0, 0.02, 0.03, 0.04) + y1n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a2n<-c(0, 0, 0.01, 0.02, 0.03) + y2n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a3n<-c(-0.04, -0.03, 0, 0.01, 0.02) + y3n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a4n<-c(-0.04, -0.03, -0.02, 0, 0) + y4n<-log(c(y[3], y[4], y[5], y[6], y[7])) + + #effects of competition + comp2<-sum(a1n*y1n) + comp3<-sum(a2n*y2n) + comp4<-sum(a3n*y3n)+ a5n<-c(-0.04, -0.03, -0.02, 0, 0) + y5n<-log(c(y[3], y[4], y[5], y[6], y[7])) © L. S. Barreto, 2017. Theoretical Ecology 295 16 Modelling, and Analysis of the Ecosystem

+ comp5<-sum(a4n*y4n) + comp6<-sum(a5n*y5n) + + #consumption of the predators y[8], y[9], and y[10] + prey7<-y[8]*c(y[3]/(50+0.2*y[3]), y[4]/(60+0.3*y[4]), y[9]/ (45+0.23*y[9])) + prey8<-y[9]*c(y[5]/(50+0.21*y[5]), y[6]/(60+0.3*y[6]), y[7]/ (70+0.3*y[7])) + prey9<-c(y[10]*y[8]/(30+0.2*y[8]), y[10]*y[9]/(40+0.3*y[9])) + + #carrying capacity of y[8] + k7<-sum(b7*c(y[2], y[3], y[4])); + + #carrying capacity of y[9] + k8<-sum(b8*c(y[5], y[6], y[7])) + #carrying capacity of y[10] + k9<-sum(b9*c(y[8], y[9])) + #faction of the biomass that falls + fr=0.01 + biom=c(y[2:10]) + m=fr*biom # fallen biomass + #nutrient contents in the biomasses + te=c(0.001,rep(0.01,5),rep(0.1,2),0.3) + + #ODE of the model + dy1.dt<-sum(te*m)-y[2]*y[1]/(50+0.4*y[1]) + dy2.dt<-y[2]*r[1]*(log(1000*y[1])-log(y[2]))-ex-m[1] + dy3.dt<-y[3]*r[2]*(8*y[4]-10^(-10))*(log(b[2]*y[2])-log(y[3])+comp2)- prey7[1]-m[2] + dy4.dt<-y[4]*r[3]*(9*y[3]-10^(-10))*(log(b[3]*y[2])-log(y[4])+comp3)- prey7[2]-m[3] + dy5.dt<-y[5]*r[4]*(log(b[4]*y[2])-log(y[5])+comp4)-prey8[1]-m[4] + dy6.dt<-y[6]*r[5]*(log(b[5]*y[2]*(2-exp(-0.003*y[7])))-log(y[6]) +comp5)-prey8[2]-m[5] + dy7.dt<-y[7]*r[6]*(log(b[6]*y[2]*(2-exp(-0.03*y[7])))-log(y[7])+comp6)- prey8[3]-m[6] + dy8.dt<-y[8]*r[7]*(log(k7)-log(y[8]))-prey9[1]-m[7] + dy9.dt<-y[9]*r[8]*(log(k8)-log(y[9]))-prey9[2]-prey7[3]-m[8] + dy10.dt<-y[10]*r[9]*(log(k9)-log(y[10]))-m[9] + + return(list(c(dy1.dt,dy2.dt,dy3.dt, dy4.dt, dy5.dt, dy6.dt,dy7.dt,dy8.dt, dy9.dt, dy10.dt))) + } > > > initials<-c(40, 400, 36, 36, 36, 36, 36, 12, 12, 4) > t.s<- seq(1, 300, by=0.1) > > > out<- ode(y=initials, times=t.s, modul) > > split.screen(figs=c(2,2)) [1] 1 2 3 4 > > screen(1) > matplot(out[,2], type='l',xlab="Time", ylab="Weight") > title("Weight of the nutrient in the soil") > > 296 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

> screen(2) > matplot(out[,3], type='l',xlab="Time", ylab="Biomass") > title("Plant biomass") > > ys<-cbind(out[,4],out[,5],out[,6],out[,7],out[,8]) > screen(3) > matplot(ys, type='l',xlab="Time", ylab="Biomass") > title("Biomasses of the herbivores") > r<-c('y3','y4','y5','y6','y7') > legend('topright',paste(r),lty=1,col=1:5, bty='n') > > ys2<-cbind(out[,9],out[,10],out[,11]) > screen(4) > matplot(ys2, type='l',xlab="Time", ylab="Biomass") > title("Biomasses of the carnivorous") > r<-c('y8','y9','y10') > legend('topright',paste(r),lty=1,col=1:5, bty='n') > > ##################### > > > y<-initials > ST2 <- runsteady(y=y,func=modul,parms=parms,times=c(0,500)) > ye<-ST2$y > H=diversity(ye[-1]) # Shannon-Weaver diversity index [1] 0.1590 5.0091 0.7431 0.7773 1.2074 1.2347 1.2341 8.4034 0.0003 2.3117 1.6917 > ye [1] 0.1590013091 5.0090621641 0.7430847673 0.7773484198 1.2074297620 [6] 1.2346934021 1.2341296936 8.4034232056 0.0002859149 2.3117477578

Figure 16.3. Dynamics of the nutrient, and of the populations in the simulated ecosystem © L. S. Barreto, 2017. Theoretical Ecology 297 16 Modelling, and Analysis of the Ecosystem

The omnivore (y8), and the plant (y2) have the largest biomasses (dominant species). For future comparisons, we calculated also the Shannon-Weaver index of diversity ( H=1,6917), using package vegan. We can confirm the stability of the fixed point with the following script:

> jac=jacobian.full(y = ye, func = modul) Function from package rootSolve > eigen(jac)$values [1] -0.68442596+0.00000000i -0.31582065+0.00000000i -0.08115262+0.12139836i [4] -0.08115262-0.12139836i -0.07539283+0.00000000i -0.06177819+0.00000000i [7] -0.05566546+0.00000000i -0.04043980+0.02024305i -0.04043980-0.02024305i [10] -0.03499474+0.00000000i

All real parts of the eigenvalues are negative. The trajectories of the system with initial values in the basin of attraction of the fixed point converge to it. Let us calculate the MAR(1):

> dat0<-out[,-1] > > N1<-dat0[,1] > N2<-dat0[,2] > N3<-dat0[,3] > N4=dat0[,4] > N5<-dat0[,5] > N6<-dat0[,6] > N7<-dat0[,7] > N8=dat0[,8] > N9<-dat0[,9] > N10=dat0[,10] > > ###################################### > ###### MAR(1) > > m<-c(dim(dat0)) > N1a<-N1[-1] > N1b<-N1[-m[1]] > N2a<-N2[-1] > N2b<-N2[-m[1]] > N3a<-N3[-1] > N3b<-N3[-m[1]] > N4a<-N4[-1] > N4b<-N4[-m[1]] > N5a<-N5[-1] > N5b<-N5[-m[1]] > N6a<-N6[-1] > N6b<-N6[-m[1]] > N7a<-N7[-1] > N7b<-N7[-m[1]] > N8a<-N8[-1] > N8b<-N8[-m[1]] > N9a<-N9[-1] > N9b<-N9[-m[1]] > N10a<-N10[-1] > N10b<-N10[-m[1]] > > ########## > fit1<-lm(N1a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit2=lm(N2a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) 298 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

> fit3=lm(N3a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit4=lm(N4a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit5<-lm(N5a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit6=lm(N6a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit7=lm(N7a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit8=lm(N8a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit9=lm(N9a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > fit10=lm(N10a ~ N1b+N2b+N3b+N4b+N5b+N6b+N7b+N8b+N9b+N10b) > > ########### > > summary(fit1)

Call: lm(formula = N1a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.059674 -0.001439 -0.001266 0.001736 0.059674

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -9.499e-02 7.308e-03 -12.997 <2e-16 *** N1b 5.051e-01 4.892e-04 1033.337 <2e-16 *** N2b 7.487e-03 9.798e-05 76.416 <2e-16 *** N3b 3.837e-01 8.410e-03 45.622 <2e-16 *** N4b -4.182e-01 9.182e-03 -45.544 <2e-16 *** N5b -3.406e-01 3.295e-02 -10.335 <2e-16 *** N6b 2.588e-01 3.029e-02 8.544 <2e-16 *** N7b 3.887e-02 1.886e-03 20.607 <2e-16 *** N8b -2.025e-03 2.132e-04 -9.497 <2e-16 *** N9b 9.079e-02 9.228e-03 9.838 <2e-16 *** N10b 1.031e-01 2.634e-03 39.133 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.004137 on 2979 degrees of freedom Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999 F-statistic: 4.207e+06 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit2)

Call: lm(formula = N2a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.36789 -0.01503 0.01269 0.01432 0.43543

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.1866200 0.0649722 49.046 <2e-16 *** N1b -0.1024583 0.0043494 -23.557 <2e-16 *** N2b 0.8743037 0.0008711 1003.717 <2e-16 *** N3b -2.9374267 0.0747658 -39.288 <2e-16 *** N4b 3.0520083 0.0816295 37.389 <2e-16 *** N5b -4.6896779 0.2929494 -16.008 <2e-16 *** N6b 4.2036121 0.2692864 15.610 <2e-16 *** © L. S. Barreto, 2017. Theoretical Ecology 299 16 Modelling, and Analysis of the Ecosystem

N7b 0.0258388 0.0167692 1.541 0.123 N8b -0.0513006 0.0018958 -27.060 <2e-16 *** N9b 1.3576618 0.0820440 16.548 <2e-16 *** N10b -0.8104718 0.0234155 -34.613 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03678 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 3.212e+07 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit3)

Call: lm(formula = N3a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.118762 -0.007156 -0.006751 0.005254 0.153243

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.9676227 0.0374541 -52.534 < 2e-16 *** N1b 0.1111215 0.0025073 44.319 < 2e-16 *** N2b 0.1952469 0.0005021 388.831 < 2e-16 *** N3b -3.3137053 0.0430998 -76.884 < 2e-16 *** N4b 3.6092920 0.0470565 76.701 < 2e-16 *** N5b 2.7902754 0.1688748 16.523 < 2e-16 *** N6b -2.2790007 0.1552339 -14.681 < 2e-16 *** N7b -0.2697391 0.0096669 -27.904 < 2e-16 *** N8b 0.0274701 0.0010929 25.136 < 2e-16 *** N9b -0.3325372 0.0472954 -7.031 2.53e-12 *** N10b 0.4004792 0.0134982 29.669 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0212 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 7.969e+06 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit4)

Call: lm(formula = N4a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.088779 -0.006291 -0.005939 0.004657 0.143016

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.7586985 0.0335835 -52.368 < 2e-16 *** N1b 0.1106741 0.0022482 49.228 < 2e-16 *** N2b 0.1769040 0.0004502 392.905 < 2e-16 *** N3b -3.9624245 0.0386457 -102.532 < 2e-16 *** N4b 4.3200856 0.0421935 102.387 < 2e-16 *** N5b 2.5298535 0.1514227 16.707 < 2e-16 *** N6b -2.0601206 0.1391915 -14.801 < 2e-16 *** 300 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

N7b -0.2506296 0.0086678 -28.915 < 2e-16 *** N8b 0.0245042 0.0009799 25.006 < 2e-16 *** N9b -0.2951938 0.0424077 -6.961 4.14e-12 *** N10b 0.3535318 0.0121032 29.210 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01901 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 8.872e+06 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit5)

Call: lm(formula = N5a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.0040913 -0.0000909 -0.0000702 0.0000813 0.0047706

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.886e-02 9.273e-04 52.690 < 2e-16 *** N1b 3.010e-03 6.208e-05 48.484 < 2e-16 *** N2b -1.054e-03 1.243e-05 -84.807 < 2e-16 *** N3b -5.353e-02 1.067e-03 -50.169 < 2e-16 *** N4b 5.897e-02 1.165e-03 50.620 < 2e-16 *** N5b 1.100e+00 4.181e-03 263.179 < 2e-16 *** N6b -1.120e-01 3.843e-03 -29.152 < 2e-16 *** N7b 2.209e-04 2.393e-04 0.923 0.356 N8b -3.242e-03 2.706e-05 -119.822 < 2e-16 *** N9b -5.764e-03 1.171e-03 -4.922 9.01e-07 *** N10b -2.542e-03 3.342e-04 -7.606 3.78e-14 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0005249 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.061e+10 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit6)

Call: lm(formula = N6a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.0043204 -0.0001039 -0.0000785 0.0000947 0.0050766

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.242e-02 9.760e-04 53.708 < 2e-16 *** N1b 3.170e-03 6.534e-05 48.510 < 2e-16 *** N2b -1.108e-03 1.309e-05 -84.641 < 2e-16 *** N3b -5.503e-02 1.123e-03 -48.994 < 2e-16 *** N4b 6.064e-02 1.226e-03 49.452 < 2e-16 *** N5b 1.340e-01 4.401e-03 30.460 < 2e-16 *** N6b 8.552e-01 4.045e-03 211.395 < 2e-16 *** © L. S. Barreto, 2017. Theoretical Ecology 301 16 Modelling, and Analysis of the Ecosystem

N7b -4.138e-04 2.519e-04 -1.643 0.100544 N8b -3.521e-03 2.848e-05 -123.616 < 2e-16 *** N9b 4.249e-03 1.232e-03 3.447 0.000574 *** N10b -2.788e-03 3.518e-04 -7.925 3.2e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0005524 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.05e+10 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit7)

Call: lm(formula = N7a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.0052462 -0.0000027 0.0000028 0.0000574 0.0054945

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.091e-02 1.341e-03 45.43 <2e-16 *** N1b 4.058e-03 8.976e-05 45.21 <2e-16 *** N2b -1.588e-03 1.798e-05 -88.35 <2e-16 *** N3b -8.679e-02 1.543e-03 -56.25 <2e-16 *** N4b 9.543e-02 1.685e-03 56.65 <2e-16 *** N5b 1.507e-01 6.046e-03 24.92 <2e-16 *** N6b -1.406e-01 5.557e-03 -25.30 <2e-16 *** N7b 9.717e-01 3.461e-04 2807.79 <2e-16 *** N8b -2.907e-03 3.912e-05 -74.29 <2e-16 *** N9b 4.566e-02 1.693e-03 26.97 <2e-16 *** N10b -5.309e-03 4.832e-04 -10.99 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0007589 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 5.969e+09 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit8)

Call: lm(formula = N8a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -0.0048604 -0.0000710 -0.0000563 0.0001245 0.0046492

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.659e-01 1.641e-03 101.080 < 2e-16 *** N1b 4.872e-03 1.099e-04 44.347 < 2e-16 *** N2b -2.195e-03 2.200e-05 -99.737 < 2e-16 *** N3b -1.727e-01 1.889e-03 -91.455 < 2e-16 *** N4b 1.894e-01 2.062e-03 91.840 < 2e-16 *** N5b 5.391e-02 7.400e-03 7.286 4.08e-13 *** N6b -9.842e-02 6.802e-03 -14.468 < 2e-16 *** 302 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

N7b 3.691e-02 4.236e-04 87.139 < 2e-16 *** N8b 9.893e-01 4.789e-05 20657.834 < 2e-16 *** N9b 5.611e-02 2.072e-03 27.073 < 2e-16 *** N10b -3.185e-02 5.915e-04 -53.841 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.000929 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 7.149e+08 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit9)

Call: lm(formula = N9a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -1.695e-03 -5.260e-06 -4.430e-06 2.092e-05 1.411e-03

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.207e-02 2.428e-04 -49.720 < 2e-16 *** N1b -1.066e-03 1.625e-05 -65.608 < 2e-16 *** N2b 3.777e-04 3.255e-06 116.047 < 2e-16 *** N3b 9.268e-03 2.793e-04 33.178 < 2e-16 *** N4b -1.032e-02 3.050e-04 -33.852 < 2e-16 *** N5b 3.531e-03 1.095e-03 3.226 0.00127 ** N6b 4.260e-03 1.006e-03 4.234 2.36e-05 *** N7b -6.205e-03 6.265e-05 -99.041 < 2e-16 *** N8b 2.781e-04 7.083e-06 39.261 < 2e-16 *** N9b 9.542e-01 3.065e-04 3112.736 < 2e-16 *** N10b 3.159e-03 8.749e-05 36.105 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0001374 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 9.484e+09 on 10 and 2979 DF, p-value: < 2.2e-16

> summary(fit10)

Call: lm(formula = N10a ~ N1b + N2b + N3b + N4b + N5b + N6b + N7b + N8b + N9b + N10b)

Residuals: Min 1Q Median 3Q Max -5.493e-04 -7.089e-05 3.633e-05 4.550e-05 8.031e-04

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.504e-03 1.693e-04 20.706 < 2e-16 *** N1b -1.246e-04 1.133e-05 -10.998 < 2e-16 *** N2b -9.697e-06 2.269e-06 -4.273 1.99e-05 *** N3b -5.773e-03 1.948e-04 -29.640 < 2e-16 *** N4b 6.260e-03 2.126e-04 29.441 < 2e-16 *** N5b -9.577e-03 7.631e-04 -12.550 < 2e-16 *** N6b 1.028e-02 7.015e-04 14.654 < 2e-16 *** © L. S. Barreto, 2017. Theoretical Ecology 303 16 Modelling, and Analysis of the Ecosystem

N7b -1.403e-03 4.368e-05 -33.116 < 2e-16 *** N8b 8.914e-04 4.939e-06 180.497 < 2e-16 *** N9b 1.698e-03 2.137e-04 7.944 2.75e-15 *** N10b 9.953e-01 6.100e-05 16316.559 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.58e-05 on 2979 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 4.296e+09 on 10 and 2979 DF, p-value: < 2.2e-16

It is clearly confirmed that ecosystems, as communities, can be modelled by MAR(1). This conclusion legitimises the application of the state space analysis, and the ecological network analysis (ENA) to the study of the ecosystem. In matrix E, the sum of column j gives the net total effect inflicted (NTEI) by component j on all other components of the ecosystem, including itself. In matrix E, the sum of row j gives the net total effect received (NTER) by component j from all other components of the ecosystem, including from itself. Subsequently, we establish the matrices of the community, and total effects:

> c1=coef(fit1);c1=c1[-1] > c2=coef(fit2);c2=c2[-1] > c3=coef(fit3);c3=c3[-1] > c4=coef(fit4);c4=c4[-1] > c5=coef(fit5);c5=c5[-1] > c6=coef(fit6);c6=c6[-1] > c7=coef(fit7);c7=c7[-1] > c8=coef(fit8);c8=c8[-1] > c9=coef(fit9);c9=c9[-1] > c10=coef(fit10);c10=c10[-1] > > > A=matrix(c( c1 , c2, c3, c4, c5, c6, c7, c8, c9, c10), nrow=10, byrow=T) > round(A,4) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] 0.5051 0.0075 0.3837 -0.4182 -0.3406 0.2588 0.0389 -0.0020 0.0908 0.1031 [2,] -0.1025 0.8743 -2.9374 3.0520 -4.6897 4.2036 0.0258 -0.0513 1.3577 -0.8105 [3,] 0.1111 0.1952 -3.3137 3.6093 2.7903 -2.2790 -0.2697 0.0275 -0.3325 0.4005 [4,] 0.1107 0.1769 -3.9624 4.3201 2.5299 -2.0601 -0.2506 0.0245 -0.2952 0.3535 [5,] 0.0030 -0.0011 -0.0535 0.0590 1.1003 -0.1120 0.0002 -0.0032 -0.0058 -0.0025 [6,] 0.0032 -0.0011 -0.0550 0.0606 0.1340 0.8552 -0.0004 -0.0035 0.0042 -0.0028 [7,] 0.0041 -0.0016 -0.0868 0.0954 0.1507 -0.1406 0.9717 -0.0029 0.0457 -0.0053 [8,] 0.0049 -0.0022 -0.1727 0.1894 0.0539 -0.0984 0.0369 0.9893 0.0561 -0.0318 [9,] -0.0011 0.0004 0.0093 -0.0103 0.0035 0.0043 -0.0062 0.0003 0.9542 0.0032 [10,] -0.0001 0.0000 -0.0058 0.0063 -0.0096 0.0103 -0.0014 0.0009 0.0017 0.9953 > E=-ginv(A) > round(E,4) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] -2.0145 -0.0423 1.7016 -1.5848 -1.6582 1.3377 0.1490 -0.0147 0.3323 0.0512 [2,] 1.5338 0.8191 -34.6237 28.4615 29.7026 -24.6980 -2.4027 0.3017 -4.1940 4.3471 [3,] 10.9411 11.7613 -189.9071 151.1247 211.6051 -177.8383 -14.8698 2.1607 -34.6076 31.3205 [4,] 10.0240 10.7552 -172.8092 137.2551 193.5143 -162.6278 -13.6048 1.9755 -31.6428 28.6308 [5,] 0.0023 -0.0037 -0.0173 0.0313 -0.9435 -0.0761 0.0036 -0.0038 0.0035 0.0102 [6,] 0.0023 -0.0041 -0.0151 0.0302 0.0865 -1.1060 0.0034 -0.0042 0.0152 -0.0113 [7,] 0.0038 -0.0043 -0.0542 0.0717 0.1085 -0.1065 -1.0253 -0.0035 0.0595 -0.0134 [8,] 0.0047 -0.0035 -0.1604 0.1812 0.0300 -0.0808 0.0408 -1.0111 0.0627 -0.0356 [9,] -0.0007 0.0018 -0.0098 0.0043 0.0287 -0.0165 -0.0084 0.0006 -1.0519 0.0072 [10,] 0.0002 0.0006 -0.0146 0.0132 0.0003 0.0020 -0.0021 0.0010 0.0000 -1.0031 > > round(eigen(A)$values,4) 304 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

[1] 0.9946+0.0069i 0.9946-0.0069i 0.9942+0.0000i 0.9740+0.0019i 0.9740- 0.0019i 0.9607+0.0138i 0.9607-0.0138i [8] 0.8620+0.0000i 0.5166+0.0000i 0.0201+0.0000i > round(eigen(E)$values,4) [1] -49.7413+0.000i -1.9358+0.000i -1.1601+0.000i -1.0407+0.015i -1.0407-0.015i -1.0267+0.002i -1.0267-0.002i -1.0058+0.000i -1.0053+0.007i -1.0053-0.007i > ########## Inflicted effects (NTEI) > Einf=c(sum(E[,1]),sum(E[,2]), sum(E[,3]), sum(E[,4]), sum(E[,5]), + sum(E[,6]), sum(E[,7]),sum(E[,8]),sum(E[,9]), sum(E[,10])) > > #################### Received effects (NTER) > Erec=c(sum(E[1,]),sum(E[2,]), sum(E[3,]), sum(E[4,]), sum(E[5,]), + sum(E[6,]), sum(E[7,]),sum(E[8,]),sum(E[9,]), sum(E[10,])) > EFT=matrix(c(Einf,Erec),nrow=2,byrow=T) > round(EFT,4) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] 20.4970 23.2800 -395.9095 315.5883 433.4742 -365.2103 -31.7161 3.4022 -71.0231 63.2833 NTEI [2,] -1.7428 -0.7526 1.6906 1.4704 -1.0139 -1.0030 -0.9637 -0.9718 -1.0446 -1.0025 NTER

The chromatic matrix that corresponds to th MTE is exhibited in figure 16.4.

Figure 16.4. Chromatic matrix that corresponds to the matrix of total effects. Positive effects: green; negative ef-

fects :red. The nutrient y1, and the herbivores y4, y5 are the components that have a positive impact on a larger

number of species. The herbivores y3, y6 are the components that have a negative impact on a larger number of species

We present a graphic of the effects inflicted by each component of the ecosystem on the other components, and on itself, in figure 16.5. The net total effects inflicted, and received are exhibited in figure 16.6. The script used is:

> windows() > nu=1:1:10 > matplot(nu,E,type='l',lw=2,xlab="Nutrient, and species", ylab="Total effect") > title("Columns of MTE") > r<-c('y1','y2','y3','y4','y5','y6','y7','y8','y9','y10') © L. S. Barreto, 2017. Theoretical Ecology 305 16 Modelling, and Analysis of the Ecosystem

> legend('topright',paste(r),lty=1,col=1:10, bty='n') > > windows() > matplot(nu,t(EFT),type='l',lw=2 ,xlab="Nutrient, and species", ylab="ETGI e EGTR") > title("ETGI e ETGR") > r<-c('ETGI','ETGR') > legend('topright',paste(r),lty=1,col=1:2, bty='n') >

Figura 16.5. Graphic of the effects inflicted by each component of the ecosystem (columns of the matrix of total effects) on the other components, and on itself. The impacts with larger absolute value are upon the first four components of the ecosystem

In figure 16.5, the trajectories of the lines corresponding to the effects of each component evince some symmetry. The sum of the positive NTEI is equal to 858.525, and the sum of the negtive NTEI is equal to -863.859. Is this a pattern characteristic of systems with fixed points that are stable nodes? Species y3 e y4 (obligatory mutualists) are the components that receive the largest positive, and negative total effects. Comparatively, the effects received by the other components are almost residuals.

Te species with greater positive effect are y4 (NTEI4=315,5883) e y5 (NTEI5=432,4742), and with greater negative impact are y3 (NTEI3=-395.9095), and y6 (NTEI6=-365.2103). The community matrix has a real dominant eigenvalue close to 1 (0.99) that mirrors the stationary state of the system. A preliminary conjecture is that the negative real parts of the eigenvalues of matrix E reflect the stability of the fixed point. 306 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.6. Graphic of the net total effects inflicted, and received by the components of the ecosystems

In figure 16.6, the NTER by the compartments are very close to zero. An issue to further clarification is the relation of this occurrence (zero NTER) with the stability of the system.

In this system, with the parametrization used, y10 is a keystone species. If the top predator is eliminated all the system collapses. From the script previously introduced we can obtain information about the functioning of the ecosystem at equilibrium. We can estimate the flows of biomass, and the flows of nutrient among the components of the ecosystem. They are represented by matrices, where the variable in the line is the donor to the variables in the columns (receivers). In these matrices the herbivores are aggregated in a sole variable H=y3+y4+y5+y6+y7, and the folowing notation is also used:N=nutrient, y1, V=plant (y2), H=herbivores, O=omnivore (y8), P1= predator y9, P2= top predator y10. First, we introduce the flows of biomass, in table 16.1.

Table 16.1. Material flows in the ecosystem, at equilibrium. FP represents the biomasses, and nutrient in the soil at equilibrium (fixed point)

N V H O P1 P2 FP 0.1590 5.0091 5.1967 8.4034 0.0003 2.3127 N 0.000000 0.015909 0.000000 0.000000 0.0e+00 0.000000 V 0.000050 0.000000 0.317763 0.498129 0.0e+00 0.000000 H 0.000520 0.000000 0.000000 0.232971 1.8e-05 0.000000 O 0.008403 0.000000 0.000000 0.000000 0.0e+00 0.613200 P1 0.000000 0.000000 0.000000 0.000053 0.0e+00 0.000017 P2 0.006935 0.000000 0.000000 0.000000 0.0e+00 0.000000

The cycle of nutrient is described in table 16.2. © L. S. Barreto, 2017. Theoretical Ecology 307 16 Modelling, and Analysis of the Ecosystem

Table 16.2. Nutrient flows in the ecosystem, at equilibrium. FP represents the nutrient in the biomasses, and in the soil at equilibrium

N V H O P1 P2 FP 0.15900 0.00501 0.05196 0.84034 0.00003 0.69352 N 0.0000000 0.0159087 0.0000000 0.0000000 0e+00 0.0000000 V 0.0000501 0.0000000 0.0003178 0.0004981 0e+00 0.0000000 H 0.0005197 0.0000000 0.0000000 0.0023297 2e-07 0.0000000 O 0.0084034 0.0000000 0.0000000 0.0000000 0e+00 0.0613200 P1 0.0000003 0.0000000 0.0000000 0.0000053 0e+00 0.0000017 P2 0.0069352 0.0000000 0.0000000 0.0000000 0e+00 0.0000000

In figure 16.7, the flowchart of the materials (table 16.1) is displayed.

Figure 16.7. Flowchart of materials in the ecosystem. Read the comma as a periode 308 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

We can also display a graph of the internal connectivity of the nutrient cycle, as it is exhibited in figure 16.8.

Figure 16.8. Graph of the internal connectivity of the nutrient cycle, at equilibrium. The size of the nodes are proportional to the nutrient content in the compartments, and the width of the arrows is proportional to the flow of nutrient. This graph emphasizes the structure, and the internal functioning of the ecosystem. This graph was obtained with package igraph (Kolaczyk e Csárdi, 2014) in R

In this simulation we started very high initial values, for reasons that will emerge in chapter 18. We can have a closer picture of the variation of the variables adding the following script:

> > initials<-c(1, 4, rep(1,5), 2, 2, 1) > out<- ode(y=initials, times=t.s, modul) > > plot(out, which=c(1,2), type = "l", lwd = 2, pch = 16, cex = 1.5, + ylab=c('Weight','Biomass'), main=c('N','V')) > > windows() > > plot(out, which=c(3:7), type = "l", lwd = 2, pch = 16, cex = 1.5, + ylab=c(rep('Biomass',5)), main=c('H1','H2','H3','H4','H5')) > > windows() > > plot(out, which=c(8:10), type = "l", lwd = 2, pch = 16, cex = 1.5, + ylab=c(rep('Biomass',3)), main=c('O','P1','P2'))

The graphics of figure 16.9 are produced. © L. S. Barreto, 2017. Theoretical Ecology 309 16 Modelling, and Analysis of the Ecosystem

Figure 16.9A. Graphs of the nutrient, and vegetation

Figure 16.9B. Graphs of the herbivores 310 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.9C. Graphs of the omnivore, and the two predators

16.4 The Effects of the Omission of Interactions

In almost all simulation studies of communities, and ecosystems, done until recent years, the only interaction modelled is the trophic one, this is, the food web. Only in the present century, the researchers started to include other interactions, namely mutualism, parasitism, and pathogenic interactions in their studies (Fontaine et al., 2011; Ings et al., 2009). One way to scrutinize the relevance of the interactions we introduced in our model, is by eliminating them, and after analyse the performance of the amputated ecosystem, maintaining always the same compartments. We implemented this procedure with the following modified ecosystems:  Without mutualism;  Without competition;  Only predation (food web). To all the simulated data provided by this three modified models we successfully fitted MAR(1). In table 16.3, we exhibit the fixed points of the complete model, and its three mentioned variants. © L. S. Barreto, 2017. Theoretical Ecology 311 16 Modelling, and Analysis of the Ecosystem

Table 16.3. Fixed point of the ecosystem, and the three variants considered. TBC= total biomass of the community. H= Shannon-Weaver index of diversity

Ecosystem Without Without Only mutualism competition predation

y1 0,1590 0,1530 0,1584 0,1530

y2 5,0091 5,3605 5,0421 5,3605

y3 0,7431 0,1448 0,7308 0,1448

y4 0,7774 0,1311 0,7680 0,1311

y5 1,2074 1,4284 1,1845 1,4284

y6 1,2347 1,4487 1,2235 1,4487

y7 1,2341 1,4115 1,2226 1,4115

y8 8,4034 8,6579 8,4289 8,6579

y9 0,0003 0,0659 0,0003 0,0659

y10 2,3118 2,3829 2,3187 2,3829 TBC 20,9212 21,0316 20,9194 21,0316 H 1,6197 1,5927 1,6862 1,5927

The results we obtain are from artificial ecosystems, and they depend on the structure, and parametrization used. The effects of the amputations are diverse. The elimination of only mutualism, and simultaneously mutualism, and competition (food web), paradoxically, evince the communities with higher total biomass. The community without competition reveals the higher diversity. The elimination of the interactions have diverse effects on the size of the populations, and nutrient in the soil, at equilibrium. Although this is an oversimplified artificial ecosystem it is already foreseeable that the studies of real communities, and ecosystems using only one interaction will give an incomplete, and eventually an erroneous picture of the studied ecosystem. The analysis of the four simulated ecosystems can be deepen with the application of the analysis of ascendency (Ulanowicz, 1986, 1997, 2004), in an innovative manner.

16.5 Ascendency

The concepts, and indices presented in this section were developed with the purpose of analyse networks including a sole interaction: the food web. If the reader is familiar with the model for ecological succession proposed by Odum (Odum, and Barrett, 2005:Table 8-1), consistent with the model of facilitation, knows that in this concept the succession reveals directionality, this is, the development of the succession is characterized by the emergence of a predictable sequence of communities that evince a set of attributes that have a fixed pattern of change. Some of them are characterized in figure 16.10. In the axis of time occurs the sequence of communities. 312 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.10. Pattern of the evolution of some attributes of the communities, along the succession. The stages of the succession move along the axis of time. To obtain these graphics in R, I adopted the scripts in Basic, displayed in Odum, and Odum (2000:257)

It is our understood that according to the concept of ecosystem development proposed by Jørgensen (2009a), and Jørgensen et al, (2007), this development has also directionality, and in a general manner the graphics in figure 16.9 can be applied to it. In this conception of ecosystem, directionality is a basic property of ecosystems. Ecosystems reveal also openness, hierarchy, connectivity among their components, and complex dynamics. From these properties it is possible to obtain 10 attributes candidates to ecological laws (Jørgensen, 2009b:5, Table 1). Directionality implies self-organization, and self-regulation, this is, due only to internal processes of the system, emerge self-sustained order, and structure. In figure 16.10 we represent a system of four components, limited by a frontier. This is, there is the internal space of the system, and the external space where the system exists. The components interact through a loop of positive effects. The network of figure 16.11 is also called a autocatalytic cycle (ACC), positive feedback loop, loop of positive directed interaction. The mutualistic interaction between two species is a simple ACC. We can say that autocatalyse is a generalized form of mutualism. © L. S. Barreto, 2017. Theoretical Ecology 313 16 Modelling, and Analysis of the Ecosystem

Figure 16.11. Autocatalyse (black arrows), and centripetality (green arrows crossing the system frontier, atracted by its components), in a system with four compartments. Red arrows represent dissipation of non usable materials or/and energy

The ACC gives origin to a spiral of growing benefits to the components of the cycle. It is the autocatalyse that provokes the emergence of directionality, and centripetality (attraction to the centre) in the ecosystem. Centripetality is the capacity of the system to attract resources, and free energy from the surroundings to its internal space, after they had crossed the system’s frontier. In figure 16.12 we attempt to present a graphical synthese of the causal chain triggered by autocatalyse. Autocatalyse promotes the increment of the organization of the ecosystem. The degree of organization is measured by ascendency, to be detailed later.

Figure 16.12. Attempt to synthesize the causal chain related to autocatalysis. Main source of this figure is Ulanowicz (2009)

The archetype of the system studied by the analysis of ascendency is illustrated in figure 16.13. A reification of this archetype is exhibited in figure 16.16. 314 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.13. Ecosystem formed by four compartments that can be biotic or non living. The other compartments can be, or not, related to the outer space as compartment 1 is

This analysis is based in the flows between the compartments or nodes of the network. The following notation is used:

Flow from node i to j: Tij T ij

Flow received by node j from all other nodes i: ∑ T ij=T. j T .j i=1

Flow sent by node i to all other nodes j: ∑ T ij=Ti . T i. j=1 n n

Sum of all flows: ∑∑ T ij=∑ T ij=T.. T .. i=1 j=1 i, j

The remaining text of this section take advantage of Scharler (2009), and Ulanowicz (2004). Below, we introduce the mathematical expressions of the eight indices that we will use. To analyse the ascendency of an ecosystem, we use a set of indices whose aim is a global evaluation of the development of the ecosystem, attained by its autocatalysis. It is underpinned by the theory of information, and the concept of conditional probability. Briefly, we will introduce the formulas of some indices calculated by function enaAscendency, of package enaR (Lau, Borrett, Hines, and Sing, 2017; Lau, Sing, and Borrett, 2017), and their ecological interpretations. For more information on this issue, see Jørgensen et al, (2007:chapter 4), Scharler (2009), Ulanowicz (2004), and references herewith. Let us recall the Shannon-Wiener index of diversity: © L. S. Barreto, 2017. Theoretical Ecology 315 16 Modelling, and Analysis of the Ecosystem

When MacArthur (1955) was seeking for a relation between the diversity, and stability of a community, replaced pi by the ratio Tij/T.., and after multiplied the obtained value by constant k, obtaining a measure of the diversity of the flows in the net work:

(16.1)

The ratio Tij/T.. is the probability that a randomly selected quantum of medium is in transit between compartments i and j. H estimates the incertitude associated to the prediction of to which Tij belongs a randomly sampled quantum, from the network. After, Rutledge et al. (1976) included in equation (16.1) the conditional probability of a quantum that had originated from component i will flow to compartment j (Tij/T.j). This alteration let them show that equation (16.1) can be decomposed as:

H = AMI + Hc (16.2)

Both components are non-negative. AMI is the average mutual information characteristic of the network, and it measures the level of organization of the way the flow occurs in the network. It is linked to the concept of structure, is an indicator of the developmental status of an ecosystem, and its formula is:

(16.3)

Hc measures the residual (conditional) diversity/freedom (known as the conditional entropy in information theory). It is written as:

(16.4)

Later, Ulanowicz (1980) made k=T.., in equation (16.3), and obtained what he named ascendency (A):

(16.5)

Ascendency is a measure of the size and organization of flows, and can be interpreted as the tightness of the constraints that channel trophic linkages. A food web with more trophic specialists, increased cycling, and higher efficiency is characterized by higher values of ascendency. Conversely, lower values for ascendency are associated to a more generalist-based food web, decreased cycling, and lower transfer efficiencies. The upper bound, to ascendency is the developmental capacity. Ascendency associates to the total activity of the system (T..) its level of organization, and measures its development. Theoretically, A increases with the channelling the material flowing into a smaller number of paths, with an increasing flow per link. This situation assumes an evolution in the direction of specialization. This statements brings to light the connectivity of the network, and its density. This issue will be approached ahead. 316 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Before we pursue, let us introduce three simple networks that makes more clear the information given by AMI, and A. We start with a network with four compartments, linked by all possible flows. All flows are equal to 3, as exhibited in figure 16.14.

Figure 16.14. Directed network with four nodes, with the maximum number of links, and all flows equal to 3

Now, we construct a function that calculates AMI, and A, assuming k=1:

ami=function(x){ + m =x + n = ncol(m) + ami = mat.or.vec(n, n) + T = sum(m) + for (i in 1:n) { + for (j in 1:n) { + if (m[i, j] == 0) { + ami[i, j] = 0 + } + else { + ami[i, j] = m[i, j]/T * log2((m[i, + j] * T)/(sum(m[, j]) * sum(m[i, + ]))) + } + } + } + print('AMI') + print(sum(ami)) + print('ASC') + print(T*sum(ami)) + } The matrix of the network in figure 16.13 is:

x=matrix(rep(3,16), nrow=4,ncol=4) © L. S. Barreto, 2017. Theoretical Ecology 317 16 Modelling, and Analysis of the Ecosystem

We apply the function to this matrix:

> ami (x) [1] "AMI" [1] 0 [1] "ASC" [1] 0

This network is undifferentiated because all nodes are equally characterized. Thus the values of A, and AMI are zero. Now, we maintain the total flow equal to 48 48 (3 x 16 = 48), and we reduce the links to half of the maximum possible number (8), all with the same flow 6, as illustrated in figure 16.15.

Figure 16.15. Directed network with four nodes, with half of the maximum number of links, and all flows equal to 6. The total flows remains 48

The matrix of this second network is:

> x=matrix(c(0,0,6,6,6,0,0,6, + 6,6,0,0,0,6,6,0), nrow=4,ncol=4, byrow=T)

Applying the function we obtain:

> ami (x) [1] "AMI" [1] 1 [1] "ASC" [1] 48

Again, we reduce the number of paths of the second network by half, and we duplicate the flow that circulates in each one (figure 16.16). The total flow remains 48. The correspondent matrix is:

> x=matrix(c(0,0,0,12,12,0,0,0, + 0,12,0,0,0,0,12,0), nrow=4,ncol=4, byrow=T) > 318 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.16. Directed network with four nodes, with one fourth of the maximum number of links, and all flows equal to 12. The total flows remains 48

For this third network we obtain:

> ami (x) [1] "AMI" [1] 2 [1] "ASC" [1] 96

From figures 16.13 to 16.15, acting constrictions give origin to a process of increasing differentiation, and specialization in the way the flow of materials is processed in the network. Favoured by autocatalysis, these changes increase the degree of self-organization, and efficiency. The evolution in specialization, and organization is mirrored by the values of AMI (012), and ascendency (04896).

Hc can be multiplied by T.. to obtain the overhead Φ of the system:

(16.6)

H (equation (16.1)) can be multiplied by T.. to obtain the development capacity, C:

(16.7)

Now, we are able to obtain from equation (16.2) the following one:

(16.8)

This equation shows that in a given ecosystem, if its development capacity is constant, its development (increasing of A) sacrifices the overhead. A high level of ascendency is associated with high perfection, but low flexibility to cope with unexpected perturbations. The development of the ecosystem is a balanced compromise between A, and Φ. © L. S. Barreto, 2017. Theoretical Ecology 319 16 Modelling, and Analysis of the Ecosystem

Presumably, mature ecosystems in very stable environments must have smaller values of overhead then ecosystems in more unstable environments. For our purposes, from function enaAscendency we obtain AMI, A (ASC), Φ (OH), C (CAP), the ratios A/C (ASC.CAP), Φ/C (OH.CAP), robustness R,and effective linkage density ELD. Let us clarify the two last indices. Robustness R estimate the capacity of the system to resist to perturbations, and disruptive impacts. Is written as:

R = -1 * A/C * log(A/C) (16.9)

Effective linkage density is calculated as a weighted geometric mean. It measures the degree of connectivity of a network. Its expression is:

(16.10)

We illustrate the use of function enaAscendency, applying it to the ecosystem Cone Springs (figure 16.17):

> library(enaR) Warning message: package ‘enaR’ was built under R version 3.4.0 > data(troModels) > m.list=troModels > > unpack(m.list[[6]])#description of the ecosystem $F PLANTS BACTERIA DETRITUS FEEDERS CARNIVORES DETRITUS PLANTS 0 0 0 0 8881 BACTERIA 0 0 75 0 1600 DETRITUS FEEDERS 0 0 0 370 200 CARNIVORES 0 0 0 0 167 DETRITUS 0 5205 2309 0 0

$z [1] 11184 0 0 0 635

$r [1] 2003 3275 1814 203 3109

$e [1] 300 255 0 0 860

$y [1] 2303 3530 1814 203 3969

$X [1] 285.0 117.0 60.0 17.0 3579.4

$living [1] TRUE TRUE TRUE TRUE FALSE

> enaAscendency(m.list[[6]]) H AMI Hr CAP ASC OH ASC.CAP OH.CAP [1,] 3.20096 1.336447 1.864513 135864.7 56725.49 79139.25 0.4175144 0.5824856 320 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

robustness ELD TD A.input A.internal A.export A.respiration [1,] 0.3646722 1.908258 2.525286 19147.85 29331.98 1051.765 7193.894 OH.input OH.internal OH.export OH.respiration CAP.input CAP.internal [1,] 6221.751 29832.46 7810.812 35274.22 25369.6 59164.44 CAP.export CAP.respiration [1,] 8862.578 42468.11 >

Figure 16.17. Network of the Cone Springs ecosystem. Dissipation losses are represented by the red colour

In the next section, we use the analysis of ascendency (the eight described indices) in an innovative manner, expanding its utility in the study of ecosystems.

16.6 The Ascendency of the Matrix of Total Positive Effects

In this section we apply the network analysis to the theory already developed for the community, and the ecosystem. It is our understanding that the elaborations we next introduce reinforce the holistic analysis of ecosystems. Jørgensen (2009b:4) stated:

The chapter Indirect Effects in Ecology focuses on perhaps the most important network property: the presence of a strong indirect effect that in many cases may even exceed the direct effect.

Salas, and Borrett (2011) observed the dominance of indirect effects in 50 trophic ecosystem networks. From these two texts, we conclude that total effects (TE = direct effects + indirect effects), must be the more important factors in shaping the structure, and dynamics of the ecosystems. Given the relevance of autocatalysis, we infer that the the matrix of total positive effects (MTPE; (total negative effects=0) deserves to be scrutinized. We are interested in the number of TPE, and how they connect the nodes in the network of the ecosystem components. We conjecture that the greater is the number of nodes corresponding to TPE, and the greater are their connectivities, the greater are the chances for the ecosystem to grow, and develop. We also admit that the ascendency analysis of the MTPE can be useful to characterize the ecosystem during its development, or to compare it with other ecosystems. We conjecture that this analysis must be more informative, and close to reality than the sole analysis of the material flows. © L. S. Barreto, 2017. Theoretical Ecology 321 16 Modelling, and Analysis of the Ecosystem

In accordance with these statements, we established the matrices of the total effects of the four ecosystems whose fixed points are inserted in table 16.1. In figure 16.17 we introduce their correspondent chromatic matrices for easier comparison. We also submitted the MTPE of these ecosystems to the analysis of ascendency. The eight relevant indices are exhibited in table 16.4. From this table we conclude the following:  The ecosystem with all interactions suggests that the greater is the variety of interactions the greater is the organization of the system (largest ASC.CAP) and its robustness;  The ecosystem without competition has its degree of development reinforced (largest ASC), and also its flexibility, and capacity of development (largest OH, and CAP). Thus competition does not favour neither adaptability, nor connectivity (largest ELD when competition is absent).  Mutualism does not favour the the degree of adaptation of the ecosystem because when it is absent the relation OH.CAP is maximum;  When the only interaction present is trophic (only food web), it is confirmed that the elimination of the other interactions causes loss of information, thus it gives wrong, and incomplete results. The absence of mutualism in this ecosystem reduces also its capacity of adaptation of (second lowest value of OH).  The relevance, and interpretation, in this context, of trophic depth is less clear to us, thus it is ignored.

Table 16.4. The analysis of ascendency of the MTPE of the four mentioned ecosystems

All Without Without Only the food web interactions competition mutualism AMI 0,9630886 0,9617592 0,5675819 0,6404051 ASC 1608,594 2044,6 1074,629 1660,262 OH 4371,033 6046,707 3917,225 6041,902 CAP 5979,627 8091,307 4991,854 7702,164 ASC.CAP 0,2690125 0,252691 0,2152766 0,2155579 OH.CAP 0,7309875 0,747309 0,7847234 0,7844421 robustness 0,3532127 0,3475987 0,3306286 0,3307791 ELD 2,476839 2,679854 2,048363 2,242732 322 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Figure 16.17. Chromatic representations of the MTE of the modelled ecosystem, and its variants. Positive values are represented by green colour, and negative values in red. In the upper row, from left to right, ecosystem with all interactions, and without competition. In the bottom row, from left to right, without mutualism, and only food web. The presence of mutualism conspicuously affects the MTE.

16.7 Conclusive Comments

The results presented in this chapter rise the following comments:  Without discontinuities, we can construct an integrated hierarchy of models for the biosystems we approached (see figure 15.2);  The same model can represent simultaneously more then one type of interaction;  It is highly probable that ecosystems can be modelled as MAR (1);  Sections 16.3, and 16.5 show that we can approach in an integrated way both the dynamics of the ecosystem, and the connected ecological networks;  Even in simple ecosystems, as the one here used, the simultaneous action of direct, and indirect effects render the prediction of the system responses to perturbations very difficult;  The simultaneous modelling of all interactions opens a window of more embracing pre- dictability, impossible with the separated modelling of each interaction; © L. S. Barreto, 2017. Theoretical Ecology 323 16 Modelling, and Analysis of the Ecosystem

 The procedures suggested to improve the modelling of communities (section 15.4) can be here applied;  Models integrating simultaneously several types of interaction open new possibilities to investigate the structure, and functioning of the ecosystem. For instance, we can use these kind of models to investigate the relations among functional, and taxonomic diversity, and the overall functioning of the ecosystem. We proposed a procedure that let us overcome the most frequently appointed insufficien- cies occurring in the present analysis of communities, and ecosystems (e.g., Ings, T. C. et al., 2009). In the next Part, we attempt to find the solutions to two important ecological problems, applying the theory we here established.

16.8 References, and Related Bibliography

Barreto, L. S., 2011. Modelling and Simulating Omnivory. Silva lusitana, 19(1):47-58. http:www.inrb.pt/inia/silva-lusitana Borrett, S. R. e M. K. Lau, 2017. Vignette: enaR. http://CRAN.R-project.org/package=enaR Canham, C.D., J.J. Cole, and W.K. Lauenroth, Compiladores, 2003. Models in Ecosystem Science. Princeton Univer- sity Press, Princeton, NJ. Chapin III, F. S., P. A. Matson, and H. A. Mooney, 2002. Principles of Terrestrial Ecosystem Ecology. Springer, New York. Csárdi, G., and T. Nepusz, 2006. The igraph software package for complex network research. InterJournal, Com- plex Systems, 1695. http://igraph.org Fath, B. D., 2012. Overview of Network Environ Analysis: A systems analysis technique for understanding complex ecological systems. Biologi Italiani, Marzo 2012:20-27. Fath, B.D., and B. C. Patten, 1999. Review of the foundations of network environ analysis. Ecosystems, 2:167–179. Fath, B. D, U. M. Scharler, R,t E. Ulanowicz, and B. Hannon, 2007. Ecological network analysis: network construction. Ecological Modelling, 208:49–55. Fontaine, C., et al., 2011. The ecological and evolutionary implications of merging different types of networks. Ecology Letters, 14: 1170–1181. doi: 10.1111/j.1461-0248.2011.01688.x Hampton, S. E. et all, 2013. Quantifying effects of abiotic and biotic drivers on community dynamics with multivariate autoregressive (MAR) models. Ecology, 94(12): 2663–2669. Holmes, E. E., E. J. Ward e K. Wills, 2012. MARSS: Multivariate Autoregressive State-space Models for Analyzing Time-series Data. The R Journal, 4(1):11-19. Holmes, E., E. Ward, and K. Wills, 2013. MARSS: Multivariate Autoregressive State-Space Modeling. R package version 3.9. Ings, T. C. et al., 2009. Ecological networks – beyond food webs. J. Anim. Ecol., 78: 253–269. Jain, S. e S. Krishna, 2001. Crashes, Recoveries, and ‘Core-Shifts’ in a Model of Evolving Networks. Proceedings of the National Academy of Sciences USA, 98:543-547. Jain, S., and S. Krishna, 2002. Large Extinctions in an Evolutionary Model: The Role of Innovation and Keystone Species. Proceedings of the National Academy of Sciences USA, 99:2055-2060. Jørgensen, S. E., 2009, (Main editor). Ecosystem Ecology. Elsevier B.V. Jørgensen, S. E. et al., 2007, (Editors). A New Ecology. Systems Perpective. Elsevier B.V. Kolaczyk, E. D., and G. Csárdi, 2014. Statistical Analysis of Network Data with R. Springer, Berlin. Lau, M.K., S.R. Borrett, D.E. Hines, and P. Sing, 2017. enaR: Tools for ecological network analysis (ena) in R. R package version 2.10.0. http://CRAN.R-project.org/package=enaR Lau, M. K., P. Sing, and S.R. Borrett, 2017. Vignette: enaR. http://CRAN.R-project.org/package=enaR Scharler, U. M., 2009. Ecological Network Analysis, Ascendency. In S. E. Jørgensen, Main editor, Ecosystem Ecology. Elsevier, Amsterdam. Páginas 57-64. Scheef L., 2013. MAR1: Multivariate Autoregressive Modeling for Analysis of Community Time-Series Data. R package version 1.0. Ulanowicz, R.E., 1980. An hypothesis on the development of natural communities. J. theor. Biol. , 85: 223–245. Ulanowicz, R. E., 1986. Growth and Development: Ecosystems Phenomenology. Springer–Verlag, New York. 324 © L. S. Barreto, 2017. Theoretical Ecology 16 Modelling, and Analysis of the Ecosystem

Ulanowicz, R. E., 1997. Ecology, the Ascendent Perspective. Columbia University Press, New York. Ulanowicz, R. E. 2004. Quantitative methods for ecological network analysis. Comput. Biol. Chem., 28:321–339. URL http://dx.doi.org/10.1016/j.compbiolchem.2004.09.001. Ulanowicz, R.E., 2009. Autocatalysis. Em S. E. Jørgensen, (Main editor), Ecosystem Ecology. Elsevier, Amsterdam. Pagess 41-43. Ulanowicz, R. E. 2011. Quantitative Methods for Ecological Network Analysis and Its Application to Coastal Ecosystems. Treatise on Estuarine and Coastal Science, 2011, Vol.9, 35-57, DOI: 10.1016/B978-0-12-374711-2.00904- 9 Weathers, K. C., D. L. Strayer, and G. E. Likens, 2013. Fundamentals of Ecosystem Science. Academic Press/Elsevier, Waltham, MA , USA. © L. S. Barreto, 2017. Theoretical Ecology 325 PART IV

PART IV

“In other words, we are not simply looking for truth, we are after interesting and enlightening truth, after theories which offer solution to interesting problems.”

Karl Popper – Objective Knowledge

Applications

We introduce in this section the following: ● A procedure to identify keystone species, and controlling factors in ecosystems; ● As an extension of the previous application, a procedure to scrutinize the sensitivity of the ecosystem components to perturbations. 326 © L. S. Barreto, 2017. Theoretical Ecology 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

17 Identification of Keystone Species, and Controlling Components in the Ecosystem

17.1 Introduction

The increasing disruptive effect of humanity on the biosphere, render the detection of keystone species, and of controlling components of the ecosystem in almost an ethical imperative for ecologists. It is not surprising that several authors had already approached this subject, such as Libralato, Christensenc, and Paulyc (2006), Smith et al. (2014), and Zhao et al. (2016). In this chapter we will establish a method to identify keystone species, and controlling components of the structure, and dynamics of ecosystems. We apply the proposed method to simulated, and real data. This chapter is underpinned by Barreto (2016:section 37.8; in press). In this chapter, we use the following R packages: MARSS (Holmes, Ward, and Wills, 2012), to obtain real data; mAr (Barbosa, 2012) to fit the MAR(1); igraph (Csárdi e Nepusz, 2006; Kolaczyk, and Csárdi, 2014) for network analysis. My theory for forest structure, and dynamics (Barreto, 2011) is an application, of the theory presented in this book, to forest stands. More specific applications are in chapters 8 (growth, regeneration, and survival indices of tree species), 9 (life-history strategies of tree species), and 11 (plant growth, and the Kleiber’s law) of the same book.

17.2 The Proposed Procedure

The procedure integrates the following concepts, and models: - The conception of communities, and ecosystems as stochastic linear dynamic systems that can be modelled by MAR(1) (Part III). - The concept of autocatalysis (sections 16.5, 16.6). - The concept of total positive effects (section 8.6). - The concept of transitivity or clustering coefficient, from network analysis (e.g., Kolaczyk, and. Csárdi, 2014). The procedure is sustained by the following conjecture:

In the network associated to the matrix of total positive effects, the nodes with high transitivity must give a strong contribution to the process of autocatalysis, and thus they have high probability of being keystone species or controlling factors.

Given the time series of the components of the community or the ecosystem, the steps of the procedure are the following ones (figure 17.1): 1. Fit a MAR(1) model to the data; 2. From the matrix of the coefficients extract the matrix of the total positive effects; 3. Use this matrix as a weighted adjacency matrix, and obtain the correspondent network; 4. Calculate the clustering coefficients of the nodes; 5. Select the nodes (components) with high coefficients as candidates to be keystone species, or controlling components of the ecosystem. © L. S. Barreto, 2017. Theoretical Ecology 327 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

Now, we introduce the two upper mentioned applications of the method.

Figure 17.1. Flowchart of the procedure to identify keystone species, and controlling components in the ecosystem

17.3 The Identification of Keystone Species

In section 16.3, after figure 16.6, we identified y10, the top predator, as a keystone species. To verify the correctness of the proposed procedure, we applied it to the four ecosystems described in section 16.4 (table 16.3). The results we obtained are displayed in table 17.1.

Table 17.1. Weighted values of the transitivity of the networks associated to the MTE of the four mentioned ecosystems. WO = without omissions; WC = without competition; WM = without mutualism; OT = only trophic interaction

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 WO 0,333 0,700 0,571 0,476 0,600 0,667 0,900 0,600 0,667 1,000 WC 0,444 0,467 0,600 0,600 0,600 0,600 0,667 1,000 0,833 1,000 WM 0.333 0.300 0.400 0.667 0.500 0.500 0.667 1.000 NaN 0.667 OT 0.476 0.600 0.476 0.800 0.800 0.524 0.667 0.800 0.667 0.833

We simulated the ecosystems WO, and WC without y10 and the result was ‘NA’ for the values of the 9 variables of each ecosystem, confirming the collapse of the simulated models. 328 © L. S. Barreto, 2017. Theoretical Ecology 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

In agreement with table 17.1, we verified that in systems without competition or mutualism, the elimination of the omnivore (y8), the simulations also report the collapse of the systems. Our conclusion is:

In appendix to this chapter, I present the R scripts of the analysis of the ecosystem without competition. I was unable to find real data with time series covering several years, and containing a keystone species. Thus my analysis is confined to simulated data.

17.4 The Identification of Controlling Components in Ecosystems

In this section, I will use data of the well known research project of Isle Royale (www.isleroyalewolf.org; Nelson, Peterson, and Vucetich, 2008). This project is the longest uninterrupted study of a predator–prey relationship in the world. The columns of the data, from left to right, are: the number of wolves (W), the number of moose (M), average temperature of January, and February (wtemp), average precipitation of January, and February (wprecip), and average temperature of July, and September (stemp). To apply the procedure to this data, the following script is used:

> library(MARSS) > library(mAr) > library(igraph) > #Data isleRoyal > royale.dat=(isleRoyal[1:53,c(2,3,4,10,6)]) > > #Obtaining MAR(1) > y=mAr.est(royale.dat,1,1) > > #The matrix of coefficients is matrix y$AHat > E=-ginv(y$AHat) # Total effects matrix > > #Obtaining total positive effects matrix > AS=E > n=1:5 > for (i in n) { + for (j in n) { + if (AS[i,j]<0) {AS[i,j]=0} + } + } > rownames(AS)=c('W','M','wtemp ','wprecip','stemp') > colnames(AS)=rownames(AS) > > #Network analysis > net.adjacency=AS > net= graph.adjacency(net.adjacency) > > #Desired transitivity > TR=transitivity(net, type="weighted") © L. S. Barreto, 2017. Theoretical Ecology 329 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

> round(TR,3) [1] 0.667 NaN 0.667 1.000 1.000 The transitivity of the nodes

The output of the script is inserted in table 17.2.

Table 17.2. The transitivity values of the components of the data of Isle Royale

W M wtemp wprecip stemp

0.667 NaN 0.667 1.000 1.000

To evaluate the results in table 17.2, I use the appreciation of the project inserted in Nelson, Peterson, and Vucetich (2008). In page 108 of this paper it is stated the following: - ‘Wolves seemed to have relatively little impact on moose abundance’, and they are the least important factor that affects short-term fluctuations in moose abundance. - ‘Climatic factors (such as summer heat and winter severity) are much more important’. In table 17.2, only two climatic factors have values of transitivity equal to 1. One parameter is measured in summer (temperature), and the other in winter (precipitation). Thus, the results in table 17.2 agree with the empirical evidence recorded in Nelson, Peterson, and Vucetich (2008:108).

17.5 Conclusive Remarks

The results obtained in sections 17.3, and 17.4 support the application of the proposed procedure to identify keystone species, and controlling components in ecosystems. The discovery of the relation between autocatalysis and total positive effects is conceptually, and theoretically relevant. The relevance of the process of autocatalysis for the structure, and dynamics of ecosystems is here corroborated, and reinforced. It is also shown that the aggregation of several different tools, and concepts in new synthesis can give rise to new insights, and operational procedures, capable of solving relevant problems. The method reveals a shortcoming: its exigence of time series of several years. Generally, ecological projects do not last so long. Probably, we must revise the funding of ecological projects, as the administrative time scale is not coincident with the one of some ecological research projects. Otherwise, we will never acquire the information we need to successfully overcome the environmental crisis. We will return to the results of this chapter in Part V, dedicated to the overall evaluation of the theory.

17.6 References, and Related Bibliography

Barbosa, S. M., 2012. mAr: Multivariate AutoRegressive analysis. R package version 1.1-2. https://CRAN.R- project.org/package=mAr Barreto, L. S., 2011. From Trees to Forests. A Unified Theory. E-book. Costa de Caparica. Barreto, L. S., 2016. Ecologia Teórica. Uma outra Explanação. III. Comunidade e Ecossistema. E-book. Costa de Ca- parica. 330 © L. S. Barreto, 2017. Theoretical Ecology 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

Barreto, L. S., in press. A Procedure to Identify Keystone Species, and Controlling Components in Ecosystem. Sub- mitted to Silva Lusitana in April, 2017. Csárdi, G., and T. Nepusz, 2006. The igraph software package for complex network research. InterJournal, Complex Systems, 1695. http://igraph.org Holmes, E., E. J. Ward, and K. Wills, 2012. MARSS: Multivariate Auto-Regressive State-Space Models for Analizing Time Series. The R Journal, 4(1):11-19. Jain, S., and S. Krishna, 2001. Crashes, Recoveries, and ‘Core-Shifts’ in a Model of Evolving Networks. Proceedings of the National Academy of Sciences USA, 98:543-547. Jain, S., and S. Krishna, 2002. Large Extinctions in an Evolutionary Model: The Role of Innovation and Keystone Species. Proceedings of the National Academy of Sciences USA, 99:2055-2060. Jordán, F., 2009. Keystone species and food webs. Philos Trans R Soc Lond B Biol Sci., 364(1524): 1733–1741. doi: 10.1098/rstb.2008.0335 Jørgensen, S. E., (Main editor), 2009a. Ecosystem Ecology. Elsevier, Amsterdam. Kolaczyk, E. D., and G. Csárdi, 2014. Statistical Analysis of Network Data with R. Springer, Berlin. Libralatoa, S., V. Christensenc, and D. Paulyc, 2006. A method for identifying keystone species in food web models. Ecological Modelling, 195:153–171.´ Nelson, M. P., Rolf O. Peterson, and John A. Vucetich, 2008. The Isle Royale Wolf–Moose Project: Fifty Years of Challenge and Insight. The George Wright Forum, 25(2):98-113. Ramsey, D., and C. Veltman, 2005. Predicting the effects of perturbations on ecological communities: what can qualitative models offer?. Journal of Animal Ecology, 74: 905–916. doi: 10.1111/j.1365-2656.2005.00986.x Salas, A. K., and S. R. Borrett, 2011. Evidence for the dominance of indirect effects in 50 trophic ecosystem networks. Ecological Modelling, 222 (2011): 1192–1204. Scharler, U. M., 2009. Ecological Network Analysis, Ascendency. Em S. E. Jørgensen, Compilador principal, Ecosystem Ecology. Elsevier, Amsterdam. Páginas 57-64. Smith, C. et al, 2014. Report on identification of keystone species and processes across regional seas. Deliverable 6.1, DEVOTES Project. 105 pp + 1, Annex. Tanner, J. E., T. P. Hughes, and J. H. Connell, 1994. Species coexistence, keystone species, and succession: a sensitivity analysis. Ecology, 75(8):2204-2219. Ulanowicz, R.E., 1980. An hypothesis on the development of natural communities. J. theor. Biol. , 85: 223–245. Ulanowicz, R.E., 2004. Quantitative methods for ecological network analysis. Computational Biology and Chemistry, 28:321 – 339. Ulanowicz, R.E., 2009. Autocatalysis. Em S. E. Jørgensen, (Main editor), Ecosystem Ecology. Elsevier, Amsterdam. Pagess 41-43. Zhao, L. et al., 2016. Weighting and indirect effects of identify keystone species in food webs. Ecology Letters, 19:1033-1040.

Appendix. R scripts for the simulation of the ecosystem without omissions, and without competition

See section 16.3 for the English comments in the scripts. Simulation without omission of any interaction:

> library(deSolve) > library(MASS) > library(rootSolve) > > modul<-function(times,y,parms) { + n<-y + + r<-c(0.05, 0.09, 0.071, 0.057, 0.055, 0.07, 0.06, 0.035, 0.02) + b<-c(20, 0.2, 0.21, 0.28, 0.29, 0.27) + b7<-c(b[1], b[2], b[3])/3 + b8<-c(b[4], b[5], b[6])/2.5 + bm7<-mean(b7)/5 + bm8<-mean(b8)/4 © L. S. Barreto, 2017. Theoretical Ecology 331 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

+ b9<-c(bm7, bm8) + + + #planta:consumo total dos herbívoros e e omnívoro + #herb & omniv + ex<-sum( y[2]*y[3]/(70+0.2*y[2]), y[4]*y[2]/(90+0.3*y[2]), y[5]*y[2]/ (90+0.3*y[2]), y[6]*y[2]/(80+0.3*y[2]), y[7]*y[2]/(75+0.3*y[2]), y[8]*y[2]/ (83+0.3*y[2])) + + #competição e predação dos herbívoros + #inicial/final<-iniciais/(bi*resouce) + + a1n<-c(0, 0, 0.02, 0.03, 0.04) + y1n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a2n<-c(0, 0, 0.01, 0.02, 0.03) + y2n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a3n<-c(-0.04, -0.03, 0, 0.01, 0.02) + y3n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a4n<-c(-0.04, -0.03, -0.02, 0, 0) + y4n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a5n<-c(-0.04, -0.03, -0.02, 0, 0) + y5n<-log(c(y[3], y[4], y[5], y[6], y[7])) + + #efeitos da competição + comp2<-0 + comp3<-0 + comp4<-0 + comp5<-0 + comp6<-0 + + #consumos dos predadores y[8], y[9] e y[10] + prey7<-y[8]*c(y[3]/(50+0.2*y[3]), y[4]/(60+0.3*y[4]), y[9]/ (45+0.23*y[9])) + prey8<-y[9]*c(y[5]/(50+0.21*y[5]), y[6]/(60+0.3*y[6]), y[7]/ (70+0.3*y[7])) + prey9<-c(y[10]*y[8]/(30+0.2*y[8]), y[10]*y[9]/(40+0.3*y[9])) + #comp. oblig mutual prey + + #capacidade de sustentação do predador e omnívoroy[8] + k7<-sum(b7*c(y[2], y[3], y[4])); + + #capacidade de sustentação do predador y[9] + k8<-sum(b8*c(y[5], y[6], y[7])) + #capacidade de sustentação do predador y[10] + k9<-sum(b9*c(y[8], y[9])) + #fração da biomassa caída + fr=0.01 + biom=c(y[2:10]) + m=fr*biom # biomassa caída + #teor de nutriente nas biomassas + te=c(0.001,rep(0.01,5),rep(0.1,2),0.3) + + #Equações diferenciais do nutriente e das espécies + dy1.dt<-sum(te*m)-y[2]*y[1]/(50+0.4*y[1]) + dy2.dt<-y[2]*r[1]*(log(1000*y[1])-log(y[2]))-ex-m[1] + dy3.dt<-y[3]*r[2]*(8*y[4]-10^(-10))*(log(b[2]*y[2])-log(y[3]))- prey7[1]-m[2] + dy4.dt<-y[4]*r[3]*(9*y[3]-10^(-10))*(log(b[3]*y[2])-log(y[4]))- prey7[2]-m[3] + dy5.dt<-y[5]*r[4]*(log(b[4]*y[2])-log(y[5]))-prey8[1]-m[4] 332 © L. S. Barreto, 2017. Theoretical Ecology 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

+ dy6.dt<-y[6]*r[5]*(log(b[5]*y[2]*(2-exp(-0.003*y[7])))-log(y[6]))- prey8[2]-m[5] + dy7.dt<-y[7]*r[6]*(log(b[6]*y[2]*(2-exp(-0.03*y[7])))-log(y[7]))- prey8[3]-m[6] + dy8.dt<-y[8]*r[7]*(log(k7)-log(y[8]))-prey9[1]-m[7] + dy9.dt<-y[9]*r[8]*(log(k8)-log(y[9]))-prey9[2]-prey7[3]-m[8] + dy10.dt<-y[10]*r[9]*(log(k9)-log(y[10]))-m[9] + + return(list(c(dy1.dt,dy2.dt,dy3.dt, dy4.dt, dy5.dt, dy6.dt,dy7.dt,dy8.dt, dy9.dt, dy10.dt))) + } > > #parms<-c(r, b, k) > initials<-c(40, 400, 36, 36, 36, 36, 36, 12, 12, 4) > t.s<- seq(1, 300, by=0.1) > > > out<- ode(y=initials, times=t.s, modul) > #matplot(out[,1], out[,-1], type="l", xlab="Tempo", ylab="N") > > > library(MASS) > library(rootSolve) > library(mAr) Warning message: package ‘mAr’ was built under R version 3.0.3 > y<-initials > ST2 <- runsteady(y=y,func=modul,parms=parms,times=c(0,5000)) > ye<-ST2$y > ye  fixed point [1] 0.1583559621 5.0421103141 0.7307663172 0.7680274823 1.1845037245 [6] 1.2234848143 1.2225728496 8.4288766952 0.0002764512 2.3187497186 > > h=out[,-1] > > g<-seq(1,2991,10) > > M=h[g,] > > y=mAr.est(M,1,1) > y$SBC [1] -14.9709 > E=-ginv(y$AHat) ># Criar a METP > AS=E > n=1:10 > for (i in n) { + for (j in n) { + if (AS[i,j]<0) {AS[i,j]=0} + + } + } > rownames(AS)=c('N','V','H1','H2','H3','H4','H5','O','P1','P2') > colnames(AS)=rownames(AS)  AS is the MTPE > library(igraph) > > teia.adjacency=AS > > teia= graph.adjacency(teia.adjacency) > TR=transitivity(teia, type="weighted") © L. S. Barreto, 2017. Theoretical Ecology 333 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

> round(TR,3) [1] 0.444 0.467 0.600 0.600 0.600 0.600 0.667 1.000 0.833 1.000  Transitivity of the nodes > > library(enaR) > > #fixed point > X=ye > > fluxos=AS > influxos <- rep(0, 10) > > defluxos <- influxos > storage=X > > modelo.ben <- pack(flow=fluxos, + input=influxos, + export=defluxos, + respiration=c(0, defluxos[2:6]/100), + storage=storage, + living=c(FALSE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE)) > > > enaAscendency(modelo.ben)  Analysis of ascendency AMI ASC OH CAP ASC.CAP OH.CAP robustness ELD [1,] 0.9617592 2044.6 6046.707 8091.307 0.252691 0.747309 0.3475987 2.679854 TD [1,] 1.947683

Simulation without competition, and without top predator (y10):

> modul<-function(times,y,parms) { + n<-y + + r<-c(0.05, 0.09, 0.071, 0.057, 0.055, 0.07, 0.06, 0.035) + b<-c(20, 0.2, 0.21, 0.28, 0.29, 0.27) + b7<-c(b[1], b[2], b[3])/3 + b8<-c(b[4], b[5], b[6])/2.5 + bm7<-mean(b7)/5 + bm8<-mean(b8)/4 + b9<-c(bm7, bm8) + + + #planta:consumo total dos herbívoros e e omnívoro + #herb & omniv + ex<-sum( y[2]*y[3]/(70+0.2*y[2]), y[4]*y[2]/(90+0.3*y[2]), y[5]*y[2]/ (90+0.3*y[2]), y[6]*y[2]/(80+0.3*y[2]), y[7]*y[2]/(75+0.3*y[2]), y[8]*y[2]/ (83+0.3*y[2])) + + #competição e predação dos herbívoros + #inicial/final<-iniciais/(bi*resouce) + + a1n<-c(0, 0, 0.02, 0.03, 0.04) + y1n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a2n<-c(0, 0, 0.01, 0.02, 0.03) + y2n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a3n<-c(-0.04, -0.03, 0, 0.01, 0.02) + y3n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a4n<-c(-0.04, -0.03, -0.02, 0, 0) 334 © L. S. Barreto, 2017. Theoretical Ecology 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

+ y4n<-log(c(y[3], y[4], y[5], y[6], y[7])) + a5n<-c(-0.04, -0.03, -0.02, 0, 0) + y5n<-log(c(y[3], y[4], y[5], y[6], y[7])) + + #efeitos da competição + comp2<-0 + comp3<-0 + comp4<-0 + comp5<-0 + comp6<-0 + + #consumos dos predadores y[8], y[9] e y[10] + prey7<-y[8]*c(y[3]/(50+0.2*y[3]), y[4]/(60+0.3*y[4]), y[9]/ (45+0.23*y[9])) + prey8<-y[9]*c(y[5]/(50+0.21*y[5]), y[6]/(60+0.3*y[6]), y[7]/ (70+0.3*y[7])) + + #comp. oblig mutual prey + + #capacidade de sustentação do predador e omnívoroy[8] + k7<-sum(b7*c(y[2], y[3], y[4])); + + #capacidade de sustentação do predador y[9] + k8<-sum(b8*c(y[5], y[6], y[7])) + #capacidade de sustentação do predador y[10] + + #fração da biomassa caída + fr=0.01 + biom=c(y[2:10]) + m=fr*biom # biomassa caída + #teor de nutriente nas biomassas + te=c(0.001,rep(0.01,5),rep(0.1,2)) + + #Equações diferenciais do nutriente e das espécies + dy1.dt<-sum(te*m)-y[2]*y[1]/(50+0.4*y[1]) + dy2.dt<-y[2]*r[1]*(log(1000*y[1])-log(y[2]))-ex-m[1] + dy3.dt<-y[3]*r[2]*(8*y[4]-10^(-10))*(log(b[2]*y[2])-log(y[3]))- prey7[1]-m[2] + dy4.dt<-y[4]*r[3]*(9*y[3]-10^(-10))*(log(b[3]*y[2])-log(y[4]))- prey7[2]-m[3] + dy5.dt<-y[5]*r[4]*(log(b[4]*y[2])-log(y[5]))-prey8[1]-m[4] + dy6.dt<-y[6]*r[5]*(log(b[5]*y[2]*(2-exp(-0.003*y[7])))-log(y[6]))- prey8[2]-m[5] + dy7.dt<-y[7]*r[6]*(log(b[6]*y[2]*(2-exp(-0.03*y[7])))-log(y[7]))- prey8[3]-m[6] + dy8.dt<-y[8]*r[7]*(log(k7)-log(y[8]))-m[7] + dy9.dt<-y[9]*r[8]*(log(k8)-log(y[9]))-prey7[3]-m[8] + + + return(list(c(dy1.dt,dy2.dt,dy3.dt, dy4.dt, dy5.dt, dy6.dt,dy7.dt,dy8.dt, dy9.dt))) + } > > #parms<-c(r, b, k) > initials<-c(40, 400, 36, 36, 36, 36, 36, 12, 12) > t.s<- seq(1, 300, by=0.1) > > > out<- ode(y=initials, times=t.s, modul) There were 50 or more warnings (use warnings() to see the first 50) © L. S. Barreto, 2017. Theoretical Ecology 335 17 Identification of Keystone Species, and Controlling Components in the Ecosystem

> out[2991,] time 1 2 3 4 5 6 7 8 9 300 NA NA NA NA NA NA NA NA NA The system collapsed > 336 © L. S. Barreto, 2017. Theoretical Ecology 18 Developmental, Structural, and Functional Sensitivities to Initial Values

18 Developmental, Structural, and Functional Sensitivities to Initial Values

18.1 Introduction

With the growth, and spread of human population, the prediction of the response of populations, communities, and ecosystems to anthropogenic impacts is an issue that must receive the attention of ecology researchers. It is our understanding that keystone species, and controlling factors are the extreme case of the sensitivity of the ecosystem to their components. Thus, the transitivity of the biotic, and non living components of the ecosystem (e.g., table 17.1) are a metric for the developmental, structural, and functional sensitivities of the ecosystem to the perturbations of each one of them, per se. This conjecture also originates a tool to distinguish ecosystems that start in different points of the basin of attraction of a given fixed point. These ecosystems have different community matrices, and thus different MTPE, although they have a common fixed point.

18.2 Analysis

Remember sections 16.5, 16.6, and particularly table 16.4. For more effortless reading, we reproduce table 17.1.

Table 18.1. Weighted values of the transitivity of the networks associated to the MTE of the four mentioned ecosystems. WO = without omissions; WC = without competition; WM = without mutualism; OT = only trophic interaction

We will analyse the sensitivity of initial values that differ only in the value of one variable, and after we compare them to the values in table 16.4, first column. The variables chosen are y1

(transitivity=0.333), y4 (0.476), y7 (0.900). Their initial values will be decreased by 1/3. Thus, y1 is reduced to 26.66667, and the other two variables to 24 (see the first script in the appendix of chapter 17). The ascendency analysis of the MTPE of the four trajectories of the system are inserted in table 18.2. The corresponding relative variations, RV ((parameter of the simulation with reduced variable-parameter of the simulation of the variant WO)/(parameter of the simulation of the variant WO)) are exhibited in table 18.3. The first script in the appendix of chapter 17 can be used to control our results. © L. S. Barreto, 2017. Theoretical Ecology 337 18 Developmental, Structural, and Functional Sensitivities to Initial Values

Table 18.2. Ascendency analysis of the MTPE of the four trajectories of the ecosystem described in the text

WO Reduced y1 Reduced y4 Reduced y7 AMI 0.9630886 0.9411782 0.9134651 0.9228698 CAP 1608.5940000 4496.6240000 7687.9330000 8801.5120000 ASC 4371.0330000 1190.4330000 1933.8040000 2220.4040000 OH 5979.6270000 3306.1910000 5754.1300000 6581.1080000 ASC.CAP 0.2690125 0.2647393 0.2515375 0.2522753 OH.CAP 0.7309875 0.7352607 0.7484625 0.7477247 Robustness 0.3532127 0.3518411 0.3471628 0.3474422 ELD 2.4768390 2.4742090 2.5651280 2.5805130

Table 18.2. The relative variations of the attributes of the ascendency analysis of the MTPE of the three new trajectories of the ecosystem, described in the text

RV(y1) RV(y4) RV(y7) AMI -0.022750139 -0.05152537 -0.04176023 CAP -0.248009282 0.28568772 0.47191656 ASC -0.259954345 0.20217034 0.38033836 OH -0.243613352 0.31642337 0.50561847 ASC.CAP -0.015884764 -0.06495981 -0.06221718 OH.CAP 0.005845791 0.02390602 0.02289670 Robusthess -0.003883213 -0.01712821 -0.01712821 ELD -0.001061837 0.03564584 0.03564584

These two tables exhibit results that can be comment as follows: ● The ascendency analysis of the MTPE of different trajectories in the basin of attraction of the same fixed point give different results for each one. This type of analysis is sensitivity to the origin of the trajectory; ● The reduction of the initial value of the nutrient causes the reduction off all eight attributes; ● The ranking of the sensitivity of the attributes AMI, CAP, ASC, OH, robustness, and ELD reflect the ranking of the transitivity of the variables in table 18.1. ● This chapter, and the previous one form a coherent conceptual wholeness that can be visualised in figure 18.1. 338 © L. S. Barreto, 2017. Theoretical Ecology 18 Developmental, Structural, and Functional Sensitivities to Initial Values

Figure 18.1. Diagrammatic representation of the integrated concepts of chapters 17, and 18

18.3 Conclusion

Given the previous analysis, our conclusion is:

The simulations executed, and the associated analysis do not invalidate our hypothesis: it is highly probable that in the networks associated to the matrix of total positive effects (MTPE), the ranking of the clustering coefficients of the nodes mirror the ranking of the developmental, structural, and functional sensitivities of the ecosystem to changes of the associated variables. © L. S. Barreto, 2017. Theoretical Ecology 339 PART V

PART V

19 Theory Evaluation

19.1 Introduction

A global characterization of the theory was given in chapter 1. Its most conspicuous trace is the existence of a hierarchy of linked models, corresponding to the hierarchy of the biosystems. The macro structure of the theory was described in section 1.4, and illustrated in figure 15.2. The conception of the theory as axiomatic was attempted in section 1.2. After this retrieval of the object of my evaluation, let me introduce it. The evaluation here presented benefits from Bunge (2005:432-468; 2009:388-400), Godfrey-Smith (2003), Mahner, and Bunge (1997:128-133).

19.2 Semantic Unity

The hierarchical structure of my theory provides conceptual unity to it, and the fundamental assumptions mentioned in section 1.2 provide consistency, homogeneity, semantic limitation, and conceptual connectivity, in a formal, and logic way. It is also verified:  The interpretation of the concepts used by the theory is exempt of ambiguity, and are of current use in the correspondent sciences.  The exposition is coherent, logic, and (I hope) also clear, although English is not my first language.  The mathematization of the theory allows the control of its internal coherence, assures its deducibility, and its non-ambiguous testing, as I did through its presentation, showing the empirical, and corroboratory evidence available, and pertinent.

19.3 Evaluation

I use the six following criteria to appraise the presented theory: C1. Empirical evidence C2. Internal consistency C3. External consistency C4. Unifying power C5. Depth C6. Fecundity The results presented in section 17.4 validate the concept of ecosystems as linear stochastic systems modelled by MAR(1). Given the hierarchical, and integrated construction of the theory, its systemicity, and deducibility, the consequence that emerges from this corroboration is the empirical validation of the entire theory (see section 1.1). The semantic unity of the theory, its mathematization, and formal deducibility guarantee its internal coherence. The several results produced by the theory, and in the context of its presentation, are also exempt of contradictions, as far I could scrutinize. The theory reveals a high degree of integration, and provides an adequate model for ecosystems. The internal links, and cross references among the several components of the book were correctly emphasized, in the text. The empirical evidence upper mentioned confers on the theory an acceptable degree of approximation, or verisimilitude. © L. S. Barreto, 2017. Theoretical Ecology 341 19 Theory Evaluation

As I do not depict any incompatibility between the theory, and any other field of knowledge, I admit that its external coherence is acceptable. The theory favours a holistic approach to the ecosystem, and in some aspects the theory has some affinity with the ecosystem ecology proposed in Jørgensen (2009). The theory gives support to the application of ecological network analysis (ENA), and state space methods to the analysis of ecosystems. Now let me comment on the unifying power. For the first time, as far as I know, the theory presents a unified conceptualization, and modelling of the organism, population, community, and ecosystem. Under the same purpose of theorization, it applies several fields of knowledge, such as evolution, ecology, mathematics, conservation biology, and network analysis. The theory exhibits the possible depth as it embraces all biosystems covered by ecology. The fecundity of the theory already realized is diagrammatically represented in figure 19.1.

Figure 19.1. Diagrammatic representation of the already accomplished fecundity of the theory. The references related to this figure can be easily identified in section 19.4

The theory let us approach successfully important problems, such as the identification of keystone species, and controlling factors in ecosystems. In Barreto (2010, 2011), I presented the first integrated theory for forests, that can be seen as a particular case, and more restricted derivation of the theory here exposed. The forest theory conferred new value on little used forest data, and opened the door to the elaboration of several forest simulators (e.g., Barreto, 2010a, 2010b), and to the clarification of several problems occurring in the area of forest structure, dynamics, yield and management. In Barreto ( 2010a) I present a simulator for Pinus pinaster stands that simulates the total biomass, its increments, and the carbon retained in both. In Barreto ( 2010b) the simulator presented, for pure, even-aged, self-thinned stands of Quercus robur, gives information about the 342 © L. S. Barreto, 2017. Theoretical Ecology 19 Theory Evaluation biomass of the components of the forest, and the mean tree; the stand density, mean dbh, and height; net primary productivity; the cycles of C, N, P, K, Ca, Mg. The other forest simulators, written in Visual Basic 6, and Scilab, mimic pure, and mixed stands with European, and North-American tree species. It is my understanding that the theory has potential to become a very useful tool in the ecological sound management of natural resources. I also admit that the theory may affect the planning of the research in ecology. To conclude the book, let me quote myself (Barreto, 2011: 291): “Humbly, I close this book with a plea. After you finished reading the book, with open mind, think about the quotations I inserted in its first pages. Next, please think how you can test, and improve my theoretical proposal or replace it. Thank you.”

19.4 References

Barreto, L. S., 2000. Pinhais Mansos. Ecologia e Gestão. Estação Florestal Nacional, Lisboa. Barreto, L. S., 2005. Pinhais Bravos. Ecologia e Gestão. E-book. Instituto Superior de Agronomia, Departamento de Engenharia Florestal, Lisboa. Barreto, L. S., 2009. Caracterização da Estrutura e Dinâmica das Populações de Lince Ibérico (Lynx pardinus). Uma Digressão Exploratória. Silva Lusitana, 17(2):193-209. Barreto, L. S., 2010. Árvores e Arvoredos. Geometria e Dinâmica. E-book, Costa de Caparica, Portugal. Barreto, L. S., 2010a. Simulação do Carbono Retido no Pinhal Bravo e da sua Acreção. Silva Lusitana, 18(1):47-58. Barreto, L. S., 2010b. Simulator SB-IberiQu. Simulator for pure, even-aged, self-thinned stands of Quercus robur, written in Scilab. Submitted to Silva Lusitana. Include in the CD Barreto, L. S., 2011. From Trees to Forests. A Unified Theory. E-book, Costa de Caparica, Portugal. Include in the CD Bunge, M. , 2009. Philosophy of Science. From Explanation to Justification. Volume II. Revised Edition. Transaction Publishers, New Brunswick, U. S. A. Godfrey-Smith, P., 2003. Theory and Reality. An Introduction to the Philosophy of Science. The University of Chicago Press, Chicago. Jørgensen, S. E., 2009, (Editor). Ecosystem Ecology. Elsevier B.V. Mahner, M., and M. Bunge, 1997. Foundations of Biophilosophy. Springer, Berlin.