Mathematical Analysis of Population Growth Subject to Environmental Change

SCIENTIA

MANU E T MENTE

MATHEMATICAL ANALYSIS OF POPULATION GROWTH SUBJECT TO ENVIRONMENTAL CHANGE

A thesis submitted for the degree of Doctor of Philosophy

By Hamizah Mohd Safuan M.Sc.(Mathematics)

Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical Sciences, The University of New South Wales, Canberra, Australian Defence Force Academy.

November 2014

PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet

Surname or Family name: Mohd Safuan

First name: Hamizah Other name/s: -

Abbreviation for degree as given in the University calendar: Ph.D (Mathematics)

School: Faculty: UNSW Canberra School of Physical, Environmental and Mathematical Sciences

Tille: Mathematical analysis of population growth subject lo environmental change

Abstract 350 words maximum: (PLEASE TYPE)

Many ecosystems are pressured when the environment is perturbed, such as when resources are scarce, or even when they are over-abundant. Changes in the environment impact on its ability lo support a population of a given species. However, most current models do not lake the changing environment into consideration. The standard approach in modelling a population in its environment is lo assume that the carrying capacity, which is a proxy for the state of the environment, is unchanging. In effect, the assumption also posits that the population is negligible compared lo the environment and cannot alter the carrying capacity in any way. Thus, modelling the interplay of the population with its environments is important lo describe varying factors that exist in the system. This objective can be achieved by treating the carrying capacity as lime- and space-dependent variables in the governing equations of the model. Thereby, any changes lo the environment can be naturally reflected in the survival, movement and competition of the species within the ecosystem. In this thesis, detailed investigations of several mathematical models for population growth are presented. Formulating the carrying capacity as being lime-dependent was the fundamental approach used lo describe a varying environment which resulted in investigating a non-autonomous equation. This approach led lo developing models that directly couple the dynamics of one or two species with their environments. To attain this, the carrying capacity was modelled as a slate-variable. In these models, the ultimate state for the ecosystem depends on the resource enrichment parameter that was found lo have significant impact on the growth of a population, leading lo either coexistence or extinction of a particular species. Other dynamical behaviours including oscillations in population have also been found to exist. Varying the carrying capacity has given a better understanding of population growth when subjected lo environmental change. This thesis serves as another platform for ecologists and biologists to investigate further the importance of a varying environment, and could be applied in future population-growth studies.

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lms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral

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i Certiﬁcate of Originality

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iv Abstract

Many ecosystems are pressured when the environment is perturbed, such as when resources are scarce, or even when they are over-abundant. Changes in the environment impact on its ability to support a population of a given species. However, most current models do not take the changing environment into consideration. The standard approach in modelling a population in its environment is to assume that the carrying capacity, which is a proxy for the state of the environment, is unchanging. In eﬀect, the assumption also posits that the population is negligible compared to the environment and cannot alter the carrying capacity in any way. Thus, modelling the interplay of the population with its environments is important to describe varying factors that exist in the system. This objective can be achieved by treating the carrying capacity as time- and space-dependent variables in the governing equations of the model. Thereby, any changes to the environment can be naturally reﬂected in the survival, movement and competition of the species within the ecosystem.

In this thesis, detailed investigations of several mathematical models for population growth are presented. Formulating the carrying capacity as being time-dependent was the fundamental approach used to describe a varying environment which resulted in investigating a non-autonomous equation. This approach led to developing models that directly couple the dynamics of one or two species with their environments. To attain this, the carrying capacity was modelled as a state-variable. In these models, the ultimate state for the ecosystem depends on the resource enrichment parameter that was found to have signiﬁcant impact on the growth of a population, leading to either coexistence or extinction of a particular species. Other dynamical behaviours including oscillations in population have also been found to exist.

Varying the carrying capacity has given a better understanding of population growth when subjected to environmental change. This thesis serves as another platform for ecologists and biologists to investigate further the importance of a varying environment, and could be applied in future population-growth studies.

Acknowledgements

First and foremost, I am grateful to the Almighty God for giving me many great opportunities in life, including this challenging yet wonderful Ph.D journey.

I am deeply thankful to Dr. Isaac Towers for his helpful supervision, generous assistance and encouragement to pursue this research that led to this thesis. His guidance and patience in explaining the key aspects of numerical methods have been extremely helpful, and without him the analysis in this thesis would have been impossible.

My sincere thanks go to Dr. Zlatko Jovanoski for his insightful questions, discussions and valuable feedback on the mathematical models and analysis in this thesis. I thank him for his friendly criticism, and for broadening my knowledge of population growth studies.

I owe very much to A/Prof. Harvinder Sidhu for his thoughtful advice, knowledge and wisdom. His guidance and assistance were invaluable in helping me acquire the necessary skills to become a Ph.D researcher. His genuine concern about my study and my family life gave me the consistent motivation to complete this thesis. My appreciation also goes to his wife, Leesa Sidhu for giving me the support and encouragement during my diﬃcult days.

I also thank Dr. Peter McIntyre and Denise Russell for their excellent suggestions in improving this thesis. Not to forget, my thanks to all PEMS staﬀ and the greater PEMS community who have given me the best support and cooperation throughout my study.

I would like to acknowledge my employer, Universiti Tun Hussein Onn Malaysia (UTHM) and The Government of Malaysia for funding my Ph.D program and living expanses for my family and I in Canberra. Without these ﬁnancial support, I would not have been able to undertake this program.

My special dedication goes to my fellow colleagues Thiansiri Luangwilai, Billie Ganendran, Fatemeh (Mona) Ziaeyan Bahri, Aduraﬁmihan Abiona, Zhenhua (Iris) Hao, Fei Li, Yanyan Liu and Zhaosu Meng for their wonderful companionship and

vii friendship. For Billie, thank you for being a wonderful friend during my happy, and occasionally diﬃcult days.

I would like to express my very great appreciation to my parents, family and friends in Malaysia. It was hard being away from them, but their constant love and care touched me in many ways. Without their support, this work would not have been possible. To my closest friends in Canberra, thank you for being an instant family to me these past years.

I am very fortunate to have my loving, understanding husband, Abid, and my dearest daughter, Husna, by my side throughout my study. Husna’s presence during this study was rather challenging, yet she is the most amazing gift I have ever had in life. I am forever in their debt for supporting me while I spent late evenings at school when I could have been spending time together with them. I would not have been able to complete this journey without their love and patience.

viii Published work

The following publications correspond to work within this thesis:

1. H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu, A simple model for the total microbial biomass under occlusion of healthy human skin, In Chan, F., Marinova, D. and Anderssen, R.S. (eds) MODSIM 2011, 19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, December 2011, pp. 733–739. http://www.mssanz.org.au/modsim2011/AA/safuan.pdf (based on part of the work in Chapter 3) 2. H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu, Coupled logistic carrying capacity model, ANZIAM J.(E), 53, pp. C142–C154, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAM/article/view/4972 (based on part of the work in Chapter 4)

3. H. M. Safuan, Z. Jovanoski, H. S. Sidhu and I. N. Towers, Exact solution of a non-autonomous logistic population model, Ecological Modelling, 251, pp. 99–102, 2013. http://dx.doi.org/10.1016/j.ecolmodel.2012.12.016 (based on part of the work in Chapter 3)

4. H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers, Impacts of biotic resource enrichment on a predator-prey population, Bulletin of Mathematical Biology, 75, pp. 1798–1812, 2013. http://link.springer.com/article/10.1007/s11538-013-9869-7 (based on part of the work in Chapter 5)

5. H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers, A two-species predator- prey model in an environment enriched by a biotic resource, ANZIAM J.(E), 54, pp. C768–C787, 2014. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/ view/6376 (based on part of the work in Chapter 5)

ix 6. H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu, On travelling wave solutions of the diﬀusive Leslie-Gower model, Manuscript submitted for publication, 2014. (based on part of the work in Chapter 6)

7. H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers, On travelling wave solutions of predator-prey model with resource enrichment, Manuscript in preparation, 2014. (based on part of the work in Chapter 6)

The following oral presentations were given based on work presented in this thesis:

1. UNSW Research Day 2011, A model for total cutaneous microbial biomass during occlusion of healthy human skin, October 2011, UNSW Canberra, Aus- tralia. (based on part of the work in Chapter 3) 2. 2011 ACT-NSW ANZIAM Joint Mini-meeting, Coupled logistic carrying capacity model, November 2011, Murramarang, New South Wales, Australia. (based on part of the work in Chapter 4) 3. The 10th Engineering Mathematics and Applications Conference (EMAC 2011), Coupled logistic carrying capacity model, December 2011, Sydney, Australia. (based on part of the work in Chapter 4) 4. The 19th International Congress on Modelling and Simulation (MODSIM 2011), A simple model for the total microbial biomass under occlusion of healthy human skin, December 2011, Perth, Australia. (based on part of the work in Chapter 3) 5. The 16th Biennial Computational Techniques and Applications Conference (CTAC 2012), A two species population growth model with varying carrying capacity, September 2012, Queensland, Australia. (based on part of the work in Chapter 5) 6. 2012 ACT-NSW ANZIAM Joint Mini-meeting, A two species population growth model with varying carrying capacity, November 2012, UNSW Kensington, Australia. (based on part of the work in Chapter 5)

x Posters were also presented based on work in this thesis:

1. Canberra Health Annual Research Meeting 2011 (CHARM 2011), A model for total cutaneous microbial biomass during occlusion of healthy human skin, June 2011, Canberra Hospital, Australia. (based on part of the work in Chapter 3) 2. UNSW Research Day 2011, A model for total cutaneous microbial biomass during occlusion of healthy human skin, October 2011, UNSW Canberra, Aus- tralia. (based on part of the work in Chapter 3) 3. UNSW Research Day 2012, A predator-prey population model with varying carrying capacity, October 2012, UNSW Canberra, Australia. (based on part of the work in Chapter 5)

The following awards were received during my candidature:

1.3 rd place of Poster Competition “Judges Selection” (UNSW Research Day 2011), October 2011. 2. Best student talk at the Joint ACT-NSW ANZIAM Meeting 2011, November 2011. 3. Funding from the CSIRO/ANZIAM student support scheme to attend the 10th Engineering Mathematics and Applications Conference (EMAC 2011), Sydney, December 2011. 4. Funding from the CSIRO/ANZIAM student support scheme to attend the 16th Biennial Computational Techniques and Applications Conference (CTAC 2012), Queensland, September 2012.

Contents

Declaration iv

Abstract v

Acknowledgements vii

Published work ix

Chapter 1 Introduction 1

1.1 Background ...... 1

1.2 Carrying capacity ...... 3

1.3 Population-growth model and carrying capacity ...... 5

1.4 Overview of thesis ...... 12

Chapter 2 Mathematical methods and tools 15

2.1 Mathematical methods ...... 15

2.1.1 Bifurcation analysis ...... 15

2.1.2 Singularity analysis ...... 16

2.1.3 Hopf bifurcation ...... 18

2.1.4 Travelling waves ...... 19

2.2 Software packages ...... 22

2.3 Conclusions ...... 24

Chapter 3 Fundamental models 25

xiii 3.1 Time-dependent carrying capacity as an explicit function of t ... 25

3.2 Carrying capacity as a state-variable ...... 26

3.3 Application of variable environmental carrying capacity ...... 28

3.3.1 Application to total microbial biomass during skin occlusion 28

3.3.2 Exact solution of a logistic model with saturating carrying capacity ...... 34

3.4 Conclusion ...... 41

Chapter 4 Single-species population-growth models with variable carrying capacities 43

4.1 Environmental carrying capacity as a biotic resource ...... 43

4.2 Modiﬁed logistic models ...... 44

4.3 Environmental carrying capacity as a simple biotic resource . . . . . 48

4.4 Conclusion ...... 55

Chapter 5 Two-species population-growth models with variable carrying capacities 57

5.1 Predator-prey models ...... 57

5.2 Modiﬁed predator-prey models ...... 58

5.3 Environmental carrying capacity as biotic resource enrichment . . . 61

5.3.1 Equilibrium and stability analysis ...... 62

5.3.2 Constant carrying capacity: c = d = e =0 ...... 62

5.3.3 Absence of inter-speciﬁc interactions: a = b = 0 ...... 64

5.3.4 Full model: a, b, c, d, e 6=0 ...... 65

5.3.5 Hopf bifurcation analysis ...... 68

5.4 Environmental carrying capacity with logistically limited biotic resource enrichment ...... 78

5.4.1 Equilibrium and stability analysis ...... 78

5.4.2 Cusp singularity analysis ...... 80

5.5 Conclusion ...... 87

xiv Chapter 6 One-dimensional spatial-temporal population-growth models 89

6.1 Travelling waves ...... 89

6.2 Single-species population growth model with biotic resource enrichment ...... 90

6.2.1 Equilibrium and stability analysis ...... 90

6.2.2 Numerical results ...... 95

6.3 Two-species population growth model with biotic resource enrichment104

6.3.1 Equilibrium and stability analysis ...... 104

6.3.2 Numerical results ...... 107

6.4 Conclusion ...... 114

Chapter 7 Concluding remarks and suggestions for future work 119

7.1 Conclusions ...... 119

7.2 Thesis highlights ...... 119

7.3 Future work ...... 122

7.3.1 Spatially independent models ...... 122

7.3.2 Spatially dependent models ...... 124

Appendix A Numerical calculation of steady-states 127

A.1 Steady-states for two-species population model with resource enrichment...... 127

A.2 Steady-states for two-species population model with logistically limited resource enrichment ...... 129

References 131

List of Tables

3.1 Time-dependent carrying capacities K(t)...... 26

3.2 Relative error sustained by approximate solutions for certain values of t...... 38

6.1 Relative errors of wave speeds from FlexPDETM and MOL relative ∗ to s2 (type II wave) for the case of constant wave speeds...... 110

A.1 Eigenvalues of Jacobian matrices for steady-states P2 and P3 of system (5.13)...... 128

A.2 Eigenvalues of Jacobian matrices for steady-states Q3 and Q4 of system (5.17)...... 130

xvii

List of Figures

1.1 Logistic growth curves with constant K for various initial conditions,

N0. The carrying capacity is K = 10...... 6

1.2 Exponential, logistic and Gompertz growth curves with N0 = 10. The carrying capacity is K = 100...... 7

2.1 Examples of bifurcation: (a) saddle-node bifurcation; and (b) transcritical bifurcation (solid line – stable steady-state, and dashed line – unstable steady-state)...... 16

2.2 A steady-state diagram exhibiting hysteresis behaviour. For γ1 <

γ < γ2, bistability occurs (solid line – stable steady-state, and dashed line – unstable steady-state)...... 18

2.3 Types of Hopf bifurcation, (a) supercritical Hopf bifurcation. (b) subcritical Hopf bifurcation (closed circle – stable periodic, and open circle – unstable periodic)...... 19

3.1 Logistic equation (1.1) with carrying capacity that is (a) linear, K(t) = 0.05t+5; (b) exponential, K(t) = 5e0.02t; (c) periodic, K(t) = 7 5×108 0.5 sin(t)+5; and (d) bi-logistic, K(t) = 5×10 + 1+e−0.032(t−350) . For bi-logistic, the data are reproduced from Meyer and Ausubel [1999]. 27

3.2 cder data with the nonlinear ﬁtted curve. Parameters are C0 = 2 2 25 nl/cm /min, Cs = 91.2 nl/cm /min, α = 1.92/day and β = 0.73. 32

3.3 Growth curves N(t) and variable carrying capacity K(t) with the es- 2 timated parameters r = 6.6/day,N0 ≈ K0 = 920 counts/cm ,Ks = 2.29 × 107 counts/cm2, α = 1.92/day, β = 0.99996 for (a) Aly et al. [1978] and (b) Pillsbury and Kligman [1967] using the proposed model...... 33

xix 3.4 Comparison between the numerical solution, Nnum, and the various approximate solutions obtained from (3.18). Parameters taken from 2 Safuan et al. [2011]: r = 6.6/day; N0 ≈ K0 = 920 counts/cm ; Ks = 2.29 × 107 counts/cm2; α = 1.92/day; β = 0.99996...... 38

3.5 A log scale plot of the relative error according to the data in Table 3.2. 38

3.6 Comparison between the numerical solution, Nnum, and various ap-

proximate solutions. The parameters are N0 = 20, Ks = 200, β = 0.9 with (a): r = 3.3, α = 1.2 and (b): r = 8.5, α = 3.1...... 39

3.7 Plot of inﬂection time, ti (days), for varying r and α. As the growth

rate, r, increases, the inﬂection time, ti, decreases...... 41

4.1 Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.2): (a) ω = 0.1; and (b) ω = 0.5, with

(n0, k0) = (0.1, 2.0)...... 46

4.2 Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.2): (a) ω = 0.1; and (b) ω = 0.5, with

n0 = k0 = 0.1...... 47

4.3 Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system 4.4 for diﬀerent initial values: (a) γ = 0.5,

n0 = 0.1 with (i) k0 = 0.4, (ii) k0 = 0.2, (iii) k0 = 0.1; (b) γ = 0.1,

0.5, 2.0, with (n0, k0) = (0.1, 2.0)...... 49

4.4 Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.6): (a) µ = 2.0; (b) µ = 1.0; (c) µ = 0.5;

(d) µ = 0.2, with n0 = k0 = 0.1 (note that diﬀerent scales are used). 52

4.5 Corresponding trajectory plots in the phase plane for Figure 4.4: (a)

µ = 2.0; (b) µ = 1.0; (c) µ = 0.5; (d) µ = 0.2. In all cases n0 = 0.1

and k0 = 0.1 (note that diﬀerent scales are used)...... 53

4.6 Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.6): (a) µ = 2.0; (b) µ = 1.0; (c) µ = 0.5;

(d) µ = 0.2, with n0 = 0.1, k0 = 2.0 (note that diﬀerent scales are used)...... 54

5.2 Examples of intraguild predation; (a) ﬁsh populations [Mylius et al., 2001]; (b) spider-mite populations [Janssen et al., 2006]...... 60

xx 5.10 Steady-state diagrams of the system (5.13) against the consumption parameter, , with α = 0.3, β = 0.5, γ = 2.5, δ = 2 for: (a) x; (b) y; (c) k...... 77

5.11 Two-parameter bifurcation diagram when φ = 0 (solid line) and φ = 2 (dashed line) with β = θ = 1. Region A – no bistability; Region B – bistability exists...... 82

5.12 Steady-state diagrams of the system (5.18) against the enrichment parameter, γ, with φ = 0, β = δ = 1 and = 3, α = 0.1 for: (a) x; (b) y; (c) k. 84

5.13 Steady-state diagrams of the system (5.18) against the enrichment parameter, γ, with φ = 0, β = δ = 1 and = 3, α = 1.5 for: (a) x; (b) y; (c) k. 85

6.6 Simulations of system (6.2) with initial conditions (a) IC1; (b) IC2;

xxi

Chapter 1

Introduction

In this thesis, we investigate several mathematical models of population growth subjected to changing environments. We start with a fundamental model which is then extended to both spatially independent and dependent models. The main aims of this thesis are to:

1. investigate changing environments by treating carrying capacities as variables in both time and space; 2. develop mathematical models to describe the interactions between one or more species sharing the same biotic resource through each individual species’ carrying capacity; 3. investigate the impacts of biotic resource enrichment on a population; 4. broaden the model to a spatial system to observe the dynamics of local dispersal and competition that will aﬀect population distribution.

1.1 Background

The feedback between a population and its environment is the underlying concept in the study of population ecology, as population growth and decay are greatly aﬀected by changes in the environment. The diversity of elements such as nutrients, food sources, weather and types of biological species are important factors in the degradation of (or improvement in) surrounding habitat and also in maintaining the sustainability of species for which the environment is unlikely to remain constant.

Besides changes in temperature and/or moisture that could correspond to changes in the environment, other biophysical factors, such as the introduction of a new species in a habitat, for instance a consumer (predator) or resource (prey), can be measured as an environmental change. Sometimes, this can ultimately modify the composition, biomass and distribution of a species in an ecosystem. For example, the bio-manipulation of an ecosystem or food-web system through modiﬁcations in

1 nutrients, zooplankton and fish populations is vital for the maintenance and en- hancement of water quality in a lake or sea. A predator species was introduced into Round Lake, Minnesota to observe its effect on the phytoplankton and zooplankton communities, as well as on the total nutrient concentrations [Shapiro and Wright, 1984]. This experiment altered the compositions of marine species in that lake after two years. It was observed that phytoplankton and zooplankton numbers decreased and, notably, a rare situation in which a herbivore species (Daphnia) became the dominant genus. This proved that changes in the primary environment of an ecosystem have a great effect and can restructure the compositions of species. Related work can be found in Chess et al. [1993] and Wahl et al. [2011].

In another example, a population living in a conﬁned habitat, such as an island, will collapse if the environment’s fundamental resources are over-exploited, for example, the reindeer populations on St. Paul Island, Alaska [Scheﬀer, 1951, Huzimura and Matsuyama, 1999] and St. Matthew Island, Alaska [Klein, 1968]. Their “boom- and-bust”population explosions illustrated how over-abundances of that species exhausted the available resources in an environment not conducive to supporting it, and eventually drove the species to extinction. A similar scenario is thought to have occurred in the human population on Easter Island, Chile [Basener et al., 2008, Bologna and Flores, 2008, Sprott, 2011], where environmental degradation was caused by over-exploitation of the island’s fundamental resources.

However, even an enriched resource can turn out to be a detrimental environment for populations. Studies of intraguild predation (a mixture of competition and predation) have found the possibility of a population being extinguished due to an environmental change (resource enrichment) [Polis et al., 1989, Polis and Holt, 1992, Holt and Polis, 1997, Mylius et al., 2001]. Increasing resources only benefits the top predator, usually an omnivore, while suppressing other populations (prey/ consumer) from the food chain. All these examples are evidence that a change in the environment can significantly affect the food-chain system.

As maintaining existing resources and the environment are crucial for the survival of populations, environmental control, management and modelling have become signiﬁcant. One approach is developing realistic mathematical models to predict populations’ behaviours when subjected to varying environments. Although it has been a challenge for mathematicians, biologists and ecologists to formulate a model that could describe all possible factors, in the long run ecologists and biologists have beneﬁtted from mathematical models designed and developed through careful analysis and various trial-and-error techniques that can describe the complex dynamics of a population and its environment. These models constantly serve as the

2 main theoretical approach for studying populations’ behaviours in a wide range of possible scenarios.

As varying environments provide a more meaningful subject for consideration than those that are constant, it is the aim in this thesis to explore and analyse population models subjected to changing environments.

1.2 Carrying capacity

Generally, the carrying capacity deﬁnes the maximal abundance of a population an environment can sustain given ﬁnite resources, and is also considered the limiting factor for an ecosystem or population. Its concept is extensively applied in many areas such as population dynamics, biology, ecology, economics, technology, agri- culture and even cultural and social studies. There are several schools of thought regarding this concept which have debated its application in population ecology and other areas [Dewar, 1984, Scarnecchia, 1990, Price, 1999, Seidl and Tisdell, 1999, del Monte-Luna et al., 2004].

Dewar [1984] explained two concepts of the carrying capacity as it being a feature of the environment or changes in a population’s growth rate and density. Although he suggested that both are unlikely to be uniformly directly related, most studies in applied ecology and mathematical ecology have regarded them as strongly linked. In fact, in some studies, the carrying capacity has been observed to be a mixture of both [Coleman, 1979a, Samanta and Maiti, 2004, W. Hsin-i and Kenerley, 2009]. Dewar [1984] agreed that no environment is unchanging, and therefore the carrying capacity should vary. McLeod [1997] argued that the concept of the carrying capacity is useful in variable environments and found that it could be adapted in a deterministic model or slightly variable environment, but not in a stochastic one.

Without carrying capacities (in other words, if population growth is not controlled), our world and its entire ecosystem would collapse due to the scarcity of resources, environmental pressures and health issues. For example, the offspring of a female housefly would cover the whole Earth’s surface in a short time if its population growth were not controlled, as it is able to reproduce over 5 trillion houseflies in a year. Also, under certain circumstances, a bacterium would form a layer a foot deep over the entire Earth’s surface in less than 2 days due to its capability to reproduce itself every 20 minutes [Miller, 2007]. These are examples of rapid exponential growth whereby populations grow indefinitely and “explode”. However as in reality, populations are finite and limited, there must be some bounds within which they increase, crowd and stabilise; their growth could be restricted by the amounts of available resources such as food, water, space and the environment.

3 Attempts have been made to estimate the Earth’s carrying capacity [Cohen, 1995, Hopfenberg, 2003, Berck et al., 2012] but, as the human population is dynamic, it is rather diﬃcult to estimate its exact carrying capacity. As humanity is capable of modifying the environment and improving the production of resources, using con- tinually developing technologies, there are deﬁnitely interactions between humans and the Earth’s human carrying capacity. The human population model proposed by Cohen [1995] depends strongly on the ability of humans to increase or decrease the carrying capacity. Therefore, it uses a variable carrying capacity whereby each additional person impacts on the carrying capacity by an amount that depends on the resources available, and his/her capability to contribute to any positive/negative production for the human population. To date, Cohen’s model remains a possible projection of human populations over the next hundred years.

To estimate the carrying capacity of non-human species, it is necessary to make some assumptions, as there are many factors that contribute to the development or destruction of the environment’s carrying capacity. Factors that aﬀect the population environment are classiﬁed as biotic or abiotic resources [Price, 1999, del Monte-Luna et al., 2004]. An abiotic resource is non-living, for example water, nutrients, temperature and even toxicants that enter the population’s environment. On the other hand, a biotic resource is one that grows, reproduces and is usually self-limiting, for example an additional living species that serves as a resource for an original population. In terms of these two factors, a constant carrying capacity may not be applicable for describing populations’ changing environments and to provide a realistic scenario, so that a variable carrying capacity should be considered when modelling them.

A review by del Monte-Luna et al. [2004] detailed the versatile concepts of the carrying capacity for populations, communities, ecosystems and the biosphere, and proposed a deﬁnition compatible with various applications as

“the limit of growth or development of each and all hierarchical levels of biological integration, beginning with the population, and shaped by processes and independent relationships between ﬁnite resources and the consumers of those resources” [del Monte-Luna et al., 2004].

From this review, the carrying capacity can be summarized as the maximal capacity of the environment to sustain a given species within the restrictions of certain biological resources (biotic and/or abiotic).

4 Mathematically, the carrying capacity is a ﬁxed parameter or a variable K in the classic logistic model discussed in the next section. In modern ecology, an ever- changing resource in an environment is an intriguing aspect to study. In this thesis, mathematical models to describe the changes in an environment through a variable carrying capacity are investigated and developed.

1.3 Population-growth model and carrying capacity

Population-growth modelling is a research area in the fields of mathematical biology and ecology due to its importance in various ecosystems. With the rich dynamics populations are capable of displaying, they make for interesting investigations. The logistic equation is an example of growth models that has been widely studied and applied in population and ecological modelling. Introduced in the 19th century, it describes population growth using a self-limitation term which serves as a correction to the unlimited growth of the Malthusian model, commonly referred to as the exponential model. The classic logistic (or Verhulst’s) equation is [Murray, 1989, Banks, 1994] dN(t) N(t) = rN(t) 1 − ,N(0) = N . (1.1) dt K 0 Equation (1.1) is autonomous as the parameters r and K are constant. Its nature is such that the properties of the solutions, for all strictly positive initial conditions, approach the constant value of the carrying capacity, K, as time, t, tends to infinity (Figure 1.1). If the population starts at a level well below the carrying capacity, K, the population first increases steadily, then rapidly, due to the linear term, until it approaches an inflection point (at which the growth rate is maximal). After passing the critical point, the population’s reproduction rate declines due to the crowding effect of individuals competing for limited resources; after experiencing this environmental pressure, the population stabilises. If the population starts at a higher value than the carrying capacity K, then population growth rate is negative, with its number decreasing and reaching a plateau. The solution to Equation (1.1) is obtained by separation of variables as

KN0 N(t) = −rt −rt . (1.2) Ke + N0(1 − e )

Another growth model widely used for modelling population growth is the Gompertz model, which is similar to the logistic equation (1.1) in terms of its sigmoid shape, clear inﬂection point and being analytically solvable [Banks, 1994].

5 15

N =15.0 0

K 10

N(t) N =5.0 0

N =2.0 0 5 N =0.8 0

N =0.2 0

0 0 2 4 6 8 10 t Figure 1.1: Logistic growth curves with constant K for various initial conditions, N0. The carrying capacity is K = 10.

The Gompertz model is written as

dN(t) K = rN(t) ln (1.3) dt N(t)

The Gompertz model is commonly applied in cancer research [Xu, 1987, Domingues, 2012, Gerlee, 2013]. The difference between the logistic and the Gompertz model is how fast the population, N(t), approaches its limiting value, as reflected by the “skewness” of the curvature [Winsor, 1932]. The Gompertz curve increases slowly initially, and gradually increases asymmetrically when approaching its asymptote, whereas the logistic curve reaches its asymptote symmetrically about the inflection point. The inflection points of the Gompertz and logistic models occur at K/e and K/2, respectively. Figure 1.2 shows a comparison of the growth curves of the exponential, logistic and Gompertz models.

As the Gompertz and logistic models possess similar shapes and properties, it is best to ﬁrst investigate the simpler model and analyse its dynamics. Thus, the logistic equation establishes the framework for the model in this thesis because it is simple and the most widely used for investigating population growth.

As discussed in the previous section, assuming that the carrying capacity is constant is not often realistic in many applications, especially in modelling populations. Thus, many studies have discussed an alternative approach by investigating the importance of time-dependent carrying capacities, K = K(t), embedded in a non-autonomous logistic equation [Coleman, 1979a, Ikeda and Yokoi, 1980, Hallam and Clark, 1981, Banks, 1994, Meyer, 1994, Meyer and Ausubel, 1999, Shepherd

6 120

K 100

N(t) 60

40 Exponential 20 Logistic Gompertz 0 0 5 10 15 20 25 30 t

Figure 1.2: Exponential, logistic and Gompertz growth curves with N0 = 10. The carrying capacity is K = 100. and Stojkov, 2007, Trappey and Wu, 2008, Rogovchenko and Rogovchenko, 2009, Grozdanovski and Shepherd, 2007, Grozdanovski et al., 2009, 2010]

dN(t) N(t) = rN(t) 1 − ,N(0) = N . (1.4) dt K(t) 0

The general mathematical properties of Equation (1.4) were introduced by Cole- man [1979a] to model a changing environment. Hallam and Clark [1981] studied the non-autonomous logistic equation in a deteriorating environment which always leads to extinction of the population. A series of works has been done since then, extending the idea of time-dependent carrying capacities to toxicants in the population and toxicants in the environment [Hallam and Clark, 1983, Hallam et al., 1983, Freedman and Shukla, 1991, Samanta and Maiti, 2004].

The effect of a nutrient enrichment in an inland sea on the fish population was studied by Ikeda and Yokoi [1980]. In their work, nutrient enrichment is described by an increase in the carrying capacity of plankton, which are the food source for small fish. In other applications, time-dependent carrying capacities have been used to represent seasonal environments [Coleman et al., 1979b, Leach and Andriopoulos, 2004, Rogovchenko and Rogovchenko, 2009]. Seasonal environments, such as those with temperature variations, can affect population reproduction rates especially in bacteria. Further, oceanic tides are responsible for fluctuations in the numbers of various aquatic organisms and their ecosystems, while rainfall influences the food availability for a population.

7 In modelling the human population in the modern world, its carrying capacity has been associated with the development of technological innovations. Studies by Meyer [1994] and Meyer and Ausubel [1999] employed a variable carrying capacity in the application of technological developments to a human population. Later, Watanabe et al. [2004] used a sigmoidal function to model the carrying capacity to compare trends in various technological goods. In another application, a similar form of the sigmoidal carrying capacity was used to model the body size of a host infected by parasites [Ebert and Weisser, 1997]. Hahnfeldt et al. [1999] and Sachs et al. [2001] introduced a carrying capacity proportional to the amount of neovascularisation in a Gompertz model of tumor growth.

A carrying capacity which depends on a natural habitat has also been studied as a changing environment. Experiments by Griﬀen and Drake [2008] demonstrated that improving a population’s carrying capacity, as demonstrated by improving its habitat quality, increased its permanence. A logistic model was used by Huang et al. [2012] to represent the rate of habitat deterioration for freshwater cetacean species. For a relatively low habitat deterioration rate, ﬂuctuations in the populations were found, whereas a sharp increase in the rate forced the population to decline and, in some cases, disappear from the system.

Gonzalez et al. [2008] modified the work of Gurney and Lawton [1996] to investigate a two-species population model consisting of a resident and an invader species. In the work of Gonzalez et al. [2008], both species competed for a habitat that was formulated as the carrying capacity of each species, and for which the habitat itself varied over time. They found that the invader species could change, adapt to and dominate the local environment due to its ability to affect the flow of resources. If the activities of a population affect its own carrying capacity, such as by enhancing or polluting resources, its carrying capacity is dependent on the population, N, itself, K = K(N), which could also involve a delay process in the system [Yukalov et al., 2012].

For these applications, it is vital that the carrying capacity is not treated as a constant. Positive changes in the environment, such as new resources or food production, increases the carrying capacity, whereas a negative change such as the advent of a toxic environment or food depletion, decreases the carrying capacity.

In this thesis, a functional form of the carrying capacity is proposed to represent the development of a population environment using a monotonically increasing function. Although Ikeda and Yokoi [1980] used this speciﬁc form to model the enrichment of nutrients in an inland sea, the carrying capacity in this thesis is incorporated in the logistic equation (1.4) and applied to bacterial populations. As, to the best of our

8 knowledge, this work has never been undertaken before, we begin by investigating a simple yet useful model. Its parameters are derived from published experimental data, independently veriﬁed using another set of data.

As well as the carrying capacity taking various time-dependent forms, K(t), the carrying capacity is formulated as a variable-dependent form. If the carrying capacity is dependent on another component, i.e. an abiotic or biotic resource which also changes with time, the system can be written in the form

dN N = rN 1 − , (1.5a) dt K dK = f(N,K), (1.5b) dt where f(N,K) is a function involving linear or non-linear interaction terms between population, N, and carrying capacity, K. In contrast to a non-autonomous equation with a time-dependent carrying capacity, a model with variable-dependent carrying capacity can be an autonomous system, and the coupled rate equations can be analysed using standard tools, such as linear stability and phase-plane analyses. Interesting dynamics can also be seen if more-complex interaction terms between the population and its carrying capacity are involved in system (1.5).

One of the simplest models has been studied by Huzimura and Matsuyama [1999] for a deer population with its primary resource (lichen). It assumed that the carrying capacity in the logistic growth of deer was resource-dependent and was depleted through the consumption of lichen by the deer. Another model that treats the carrying capacity as a state-variable was studied by Thornley and France [2005]; the carrying capacity in this case depended on the current values of environmental variables that could cause the ﬁnal value of the carrying capacity to increase or decrease depending on parameter settings.

In this thesis, we focus on the concept of an environmental carrying capacity as a biotic resource. The population alters its carrying capacity by interacting with its environment. To model this, the reproductive and death terms of the biotic resource should be included, in addition to the growth equation of the population. Although the idea of a biotic resource has been included in Huzimura and Matsuyama [1999] as a linear term, the eﬀect of the nonlinear terms are analysed here and the results presented. The results obtained in our work are compared to those found in their work and that of Thornley and France [2005].

There are studies that took into account the over-exploitation of the primary resources, such as trees on an isolated island, that led to the collapse of the society

9 inhabiting that island. A study by Basener and Ross [2005] incorporated limited self-replenishing resources, such as trees and plants, in the carrying capacity of the human population living on Easter Island. By drastically harvesting this abundant resource, eventually its supply deteriorated and the population was driven to extinction. Degradation of the environment can also be exacerbated by the introduction of other inhabitants, such as a pest population, for example, rats [Basener et al., 2008, Sprott, 2011]. In another study, the model in Sprott [2011] is rewritten as

dX X = r X 1 − , (1.6a) dt 1 K dY Y = r Y 1 − , (1.6b) dt 2 K dK = f(X,Y,K), (1.6c) dt where f(X,Y,K) involves interactions among humans, X, rats, Y , and trees, K. In the work of Basener and Ross [2005], a periodic solution exists in a certain range of parameters that demonstrates persistent ﬂuctuations between species. Sprott [2011] found that, in a narrow range of parameters, periodic solutions and chaos existed in the system, with chaos having the characteristic of random population ﬂuctuations.

Another situation that is related to a resource-dependent carrying capacity is a predator-prey type model. Assuming that the carrying capacity of a predator’s environment is proportional to the abundance of its prey, we can write a system of ratio-type predator-prey equations. This type of model introduced by Leslie [1958] and Leslie and Gower [1960], and work related to it has continue to the present day [Korobeinikov, 2001, Wang et al., 2003, Seo and Kot, 2008, Chen et al., 2009, Chen and Chen, 2009, Li and Li, 2013, Yang and Li, 2014]. Although, generally, the model consists of a predator and prey with nonlinear interaction terms, it is not restricted to only two species but can be extended to three or more in a food-chain model [Gakkhar and Naji, 2003, Korobeinikov and Lee, 2009].

Most studies involving predator-prey systems have presumed that the carrying capacity of a predator is dependent on only its prey, without considering the carrying capacity of the prey, which is also dependent on another species, that is the base resource. Even though there exist intraguild models (in which the predator and prey share the same resource), they do not incorporate variable carrying capacities [Mylius et al., 2001, Diehl and Feissel, 2000, 2001, Hin et al., 2011].

10 In this thesis, we develop models of two species with carrying capacities as functions of a shared biotic resource, using the concept of a ratio-dependent type of predator- prey models. The general model can be written as

dX X = r X 1 − − aF (X)Y, (1.7a) dt 1 pK dY Y = r Y 1 − + bF (X)Y, (1.7b) dt 2 qK dK = f(X,Y,K). (1.7c) dt

Stability and bifurcation analyses of model (1.7) are performed, focussing on an environmental change whereby the enrichment of the resource has a great eﬀect on perturbing the system’s behaviour from one stable steady-state to another in the system. Other dynamics found in this system are discussed and presented.

Non-spatial models, such as ordinary differential equations (ODEs), which depend on a single variable serve as primary and significant platforms for investigating dynamical systems. To include a spatial dimension, partial differential equations (PDEs) must be considered. Applying them to ecological modelling is well established, as reviewed by Holmes et al. [1994] while more detailed explanations of diffusive and non-diffusive population models were discussed by Jungel [2010]. The Fisher equation is a classic example of a PDE for a single-species population, " # ∂u(r, t) u(r, t) = D∇2u(r, t) + ru(r, t) 1 − , (1.8) ∂t K where r = (x, y, z)|, u(r, t) is the time-dependent population density, and D the diffusion coefficient. Equation (1.8) has a travelling wave solution with a specific wave speed. The equation was introduced to describe the spreading of a favoured gene in a population [Fisher, 1937]. The wavefront solution provides information on how populations disperse over space. For a system with a two-species population, Dunbar [1983] investigated a diffusive predator-prey model that gave rise to a travelling wave solution in which the travelling wave moved with a constant shape and speed.

In this thesis, a travelling wave solution for a predator-prey ratio-dependent type of model with diffusion is investigated. The wave speed and the effects of the diffusivity parameters for each species over a spatial domain are analysed. For a food-chain model that involves more than two species, the ODE model is extended to a PDE system, and the travelling wave solution that arises in the predator-prey model with

11 a shared-resource model is investigated. Numerical solutions and simulations are presented to illustrate the system’s dynamics.

1.4 Overview of thesis

The aim of this thesis is to develop and analyse population-growth models subject to changing environments, modelled through variable carrying capacities. It is organised as follows:

Chapter 2 introduces the mathematical methods and tools that are used to investigate and analyse the mathematical models in this thesis, and provides a brief introduction to bifurcation analyses, in particular cusp singularity and Hopf bifurcation. Software packages which are used to analyse the models are also presented in this chapter.

Chapter 3 introduces the fundamental, non-autonomous logistic model, and various functional forms of the carrying capacity, K(t). A logistic model with a saturating carrying capacity is developed and applied to describe the total microbial biomass under an occlusion of healthy human skin. Data from published work are used to inform modelling decisions, and successfully veriﬁed and published in Sa- fuan et al. [2011]. A derivation of the exact solution for the developed model is also discussed in this chapter, based on Safuan et al. [2013a].

Chapter 4 introduces modiﬁed logistic models which consider the carrying capacity as a state-variable. A logistic model limited by its carrying capacity to represent a simple biotic resource, based on Safuan et al. [2012], is also discussed.

Chapter 5 introduces two-species population-growth models with variable carrying capacities. The concepts of the ratio-dependent type model, as well as the food-chain model are applied to modified predator-prey systems, with the carrying capacity modelled to represent a third species — a biotic resource. The carrying capacity is taken to be proportional to a biotic resource, and both species can directly alter the density of the biotic resource by interacting with it. The proposed models can be linked to intraguild predation models in which the predator and prey share the same resource. Taking the enrichment parameter of the resource as the bifurcation parameter, Hopf bifurcations are found for some parameter ranges, which generate solutions that possess limit-cycle behaviour. An analysis of the Hopf bifurcation was given in Safuan et al. [2013b]. In a slightly different model, the bistability in certain parameter regions describes the transition from a bene- ficial to a detrimental environment [Safuan et al., 2014b]. Special cases of these systems are examined, and the possibility of permanence and (or) extinction of species presented.

12 Chapter 6 introduces a one-dimensional spatially dependent population-resource model. The existence of travelling wave solutions and the minimum wave speed relationship are investigated. A linear stability analysis is performed in addition to a full numerical simulation of the model. Using the method of lines, the PDE systems are numerically solved for several types of initial conditions, and the corresponding travelling wave solution is demonstrated, as discussed in [Safuan et al., 2014c] (Manuscript submitted for publication). A similar analysis is applied to the diﬀusive two-species population model with a shared biotic resource, whereby the travelling wave solution of the model is studied and its dynamics investigated [Safuan et al., 2014a] (Manuscript in preparation).

Chapter 7 concludes this thesis by summarising all the main results from the study and discussing possible extensions to the models presented.

Chapter 2

Mathematical methods and tools

In this chapter, the mathematical tools, programs and software packages that are used throughout this thesis to analyse the mathematical aspects of various population models are introduced. Some parts of this chapter are discussed in Safuan et al. [2014b].

2.1 Mathematical methods

In this section, the bifurcation and singularity analyses used in Chapter 5, and the travelling wave solution applied in Chapter 6 are brieﬂy introduced.

2.1.1 Bifurcation analysis

Bifurcation is deﬁned as “splitting into two”, and is used to describe a change that occurs in a system by varying certain parameters [Arnol’d, 1994]. There are many types of bifurcation that can exist in dynamical systems, such as saddle-node, transcritical and Hopf.

Consider a one-dimensional system in the form

dx = f(x, γ), dt where x is a state-variable and γ the parameter to be varied. This parameter, γ, is referred to as a bifurcation parameter. By varying the value of this bifurcation parameter, diﬀerent types of dynamics emerging in the steady-state solution may be observed. Bifurcation diagrams play an important role in helping to graphically illustrate these dynamics.

Saddle-node bifurcation

A saddle-node bifurcation is a type of local bifurcation that occurs when a zero of a vector ﬁeld has a linearisation with zero determinant (at least one of the eigenvalues

15 is zero) [Hubbard and West, 1995]. The creation and destruction of a steady-state is another way of deﬁning it. This type of bifurcation is often also referred to as a turning point, limit point or fold bifurcation. Figure 2.1a shows an example of a saddle-node bifurcation, with the point located at γ = 0 marking the turning point between the stable (solid line) and the unstable (dashed) steady-states.

1 1

0.5 0.5

x x 0 0

−0.5 −0.5

−1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 γ γ (a) (b)

Figure 2.1: Examples of bifurcation: (a) saddle-node bifurcation; and (b) transcritical bifurcation (solid line – stable steady-state, and dashed line – unstable steady-state).

Transcritical bifurcation

A transcritical bifurcation occurs when two steady-state branches collide at an intersection referred to as a bifurcation point (Figure 2.1b.). The stability along each branch changes after passing through γ = 0. In Figure 2.1b, one of the steady-states is stable if the bifurcation parameter, γ, is less than zero, but becomes unstable otherwise, while the opposite applies for the other steady-state branch.

2.1.2 Singularity analysis

Singularity theory [Golubitsky and Schaeffer, 1979, 1985a,b] is a convenient approach to determine regions which show different steady-state behaviours. Gener- ally we start by reducing a physical system to a non-linear implicit scalar equation, sometimes called a singularity function, which helps to determine the behaviour of the steady-state solution. Consider a system of two differential equations in the form

dx = f(x, y, γ, p), (2.1a) dt dy = g(x, y, γ, p), (2.1b) dt

16 where x and y are the state-variables of the system, γ the main primary bifurcation parameter, and p all other parameters, which are often called secondary bifurcation parameters. Setting both equations to zero, and re-arranging for y = y(x) from any of the two equations and substituting it into the other equation, a steady-state solution of the system (2.1) is now written in the form of a reduced singularity function, G [Gray and Roberts, 1988],

G(x, γ, p) = 0. (2.2)

Using Equation (2.2), various properties of bifurcations such as limit point, cusp, isola, among others can be investigated. A steady-state diagram can be plotted in a graph of x versus γ for ﬁxed p. The objective is to identify the diﬀerent types of steady-state behaviour and their locations in the bifurcation diagrams, and is discussed in Chapter 5.

Cusp singularity

The cusp singularity is found when G satisﬁes the conditions

G = Gx = Gxx = 0 , with Gγ 6= 0 and Gxxx 6= 0 . (2.3)

When a cusp curve is crossed, the number of limit points (or turning points) on the steady-state curve changes, that is, limit points appear and disappear when moving from one region to the next. The cusp curve splits two regions, one of which has a hysteresis feature. Figure 2.2 shows an example of hysteresis which occurs when there are two stable steady-state branches as well as an unstable one between a set of parameters γ1 < γ < γ2. In this region of γ, the solution approaches one of the two attractors, depending on the starting point, a feature which is an indication of bistability.

The occurrence of bistability in population dynamics is not uncommon. For example, Malka et al. [2010] modelled the dynamics of bacterium and phagocyte populations, and found bistability regions that indicate a healthy or an infected state of a host. Elf et al. [2006] studied the response of bacteria to antibiotics, and found bistable behaviour in the bacterial growth rate. Studies by Dubnau and Losick [2006] and Santill´an[2008] also discussed bistability. In these studies, the choice of initial conditions has a great impact on establishing a desired or undesired state of the population.

17 1

0.5

x 0

−0.5

−1 γ 0 γ γ 1 2

Figure 2.2: A steady-state diagram exhibiting hysteresis behaviour. For γ1 < γ < γ2, bistability occurs (solid line – stable steady-state, and dashed line – unstable steady-state).

2.1.3 Hopf bifurcation

A Hopf bifurcation occurs when a pair of complex eigenvalues crosses the imaginary axis, and can only arise in a system with two or more equations. It is a local bifurcation that results in a periodic solution branch emanating from an equilibrium point [Kuznetsov, 1998]. The condition for a Hopf bifurcation is the set of p satisfying

d [tr(J)] f = g = tr(J) = 0, 6= 0 and det(J) > 0, dγ where J is the Jacobian matrix obtained by linearising about the critical points of the system (2.1). A Hopf bifurcation can be either super– or subcritical. If a stable periodic solution branch emanates from the Hopf point, then the Hopf bifurcation is said to be supercritical (Figure 2.3a). As a result, the system behaviour changes from stationary to a persistent oscillation. Conversely, if an unstable solution branch originates from the Hopf bifurcation, then it is said to be subcritical (Figure 2.3b).

Hopf bifurcations appear in a vast number of ecological applications in which oscillations or periodic phenomena are observed. For example, some predator-prey systems illustrate oscillatory dynamics whereby resource enrichment leads to destabilisation of the system and cause them to switch to limit cycles [Fussmann et al., 2000, Olivares and Palma, 2011].

18 1 1

0.5 0.5

x x 0 0

−0.5 −0.5

−1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 γ γ (a) (b)

Figure 2.3: Types of Hopf bifurcation, (a) supercritical Hopf bifurcation. (b) subcritical Hopf bifurcation (closed circle – stable periodic, and open circle – unstable periodic).

2.1.4 Travelling waves

The standard approach for travelling wave analysis is to convert a PDE problem to an associated ODE system using an appropriate substitution. Analysing the ODE system provides the analytical solution to the original PDE problem. Numer- ical approach such as the relaxation method is applied to the corresponding ODE system.

Consider a one-dimensional PDE system

∂u(x, t) ∂2u(x, t) = D + F u(x, t), v(x, t), (2.4a) ∂t 1 ∂x2 1 ∂v(x, t) ∂2v(x, t) = D + F u(x, t), v(x, t), (2.4b) ∂t 2 ∂x2 2 where u(x, t) and v(x, t) are populations with space variable x and time variable t, and D1 and D2 the diﬀusion coeﬃcients of u and v, respectively. F1 and F2 are the terms that describe the interactions between u and v.

Let the travelling wave solution have the form u(x, t) = u(ζ), and v(x, t) = v(ζ), where ζ = x−st is a moving frame with speed s. Then, substituting them in system (2.4) and using dimensionless variables gives the corresponding dimensionless ODE formulation

00 0 δu + su + G1(u, v) = 0, (2.5a)

00 0 v + sv + G2(u, v) = 0, (2.5b)

19 where δ is the ratio of D1 and D2. The functions G1 and G2 are the respective transformations of the terms F1 and F2. From system (2.5), the steady-states and their stability are investigated. The possible minimum wave speed is generally determined to ensure non-negative solutions, especially when considering population models.

Propagation of travelling population waves has been studied by others [Okubo, 1980, Dunbar, 1983, Murray, 1989, Owen and Lewis, 2001, Huang and Weng, 2013] to investigate how populations disperse spatially. If one species does not move, another species has the advantage if it moves and catches the sedentary population. The waves recognised for this type of situation are sometimes called waves of pursuit and evasion in predator-prey systems.

Relaxation method

Relaxation or collocation is a method to seek solutions for the PDE based on finite- difference schemes [Quarteroni and Valli, 2009], and is an iterative method. Starting with an initial guess, the solution is allowed to “relax” towards the true solution while, at the same time, reducing the errors. We begin with the simplest initial guess function   α1, ζ < ζ0, β1, ζ < ζ0, u(ζ) = v(ζ) = (2.6) α2, ζ > ζ0, β2, ζ > ζ0, where u(ζ) and v(ζ) are step functions. From the governing equations (2.5), the first and second derivatives are discretised using a centered-difference scheme as

u − u u − 2u + u u0 = n+1 n−1 , u00 = n+1 n n−1 , 2h h2 v − v v − 2v + v v0 = n+1 n−1 , v00 = n+1 n n−1 , 2h h2 respectively, where h is the step size. Backward- and forward-diﬀerence schemes are other options available for discretising the derivatives. We write     0 1 0 −2 1      −1 0 1   1 −2 1  1   1   A =  ......  ,B =  ......  ,  . . .  2  . . .  2h   h    −1 0 1   1 −2 1      −1 0 1 −2

20 " # H H = 1 , H2 where A and B are the diﬀerentiation matrices , and H is

H1 = δBu + sAu + G1(u, v), (2.7a)

H2 = Bv + sAv + G2(u, v), (2.7b)

with G1 and G2 from the system (2.5). Thus, the vector solution is   u1  .   .       un  Wn =    v   n+1   .   .    v2n such that −1 Wn+1 = M H(Wn), where M is a block diagonal matrix constructed from vectors u and v. This method is used in Chapter 6 to obtain the travelling wave solution in diﬀusive population models.

Method of Lines (MOL)

The MOL is a numerical approach to obtaining solutions to PDEs [Schiesser, 1991]. This scheme involves replacing partial derivatives in the spatial variables by ﬁnite- diﬀerence approximations. As a consequence, only time derivatives are present and as such the PDEs are “converted” to a system of ODEs. Hence all that is required is a good ODE solver such as those available in software packages like MATLAB R , which is discussed in the next section.

In our simulations, the spatial domain is divided into regions with uniform step size. A Neumann boundary condition ∂u/∂n = ∂v/∂n = 0 is used where n is the outward unit normal vector to the boundary, which reflects the fact that no population can escape across the boundary. Initial conditions are set to be step functions in most simulations. To calculate the wave speed, peaks of the waves are traced for two different time steps. By following the peaks, the speed is calculated by dividing the distance by the time difference between the peaks. This method is applicable for constant speed, since it is easy to trace waves with the same speed. However, to trace waves with non-constant speeds, a more accurate method and careful analysis are

21 needed. In our numerical calculation, we use the algorithm developed by Fornberg [1988, 1998] to generate and compute ﬁnite-diﬀerence weights at each grid point. The algorithm works for equispaced grids and also for arbitrarily spaced grids.

2.2 Software packages

Mathematical modelling, especially when considering nonlinear systems, is quite challenging. If it is not possible to obtain an analytical solution, numerical analyses must be used for solving complex systems. The following software packages are ones we found useful in this research.

MATLAB R

MATLAB R [MathWorks, 2008] is a powerful computing tool widely used to analyse and explore mathematical models in a huge range of applications. Most of plots in this thesis were made with MATLAB R . Numerical simulations in Chapters 3 and 4, and the steady-state and periodic solutions in Chapter 5 were found and veriﬁed using inbuilt MATLAB R ODE solver routines such as ODE15s, ODE23 and ODE45.

ODE45 is based on a fourth/fifth order Runge-Kutta formula. It is a one-step solver, similar to ODE23 [Redfern, 1998, Moler, 2008]. In general, ODE45 is the first solver to be used to solve ODE system before trying other solvers. ODE23 uses the second and third order Runge-Kutta formulas. In most cases, ODE45 is more accurate and more efficient than ODE23. ODE23 is used to solve problems with crude error tolerances or for solving moderately stiff problems. ODE15s is a multi-step solver that solves stiff ODE systems, and is used if ODE45 fails or is found to be inefficient to solve the system.

MATLAB R is capable of solving PDE problems in one, two and three dimensions provided the correct toolbox is available in the package. The spatially dependent models in Chapter 6 were solved using numerical routines such as the ﬁnite- diﬀerence method and MOL with the algorithms we developed in MATLAB R .

MAPLE

MAPLE [Maplesoft, 2008] is another eﬃcient computing package for analysing mathematical models. Its capability for interactive symbolic computation is very useful and it is able to generate numerical and graphical representations. This software was used for the greater part of the singularity analysis in Chapter 5. Some

22 parts of the analytical work in these chapters were veriﬁed using this software, especially the analytical form of the nontrivial eigenvalues of the steady-states.

G3DATA

G3DATA [Frantz, 2000] is a freeware executable tool for extracting data from graphs found in articles. With these extracted data, the plots can be redrawn and further analysis can be done, particularly to compare the analytical solutions with the experimental data. The use of this software is discussed in Chapter 3.

XPPAUT

XPPAUT [Ermentrout, 2010] is a freeware tool for analysing and simulating dynamical systems. This package was found to be extremely useful in obtaining the steady-state plots as well as periodic-solution branches. The stability of the steady- state and of periodic-solution branches are easily determined by XPPAUT’s inbuilt AUTO [Doedel, 2007] program which allows for path-following and bifurcation analysis. This software was extensively used in Chapter 5.

FlexPDETM

FlexPDETM [PDE Solution Inc., 2011] is a script-based numerical solver designed to solve a PDE problem. Although it is a commercial package, a free student version (with restricted mesh size) is also available. We used this software to solve spatially dependent models such as the travelling waves in Chapter 6. Similar to MATLAB R , it is able to solve the PDE problem in one, two and three dimensions through its automatic inbuilt computational technique – adaptive mesh refinement. This method adapts the finite-element mesh in areas with large errors. It iterates the mesh cells and if any exceed this specified tolerance level, FlexPDETM recomputes the solution. As a result, the accuracy of the solution is guaranteed and errors are minimised to the desired relative error tolerance level. The error tolerance level (errlim) depends on the type of the PDE problem. The script codings for the governing equations are easy to write and the graphical output interface allows several plots to be analysed simultaneously while the computations take place. The data can be extracted from the solution and exported to other numerical software for further analysis. As an independent solver for the PDE system, FlexPDETM can be used as a validation of the solutions obtained via MOL or finite differences.

23 2.3 Conclusions

This chapter presents the analytical and numerical methods, and software packages used in the analysis in this thesis. In the next chapter, fundamental models are presented and investigated. An application of the model is also discussed and veriﬁed.

24 Chapter 3

Fundamental models

In this chapter, we introduce some mathematical models for studying changing environments. Incorporating a time-dependent carrying capacity in growth models is a practical approach for dealing with variations in environments. A varying environment certainly inﬂuences the dynamics of a population either positively or negatively. We investigate several types of variable carrying capacities in the logistic equation (1.1); this equation then becomes non-autonomous, and more analysis is required. A carrying capacity that represents a rich environment for a population is developed and analysed, and the corresponding exact solution is discussed in Section 3.3. A possible application of the model is discussed using published experimental data. Some of the work presented in this chapter appears in Safuan et al. [2011] and Safuan et al. [2013a].

3.1 Time-dependent carrying capacity as an explicit function of t

A changing environment may be characterised by a variable carrying capacity and possibly a variable intrinsic growth rate. One way to model such a scenario is using the non-autonomous logistic equation

dN(t) N(t) = r(t)N(t) 1 − . (3.1) dt K(t)

The general mathematical properties of Equation (3.1) were deduced by Coleman [1979a]. As the main focus of this thesis is investigating changing environments, we discuss only changes in the carrying capacities of all models. Table 3.1 sum- marises several functional forms of carrying capacities that have been used in the literature. By using a time-dependent carrying capacity, K(t), the logistic equation is non-autonomous. However, as opposed to the classic logistic equation for which an exact solution can be found, determining an analytical solution of the non- autonomous equation is often not possible. Exact solutions exist for only a limited number of cases [Banks, 1994]. For a logistically varying carrying capacity, an exact

25 Table 3.1: Time-dependent carrying capacities K(t). K(t) References K(t) = a + b sin(ct + φ) Coleman et al. [1979b]; Leach and Andriopoulos [2004]; Rogovchenko and Rogovchenko [2009]

K(t) = a + b(1 − e−ct) Ikeda and Yokoi [1980]

−bt K(t) = K1 + K2/(1 + ae ) Meyer [1994]; Meyer and Ausubel [1999]

−bt K(t) = K1K2/[(K2 − K1)e + K1] Ebert and Weisser [1997]

−bkt K(t) = Kk/[1 + ake ] Watanabe et al. [2004]

solution can be found [Meyer, 1994, Meyer and Ausubel, 1999], while Shepherd and Stojkov [2007], Grozdanovski and Shepherd [2007], Grozdanovski et al. [2009] and Grozdanovski et al. [2010] used a multi-scaling technique to understand a carrying capacity that varied slowly relative to a changing population.

3.2 Carrying capacity as a state-variable

The carrying capacity, K(t), in Equation (3.1) can be viewed as a solution to a ﬁrst- order diﬀerential equation. For example, the constant carrying capacity, K = k, in the classic logistic equation (1.1) is a solution to

dK = 0,K(0) = k. (3.2) dt

Equation (3.2) describes a carrying capacity that does not change with time. If K is changing at a constant rate, a, this equation becomes

dK = a, K(0) > 0. (3.3) dt

In the absence of a self-reproduction growth term, the carrying capacity in Equation (3.3) is assumed to be an abiotic resource constantly provided to the system (a > 0). The solution K(t) to Equation (3.3) is a linear function (Figure 3.1a).

Another form representing an increasing function is

dK = aK, a > 0, (3.4) dt

26 8 14

7 12 6 K(t) 10 5 8 4 N, K N, K 6 K(t) 3 N(t) 4 2 N(t) 1 2

0 0 0 10 20 30 40 50 0 10 20 30 40 50 t t (a) (b)

7 x 10 6 6 K K(t) 2 5 5

4 4

3 N(t) 3 K(t) N(t) N, K N, K

2 2

1 1 K 1

0 0 0 10 20 30 40 50 0 100 200 300 400 500 t t (c) (d)

Figure 3.1: Logistic equation (1.1) with carrying capacity that is (a) linear, K(t) = 0.05t+5; (b) exponential, K(t) = 5e0.02t; (c) periodic, K(t) = 0.5 sin(t)+5; and (d) 7 5×108 bi-logistic, K(t) = 5 × 10 + 1+e−0.032(t−350) . For bi-logistic, the data are reproduced from Meyer and Ausubel [1999].

27 which has an exponential solution (Figure 3.1b). Compared with Equation (3.3), the carrying capacity in Equation (3.4) has a growth rate, a, which represents a biotic resource that is replicated. For these two cases, K(t), is linearly and exponentially increasing, respectively, without bound. However, to model a realistic population, a carrying capacity that has an upper limit is usually chosen since, in reality, resources are limited.

If external factors, such as rain or temperature, change in a periodic manner, an obvious way to model the carrying capacity is to use the sinusoidal function

dK = a cos(bt + c). (3.5) dt

The solution to Equation (3.5) is periodic (Figure 3.1c). This type of carrying capacity has been extensively applied in population dynamics to describe ﬂuctuating environments [Coleman et al., 1979b, Leach and Andriopoulos, 2004, Rogovchenko and Rogovchenko, 2009].

Meyer [1994] and Meyer and Ausubel [1999] introduced a carrying capacity as a logistic function dK K = aK 1 − , (3.6) dt Ka where Ka is a positive constant. As the carrying capacity varies logistically in the logistic growth function Equation (3.1), this model is called the bi-logistic model. To ensure an asymptotically nonzero carrying capacity, Equation (3.6) is rewritten as dK K − K1 = a (K − K1) 1 − . (3.7) dt K2 The purpose of writing it in this way is so the carrying capacity increases from a lower level K1 to an upper level K2. This model has been used in forecasting technological development of a human population [Meyer, 1994, Meyer and Ausubel, 1999].

3.3 Application of variable environmental carrying capacity

In this section, we treat the environmental carrying capacity as a state-variable to describe the total microbial biomass under a skin occlusion, and later derive the exact solution to the associated non-autonomous logistic equation.

3.3.1 Application to total microbial biomass during skin occlusion

Occlusive treatments, such as the application of bandages to wounds, are ancient in origin but have only been the subject of dermatological experiments since the

28 late 1960s. The goal of these experiments has been to investigate the signiﬁcance of covering part of the skin with certain permeable or impermeable materials. As a therapy, occlusions have been successfully applied to atopic dermatitis, psoriatic patients and also to wounds, ulcers and cutaneous eruptions [Braham et al., 2010].

Several experiments have sought to measure the physiological changes that take place on the surface of the skin while it is occluded for a period of time [Aly et al., 1978, Hartmann, 1983, Faergemann et al., 1983]. Aly et al. [1978] provides the most complete and comprehensive data, recording over five consecutive days the total microbial density, trans-epidermal water loss (tewl), carbon dioxide emission rate (cder) and skin-pH after a vinylidene polymer plastic film (Saran Wrap c ) was placed on the forearms of ten healthy volunteers. Hartmann [1983] also studied the effect of a three-day occlusion placed on the forearm of 26 volunteers, while Faergemann et al. [1983] experimented with the effect of an eight-day skin occlusion on the back of ten volunteers.

These experiments demonstrate that the epidermis undergoes changes when the skin is occluded. Water from the deeper layers of the skin diﬀuse through the stratum corneum (sc) and is trapped by the occlusive dressing, preventing the evaporation of water from the skin’s surface. As the epidermal membrane becomes hydrated, the sc becomes more permeable, which in turn results in an increase in the tewl. In the study by Aly et al. [1978] the tewl increased from 0.56 mg/cm2/hr to 1.87 mg/cm2/hr. This increase in tewl is evidence of an increased skin-moisture content of the epidermis and skin surface. Hartmann [1983] found that the skin- moisture content increased from a low 20% and saturated at 75% after only three days of occlusion of the skin. According to the measurements of Faergemann et al. [1983], the tewl reading remained relatively steady after the third day.

The skin also expels carbon dioxide to the atmosphere. Without any pressure on its surface, the skin releases carbon dioxide constantly at a low rate [Agache et al., 2004]. A hydrated and thus more permeable skin has a higher diﬀusion rate of carbon dioxide. In Aly et al. [1978] the cder readings quickly rose to three times their base level by the ﬁrst day and saturated thereafter.

Besides tewl and cder, changes in skin-pH also occurred during occlusion. The skin surface pH increased steadily from 4.38 to 7.05. The occlusion inhibits the evaporation of eccrine perspiration, thus elevating the pH. Faergemann et al. [1983] also found the pH to rise rapidly during the ﬁrst three days, then remain fairly constant until the eighth day. Other factors that might be important to the pH change are the changes to the composition of sweat, sebum, lipids, fatty acids and

29 nutrients [Schmid and Korting, 2006] during the occlusion. These latter quantities have not been measured because they are complex and not easily interpreted.

Under an occlusion, the normally harsh skin environment is transformed to one more favourable, with enriched nutrient ﬂows resulting in an increase in the density of micro-organisms found there. Aly et al. [1978] measured an almost four orders of magnitude increase in the total microbial density in the ﬁrst two days after the application of the occlusion, and it remained approximately constant for the remaining three days of the experiment. Hartmann [1983] found similar rises in the microbial density after a three-day occlusion of the skin, but Faergemann et al. [1983] reported only a two orders of magnitude increase for microbes residing on the back.

We are interested in modelling the growth of the total microbial biomass under an occlusion. Many physiological factors impact upon bacterial counts; our mathematical model attempts to incorporate one or more of these factors.

In the pre-occluded skin, the environment is relatively constant, and the microbes under the skin receive a continuous source of nutrients. The density of microbes is 2 in equilibrium with its environment, K ≈ N0 = 920 counts/cm . A common model to describe such a population is the Verhulst or logistic equation (1.1).

When an occlusion is applied to the skin, the environment beneath it begins to change to one that is generally more favourable for microbial growth. A changing environment may be characterised by a varying carrying capacity and possibly a varying intrinsic growth rate. The appropriate way to model such a scenario is the non-autonomous logistic equation (3.1)

In what follows, we assume the intrinsic growth rate, r, is constant and seek a monotonically increasing function, K(t), that begins with carrying capacity K(0) = K0,

K0 ≈ N0, representing the unoccluded skin with baseline values for tewl, cder, skin-moisture, skin-pH and nutrient ﬂow rates. K increases to a new and larger carrying capacity, Ks, representing an environment with elevated but constant values for the aforementioned physiological factors. Meyer [1994] and Meyer and Ausubel [1999] introduced a logistic form of the carrying capacity, but a much simpler model that meets our criteria is expressed as [Safuan et al., 2011]

dK = −α(K − K ), (3.8) dt s where α is the saturation constant. The solution for the carrying capacity is

−αt K(t) = Ks 1 − βe , (3.9)

30 where β = 1 − K0/Ks. A carrying capacity of this form has previously been used to model ﬁsh stocks in a nutrient-enriched environment [Ikeda and Yokoi, 1980].

It is possible to use the data for the microbial densities to estimate r, K0 and Ks but not the value of α. For this purpose we take a diﬀerent approach. From the discussion of the previous section, the cder and tewl co-evolve with the environment. This means that the changes in the carrying capacity correlate positively with the changes in both cder and tewl. Therefore, we assume that the functional form for both of these quantities is given by expressions similar to that of Equation (3.9), with the same parameter α.

Here we use the cder data taken from Aly et al. [1978]. Let C(t) be the cder −αt at time t, the time evolution is then given by C(t) = Cs (1 − βce ), where βc =

1 − C0/Cs. This requires the estimation of two further parameters, C0 and Cs. To be clear, we use the cder data to estimate the value of α, which in turn is used in the expression for K(t).

The cder and total microbial density data from Aly et al. [1978] were extracted using G3DATA [Frantz, 2000], and are reproduced in Figure 3.2 and Figure 3.3a, respectively. There are some inconsistencies between the graphs and the text in Aly et al. [1978]. Throughout this discussion we will assume that the data from the graphs are correct while the text is in error.

2 Using the cder observations of Figure 3.2, we set C0 = 25 nl/cm /min and estimate 2 Cs ≈ 91.2 nl/cm /min, it being the average of the observations over the last three days of the occlusion. A nonlinear least-squares fit was then used on the cder data to estimate α. We obtained α = 1.92/day, with a 95% confidence interval of (0.29, 3.54). The large confidence interval is a consequence of the small number of observations. The fitted curve is shown in Figure 3.2. The model appears to adequately represent the observed data.

We now turn our attention to the microbial density data. Since we are dealing with a general biomass and not a speciﬁc bacterium, and because in vivo bacterial growth rates are not known, we must estimate r for the data of Figure 3.3a.

The intrinsic growth rate, r ≈ ∆ ln N/∆t, can be determined from the so-called exponential growth phase of the growth curve. By taking the average slope from the ﬁrst and second days relative to the pre-occlusion microbial density, we get 2 7 r = 6.6/day. Further, we set K0 = 920 counts/cm , and estimate Ks = 2.29 × 10 counts/cm2, it being the geometric mean of the microbial density over the last three days. No attempt was made to ﬁnd standard errors for these estimates due to the small size of the data set.

31 120

100

80 /min) 2

40 CDER (nl/cm

20 Data from Aly et al. (1978) Nonlinear fit with α = 1.917 0 0 1 2 3 4 5 Time (days)

Figure 3.2: cder data with the nonlinear ﬁtted curve. Parameters are C0 = 2 2 25 nl/cm /min, Cs = 91.2 nl/cm /min, α = 1.92/day and β = 0.73.

Incorporating Equation (3.9) into Equation (3.1) leads to

dN(t) h N(t) i = rN(t) 1 − −αt . (3.10) dt Ks(1 − βe )

The numerical solution of Equation (3.10) with the estimated parameters is shown in Figure 3.3a. It is apparent that the model does a good job in describing the growth of the microbial biomass. What is also clear is that this model predicts the very rapid increase in the environment’s carrying capacity.

The ability to validate a model with an independent set of data is vitally important. To validate our model, we use the data of Pillsbury and Kligman [1967] (report released by the US Army in 2000). They reported the total microbial density on forearm skin under occlusion for a period of more than two weeks. The data are reproduced in Figure 3.3b. As with other experiments mentioned so far, a four orders of magnitude increase in cutaneous microﬂora resulted within the two days of the application of an occlusion. The geometric mean was calculated for each of the days for which the microbial density was measured. Using the same model with the same parameters as Aly et al. [1978], our model predicts the total microbial biomass remarkably well.

The carrying capacity, K(t) in the model (3.10) encompasses the adjustment of the environment resulting from the changes on the skin surface due to the application of an occlusion.

From the model we deduce that there exist two characteristic time scales: one is related to the increase in the carrying capacity, the other to the increase in

32 (a)

(b)

Figure 3.3: Growth curves N(t) and variable carrying capacity K(t) with the 2 estimated parameters r = 6.6/day,N0 ≈ K0 = 920 counts/cm ,Ks = 2.29 × 107 counts/cm2, α = 1.92/day, β = 0.99996 for (a) Aly et al. [1978] and (b) Pills- bury and Kligman [1967] using the proposed model.

33 the population density. From Equation (3.8), the quantity |K(t) − Ks| represents the environment’s instantaneous excess capacity, and the time required to halve this capacity is t1/2 = ln 2/α ≈ 8.7 hours; the doubling time for the microbial biomass is t1/2 = ln 2/r ≈ 2.5 hours. This suggests that by the ﬁrst 8.7 hours after the occlusion is placed on the skin, the excess capacity is decreased (carrying 7 2 capacity increased) by approximately Ks/2 ≈ 10 counts/cm . In the meantime the microbial density has experienced only a ten-fold increase. Initially, the environment under the occlusion is developing at an extremely fast rate, enabling the growth of the total skin microﬂora without restraint. This feature of the model is evident from Figures 3.3a and 3.3b.

There are several limitations in this study. The model possibly could be improved if we acquire a greater number observations of physiological effects for a longer period of occlusion time. Better quality cder data, for example, would provide a better estimate for α. Of course, it might be possible to use the tewl data to find another independent estimate of α. However, this was not done here because of a problem with the quality of the data in Aly et al. [1978]. The tewl was measured about 20 minutes after the occlusion was removed. Pinto and Rodrigues [2005] recorded the tewl immediately after the removal of a one-day occlusion of the forearm for 30 minutes. It was found that the tewl fell to less than half its maximum level within a 20-minute interval. This demonstrates that the tewl data in Aly et al. [1978] grossly underestimate the true value, at least for the first and perhaps the second day; any estimate of α based on these data would also have been an underestimate.

In experiments, the intrinsic growth rate depends on temperature, water content, pH and CO2, to mention a few [Denyer et al., 1990, Ratkowsky and Ross, 1995]. Our assumption of a constant r is certainly not valid, but how the effects of an occlusion are reflected in an increase in r is not possible to ascertain. Furthermore, we are dealing with a biomass consisting of four different species of bacteria, and for each in vivo growth rates are not known.

Future work will need to focus on the interaction between the dominant bacterial species present on the skin. Aly et al. [1978] and Pillsbury and Kligman [1967] present an interesting dynamic between the major bacterial species of the skin. A better understanding of these interactions would assist in building a model of the time evolution of bacterial growth in a wound, with obvious health implications.

3.3.2 Exact solution of a logistic model with saturating carrying capacity

For a non-autonomous equation, an exact solution is usually impossible due to the varying carrying capacity or the growth rate. Sometimes when the exact solution can be found, it is only valid for speciﬁc values of the parameters, not a range. Thus,

34 the use of numerical methods are necessary to solve the problem. It is worthwhile ﬁnding an approximate or exact solution of Equation (3.1) so that it can be applied for any range of parameters. Recently, Safuan et al. [2013a] presented the exact solution of Equation (3.10), which is discussed in this section. Equation (3.10) has the solution N ert N(t) = 0 . (3.11) Z t rx rN0 e 1 + −αx dx Ks 0 (1 − βe )

The exact solution of the non-autonomous equation can be derived by noting

Z t erx −αx dx = F (t) − F (0) , (3.12) 0 1 − βe where

ert r r F (t) = F 1, − ; 1 − ; βe−αt . (3.13) r 2 1 α α

The function 2F1 is a hypergeometric function, which converges for α > 0 and 0 < β < 1. It can also be expressed as the series

∞ −αt n r r X (1)n(−r/α)n (βe ) F 1, − ; 1 − ; βe−αt = , (3.14) 2 1 α α (1 − r/α) n! n=0 n where (a)n is a Pochhammer symbol: (a)n = a(a + 1) ··· (a + n − 1) [Abramowitz and Stegun, 1972].

Provided the initial population, N0, and the saturation level, Ks, can be obtained from experimental data, parameter β can be estimated. The parameters r and α are estimated to ﬁt the data. We take r = 7/day and α = 2/day (values close to Safuan et al. [2011]), and obtain

Z t 7x 7t 5t 2 3t e e βe β e 3 t −2x dx = + + + β e 0 1 − βe 7 5 3 √ β7/2 et − β + ln √ − C , (3.15) t 0 2 e + β where √ ! 1 β β2 β7/2 1 − β 3 C0 = + + + β + ln √ . 7 5 3 2 1 + β By taking the parameters r and α to be integers, a ﬁnite number of terms in the solution is obtained. However, this is inconvenient for most applications, as the parameters are rarely integers in experimental data.

35 Alternative approach

Suppose r > 0 and α > 0. Since 0 < β < 1, we may formally write

∞ 1 X = βne−nαx , x ≥ 0 . (3.16) 1 − βe−αx n=0

The integral in the denominator of (3.11) then becomes

∞ Z t erx Z t X dx = βne(r−nα)x dx, 1 − βe−αx 0 0 n=0 ∞ X βn = e(r−nα)t − 1 . (3.17) r − nα n=0

Accordingly, the exact solution is

KsN0 N(t) = ∞ . (3.18) X βn K e−rt + aN e−nαt − e−rt s 0 r − nα n=0

Although this solution is exact for t ≥ 0, for practical applications the series solution needs to be truncated. Note that for some n ∈ N, r − nα < 0, so the index of the exponential in Equation (3.17) is negative and this term’s contribution to the series solution falls oﬀ quickly for increasing t. Consequently, truncating the series at the (n − 1)th term incurs a maximal error of O(βn).

Special cases

For α = 0, K(t) = K0 for all t. The solution for this case is given by Equation (1.2) with K replaced by K0,

K0N0 N(t) = −rt −rt . (3.19) K0e + N0(1 − e )

As α → ∞, the carrying capacity is a step function (K0 < Ks),   K0, t ≤ 0, K(t) = (3.20)  Ks, t > 0.

36 Suppose N(t1) = N1 < K0 for some t1 < 0. The solution is

 N1K0  −r(t−t ) −r(t−t ) , t ≤ 0 ,  K0e 1 + N1(1 − e 1 ) N(t) = (3.21)  KsN0  , t > 0 ,  −rt −rt Kse + N0(1 − e ) where N1K0 N0 = rt rt . (3.22) K0e 1 + N1(1 − e 1 )

Further, if for t < 0 the population is in equilibrium with its environment, then

K0 ≈ N(t1) = N1 implies K0 ≈ N0. The solution for N(t) can then be written in terms of the carrying capacities K0 and Ks,

KsK0 N(t) ≈ −rt −rt . (3.23) Kse + K0(1 − e )

Figure 3.4 shows the comparison between the numerical solution of Equation (3.11),

Nnum, and the various approximate solutions generated by Equation (3.18) for the same parameters used in Safuan et al. [2011]. Napp1,Napp2, Napp3, Napp4 and Napp5 represent one (open circle), two (cross), three (closed circle), four (left arrow) and ﬁve (right arrow) terms retained in the approximation, respectively. It can be seen that, with one term the approximation is in good agreement with the numerical result. With more terms in the approximation, even better agreement is achieved. Due to the large scale of N, diﬀerences in the curves are not clearly visible.

To determine the accuracy of the various approximate solutions, relative errors were calculated (Figure 3.5). The maximum relative error at t = 2 for the ﬁve-term approximation (retaining terms up and including n = 4) is only 0.06%, four terms (0.06%), three terms (0.07%), two terms (0.18%) and with only one term (3.2%). These values indicate that our approximations are reasonably good. Maximum relative error is clearly reduced by using more terms in the solution (3.18). According to our simple criterion, the series may be terminated at the (n − 1) term when a − nc < 0. Here, the series may be terminated at n = 3 (Napp4).

Other examples, which more clearly demonstrate the diﬀerences among the various approximate solutions, are given in Figure 3.6. In these two examples, approximate solutions up to n = 3 are used. The Napp1 solution (open circle) deviates substantially from the numerical solution over the critical intermediate values of t. Increasing the number of terms in the approximation reduces the relative error and more closely resembles the numerical results.

37 7 x 10 2.5 ) 2

2 N num N app5 1.5 N app4 N app3 1 N app2 N app1 0.5 N (Microbial biomass, counts/cm

0 0 1 2 3 4 5 t (Days)

Figure 3.4: Comparison between the numerical solution, Nnum, and the various approximate solutions obtained from (3.18). Parameters taken from Safuan et al. 2 7 2 [2011]: r = 6.6/day; N0 ≈ K0 = 920 counts/cm ; Ks = 2.29 × 10 counts/cm ; α = 1.92/day; β = 0.99996.

Table 3.2: Relative error sustained by approximate solutions for certain values of t.

t Napp1 Napp2 Napp3 Napp4 Napp5 1 4.64e-3 1.23e-3 2.61e-3 3.02e-3 3.18e-3 2 3.18e-2 1.77e-3 6.75e-4 6.11e-4 6.02e-4 3 4.74e-3 3.77e-5 1.14e-5 1.11e-5 1.11e-5 4 6.56e-4 4.63e-5 4.69e-5 4.69e-5 4.69e-5 5 1.09e-4 3.95e-6 3.93e-6 3.93e-6 3.93e-6

−1 10 N app5 N −2 app4 10 N app3 N app2 −3 10 N app1

−4 10 Relative Error

−5 10

−6 10 1 2 3 4 5 t (Days)

Figure 3.5: A log scale plot of the relative error according to the data in Table 3.2.

200

N 150 num N app4 N

N app3 100 N app2 N app1 50

0 0 1 2 3 4 5 t (a)

200

N 150 num N app4 N

N app3 100 N app2 N app1 50

0 0 1 2 3 4 5 t (b)

Figure 3.6: Comparison between the numerical solution, Nnum, and various approximate solutions. The parameters are N0 = 20, Ks = 200, β = 0.9 with (a): r = 3.3, α = 1.2 and (b): r = 8.5, α = 3.1.

39 Inﬂection point

The inflection point is an important property in population growth. It determines the time, ti, and population, Ni, when the growth rate is maximal. It is obtained by setting N 00(t) = 0. An inflection point is sometimes important in an experiment. Tracy [1999] used the inflection point to determine the maturity stage of a chuckwalla (large lizard) population, and in an experiment by Cronmiller and Thompson [1980], the mean weight gain of a red-winged blackbird was measured at the inflection point.

For the non-autonomous logistic equation (3.1) the derivation of an algebraic form of the inﬂection point is not straightforward, as K is no longer constant. Requiring N 00(t) = 0 leads to N 2N 2 − 3NK + K2 + K0 = 0. (3.24) r

For the limiting case α → ∞, K(t) → Ks for t > 0, the above equation reduces to

2 2 2N − 3NKs + Ks = 0, giving N = Ks,Ks/2. The inﬂection point occurs at population

K N = s . i 2

Substituting this into Equation (3.23) leads to

1 Ks ti ≈ ln − 1 . (3.25) r K0

Using the approximate solution (3.18) and keeping all parameters as in Safuan et al. [2011], Figure 3.7 is obtained, a contour plot of the inﬂection time, ti, over the domain 1 ≤ r ≤ 8 and 0.1 ≤ α ≤ 2.

For a ﬁxed value of α, as expected, the value of ti decreases as the intrinsic birth rate, r, increases. For larger values of α (α > 1), the contour lines appear to be parallel. The carrying capacity is well approximated by a step function, and the value of ti is essentially independent of α. The value of a that leads to the given ti can be estimated from Equation (3.25).

For a ﬁxed r, ti increases, although slightly, as α increases. This is due to the development of the environment’s carrying capacity, thus delaying the time to reach the maximal growth rate.

40 For small to intermediate values of α, the contour lines show a slight curvature. The curvature is more pronounced for larger values of r but smaller values of α. As the value of α increases, the environment’s carrying capacity reaches saturation faster and in order to maintain ti at a ﬁxed value r must increase, until it reaches its asymptotic value expressed through the inﬂection time in Equation (3.25).

Figure 3.7: Plot of inﬂection time, ti (days), for varying r and α. As the growth rate, r, increases, the inﬂection time, ti, decreases.

3.4 Conclusion

A saturating carrying capacity is introduced to describe a positively changing environment that leads to increased numbers in a population. The model is then applied to describe the growth of a microbial biomass under an occlusive patch on the skin. The model provides a remarkably good ﬁt to existing data. This model is the ﬁrst to mathematically describe bacterial growth under an occlusion.

For the same form of the carrying capacity, an exact solution of the non-autonomous logistic equation is derived using a straightforward algebraic method. Instead of a complex solution based on an inﬁnite series, an approximate solution based on a simple criterion is used for practical applications. A good agreement with numerical simulations is possible, even though only a small number of terms is retained in the approximate solution. The relative errors presented validate the accuracy of the approximate solutions. These results can be used for any parameter values of the growth rate, r, and saturation rate, α, with 0 < β < 1.

The model developed in this chapter provides a fundamental understanding of a changing environment as represented by a time-dependent carrying capacity. In the next chapter, variable-dependent carrying capacities and how they inﬂuence population dynamics are investigated.

Chapter 4

Single-species population-growth models with variable carrying capacities

Chapter 3 discussed the idea of modelling a varying environment via a time-dependent carrying capacity. This requires that an appropriate functional form be chosen that models the scenario under consideration. Chapter 3 also mentions that an alternative approach is to treat the carrying capacity as a state-variable. This provides for a more flexible and possibly a more realistic approach to incorporate changes in the carrying capacity. If the carrying capacity depends on a resource which changes with time, and the population has the effect of degrading or improving the environment, both the carrying capacity and population have to be formulated as a system which is autonomous. Therefore, an equation is added to Equation (1.1) to represent the rate of change of a varying resource, which could involve linear or nonlinear interaction terms. These two equations form a coupled system of differential equations in which the equilibrium, stability and phase plane can be analysed.

Some modiﬁed logistic models that have been studied by others are discussed in Section 4.2. The models are compared to the system presented in Section 4.3 in which we consider the environmental carrying capacity as a simple biotic resource. Some of the work presented in this chapter appears in Safuan et al. [2012].

4.1 Environmental carrying capacity as a biotic resource

The carrying capacity has commonly been represented as the maximum population an environment can sustain. Some studies have regarded it as the equilibrium density of the resources, and others as a measure of environmental productivity [Mylius et al., 2001, Diehl and Feissel, 2001]. These studies considered the carrying capacity as a constant quantity in order to describe the measure and capability of a system for a given population. However, the quantity used usually did not represent any speciﬁc resource, but was only a number that established the upper limit of the population. If a population size is below an environment’s carrying capacity,

43 this indicates that the environment has a relatively high resource supply/demand ratio which results in population increase [Taylor et al., 1990]. On the other hand, a population size exceeding the carrying capacity implies that the environment’s resource are being exhausted, and the population will eventually decline until it reaches the stable equilibrium (carrying capacity), a feature called an overshoot.

However, in this thesis, we consider the carrying capacity as a (biotic) resource representing the environment. The reason for doing this is that this quantity can be measured, the dynamics between the population and its resource can be seen more clearly, and, more importantly, the carrying capacity is varying rather than constant.

The availability of a resource has a great influence on setting a population’s carrying capacity, and comes in a wide range of types for sustaining population growth. Examples of biotic resources are plants, crops, birds, fish and phytoplankton, which are generally finite and limited. As an increasing population becomes larger and more crowded, its members have to compete with each other and possibly with other species for the remaining resources.

4.2 Modiﬁed logistic models

There are several examples of a logistic-type equation being coupled to an equation for the carrying capacity. Huzimura and Matsuyama [1999] studied a model in which the rate of change of the carrying capacity decayed proportionally to the population size. The model assumed that the carrying capacity, K, was a function of the vital resource, which is a resource-dependent carrying capacity. This model was applied to a deer population in a conﬁned environment (an island). The deer population is dependent on the main resource available (lichen), with the model represented as

dN N = aN 1 − , (4.1a) dt K dK = −bN, (4.1b) dt where a is the population growth rate and b the consumption rate of the resource by the population. In their work, they didn’t include the reproduction rate of the resource, as it was relatively small compared with the consumption rate.

44 System (4.1) can be rescaled using a set of substitutions n = bN, k = bK, τ = at, such that

dn n = n 1 − , (4.2a) dτ k dk = −ωn, (4.2b) dτ where ω = b/a is the ratio of the consumption rate to the population growth rate.

Figure 4.1 shows the eﬀect of two values of ω with the initial condition (n0, k0) = (0.1, 2.0). For ω = 0.1, the consumption rate is one tenth of the growth rate, so that the resource carrying capacity, k, is slowly decreasing due to consumption by the population (Figure 4.1a). As a result, n grows gradually and reaches a maximum, before decreasing due to overpopulation and resource exhaustion and, eventually, the population is driven to extinction after all the resources are consumed over a long period of time (τ = 60).

For the case ω = 0.5, which represents a consumption rate that is half the growth rate, the resource is consumed at a faster rate, which exhausts the available resource in a shorter period of time (τ = 15). Notice in Figure 4.1b that the maximum population reached is lower than that in the case of ω = 0.1. In a diﬀerent scenario, in which the population and the carrying capacity start at the same values (n0, k0) = (0.1, 0.1), n and k decrease as shown in Figure 4.2.

Another model that links the dependence of a population on its carrying capacity was proposed by Thornley and France [2005], later modiﬁed by Thornley et al. [2007], and can be written as

dN N = aN 1 − , (4.3a) dt K dK = −b(K − N), (4.3b) dt where the positive constants a and b are the population growth rate and environmental development rate, respectively. As parameter b is positive, it describes the natural loss of the carrying capacity (for N < K), but at the same time allowing the population, N, to contribute to the carrying capacity. It should be noticed that in Equation (4.3b), there is no growth term in K except for the contribution of the population N. An important feature of this model is that the asymptotic value of K is not known a priori as it is in the logistic equation (1.1), where K is a ﬁxed parameter. However, the model has inﬁnitely many stationary solutions along the line N = K. Therefore the initial condition N(0) = K(0) will result in no dynamical behaviour, which is unfortunate for the case where the population

45 2

1.5

1 ω = 0.1 n, k

0.5

0 0 10 20 30 40 50 60 τ (a)

1.5

1 ω = 0.5 n, k

0.5

0 0 5 10 15 τ (b)

Figure 4.1: Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.2): (a) ω = 0.1; and (b) ω = 0.5, with (n0, k0) = (0.1, 2.0).

46 0.1

0.08

0.06 ω = 0.1 n, k 0.04

0.02

0 0 10 20 30 40 50 60 τ (a)

0.1

0.08

0.06 ω = 0.5 n, k 0.04

0.02

0 0 5 10 15 τ (b)

Figure 4.2: Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.2): (a) ω = 0.1; and (b) ω = 0.5, with n0 = k0 = 0.1.

47 is initially in equilibrium with its environment. Eﬀectively, within this model the choice of the initial conditions determines the asymptotic carrying capacity.

System (4.3) can be rescaled using a set of substitutions n = bN, k = bK, τ = at, such that

dn n = n 1 − , (4.4a) dτ k dk = −γ(k − n), (4.4b) dτ where γ = b/a is a ratio of the environmental development rate to the population growth rate. System (4.4) has steady-states at (n∗, k∗) = (0, 0) and n∗ = k∗. The origin is always a saddle point, since the eigenvalues of the Jacobian matrix ∗ ∗ are λ1,2 = 1, −γ, and n = k is a stable critical line with eigenvalues of λ1,2 = 0, −(γ + 1). With zero and negative real eigenvalues, all trajectories will approach the critical line for a given set of initial conditions. The system shows that, by providing speciﬁc values for the initial conditions, the solution will always move to another stable solution (see curves (i) and (ii) in Figure 4.3a).

When the initial conditions start at the same value, for example n0 = k0 = 0.1, as no growth occurs for either the population or its environment, they remain the same for all time. This can be seen in curve (iii) in Figure 4.3a. This system requires the initial value of the carrying capacity to be larger than that of the initial population in order to observe the population growth. If the initial carrying capacity value is lower than the initial population, the population decreases toward the carrying capacity. Figure 4.3b demonstrates the eﬀect of diﬀerent parameter values of γ for

ﬁxed initial conditions, with k0 > n0.

4.3 Environmental carrying capacity as a simple biotic resource

Unlike the models proposed by Huzimura and Matsuyama [1999] and Thornley and France [2005], we study a modiﬁed coupled logistic carrying capacity model that incorporates dependence of a population and its environment using a nonlinear interaction term [Safuan et al., 2012]. This model assumes a standard logistic equation for the population, N, with a constant growth rate a. The carrying capacity, K, is assumed to change at a rate proportional to the present value of the resource, with proportionality constant, b, the development rate.

48 0.4

0.35

0.3 γ = 0.5 (i) 0.25

n, k 0.2 (ii) 0.15 (iii) 0.1

0.05

0 0 2 4 6 8 10 τ

(a)

γ=0.1 1.5

n, k 1

γ=0.5

0.5 γ=2.0

0 0 2 4 6 8 10 τ

(b)

Figure 4.3: Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system 4.4 for diﬀerent initial values: (a) γ = 0.5, n0 = 0.1 with (i) k0 = 0.4, (ii) k0 = 0.2, (iii) k0 = 0.1; (b) γ = 0.1, 0.5, 2.0, with (n0, k0) = (0.1, 2.0).

49 The model (4.3) is modiﬁed to

dN N = aN 1 − , (4.5a) dt K dK = bK − cKN. (4.5b) dt

Here, the carrying capacity is assumed to be directly proportional to the resource to make a comparison with the model studied in Thornley and France [2005]. The population depletes the carrying capacity through an interaction term with the rate, c. Instead of the population independently contributing to the development of the carrying capacity, in the proposed model the dependence of the carrying capacity on population is through an interaction with the population.

Meyer [1994] and Meyer and Ausubel [1999] introduced the carrying capacity func- 2 2 tion as dK/dt = αK − αK /Kα in their models. The nonlinear term K ensures the carrying capacity is self–limiting. If the population has an inﬂuence on the carrying capacity or vice versa, this can be viewed as an interspeciﬁc relationship between the population and its resource via the carrying capacity. Thus, in the proposed model, the nonlinear term KN in Equation (4.5b) represents this interaction. Equation (4.5b) can also be written as dK/dt = bK(1 − αN), where α = c/b. By introducing the non-dimensional parameters

n = αN, k = αK and τ = at, the dimensionless form of model (4.5) is

dn n = n 1 − , (4.6a) dτ k dk = µk(1 − n), (4.6b) dτ where µ = b/a is the ratio of the carrying capacity development rate to the population growth rate. By equating system (4.6) to zero, two steady-states are obtained at (n∗, k∗) = (0, 0) and (n∗, k∗) = (1, 1). To examine the stability properties in the neighbourhood of the critical points, the Jacobian matrix is used:

 2n n2  1 −  k k2  J =   .   −µk µ(1 − n)

50 At (0, 0), the Jacobian has the eigenvalues λ1,2 = 1, µ, and the origin is always an unstable node as a, b > 0. For the second critical point, the Jacobian eigenvalues are p λ1,2 = (−1 ± 1 − 4µ)/2.

Therefore, point (1, 1) is a stable spiral if µ > 1/4, or a stable node otherwise.

System (4.4) is autonomous and can be solved exactly. However, although only numerical solutions can be obtained for system (4.6) due to its nonlinearity, it has the advantage that there are stable stationary solutions for any initial conditions (n, k) = (n0, k0) so that any combination of n0 and k0 will lead to the same equilibrium solution. Consider the ﬁrst case, in which the population and its carrying capacity (resource) are the same, n0 = k0, which is a problematic initial condition for system (4.4). Figure 4.4 shows the behaviour of solutions of n and k. When µ > 1/4, oscillations between n and k are expected, as demonstrated by the linear stability analysis. Overshoots for n and k can be seen, and are more pronounced if the value of µ is much greater than 1/4.

For µ = 2, b = 2a, the development rate is double the growth rate and as a result, the carrying capacity shows an initial rapid increase while there is slower growth in the population. This increase in the carrying capacity, indicative of an improved environment, is conducive results in faster population growth. When the carrying capacity decreases, due to the interaction with the population, the population will eventually have to fall. Moreover, an overpopulated environment exhausts available resources, thus lowering the carrying capacity. With a population decrease, the resource becomes richer once more, which allows the population to slowly grow back; the system therefore exhibits repeated (damped) oscillations until the population and carrying capacity reach stable values.

Figure 4.4b illustrates that, if the population growth rate is equal to its carrying capacity development rate (µ = 1, b = a), then the diﬀerence between n and k in the early stage of growth is smaller compared to when µ = 2. A small diﬀerence can be seen when µ = 0.5 in Figure 4.4c, with the carrying capacity is growing slightly faster than that of the population. As µ decreases (µ = 0.2 < 1/4), the population and the carrying capacity gradually increase to the equilibrium level (Figure 4.4d). The associated phase-plane plots are shown in Figure 4.5.

For the second case, in which an initially rich environment is assumed, it can be seen that a huge increase in the carrying capacity is followed by an increase in the population, as shown in Figure 4.6. Once the carrying capacity starts to decrease, the population density also starts to decline. It can be observed that the variation in the carrying capacity is very large compared with that in the population growth.

51 For example, for µ = 2, the carrying capacity peaks at about 36, before decaying (Figure 4.6a), but, when µ = 0.2, the peak is about 12 times smaller than for µ = 2 (Figure 4.6d).

7 2

6 1.5 5 µ 4 µ=2.0 =1.0 1 n, k 3 n, k

2 0.5 1

0 0 0 5 10 15 0 5 10 15 20 τ τ (a) (b)

2 1.5

µ=0.5 µ=0.2 1.5 1

1 n, k n, k

0.5 0.5

0 0 0 5 10 15 20 0 10 20 30 40 τ τ (c) (d)

Figure 4.4: Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.6): (a) µ = 2.0; (b) µ = 1.0; (c) µ = 0.5; (d) µ = 0.2, with n0 = k0 = 0.1 (note that diﬀerent scales are used).

52 7 2

1.8 6 1.6

5 1.4

1.2 4 ) ) τ τ 1 k( k( 3 0.8

2 0.6

0.4 1 0.2

0 0 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n(τ) n(τ) (a) (b)

1.1 1

1 0.9

0.9 0.8 0.8 0.7 0.7 0.6 ) ) τ 0.6 τ k( k( 0.5 0.5 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n(τ) n(τ) (c) (d)

Figure 4.5: Corresponding trajectory plots in the phase plane for Figure 4.4: (a) µ = 2.0; (b) µ = 1.0; (c) µ = 0.5; (d) µ = 0.2. In all cases n0 = 0.1 and k0 = 0.1 (note that diﬀerent scales are used).

53 40 10

35 8 30 µ 25 =2.0 6 µ=1.0 20 n, k n, k 15 4

10 2 5

0 0 0 2 4 6 8 10 0 5 10 15 τ τ (a) (b)

5 3

2.5 4

µ=0.5 2 µ=0.2 3 1.5 n, k n, k 2 1

1 0.5

0 0 0 5 10 15 20 0 5 10 15 20 τ τ (c) (d)

Figure 4.6: Logistic growth curve, n, (solid line) and its carrying capacity, k, (dashed line) in system (4.6): (a) µ = 2.0; (b) µ = 1.0; (c) µ = 0.5; (d) µ = 0.2, with n0 = 0.1, k0 = 2.0 (note that diﬀerent scales are used).

54 4.4 Conclusion

The concept of the carrying capacity as an environment’s maximal load defines the limited growth of a population. However, the difficulty of determining its exact form often means that certain assumptions have to be made. The contribution of environmental enrichment to the carrying capacity, due to an increase in a biotic resource, is proposed in the model (4.5). The coupled model encompasses the direct dependence of the growth equations on carrying capacity. Instead of assuming the carrying capacity is an independent parameter, the coupled model provides a physical idea of the inter-relationship that coexists between the environment and its population. By connecting them, the effect of a variable carrying capacity on the population’s growth can be understood.

The results show that an increase in the carrying capacity encourages population growth. The main parameters in system (4.5) are the population growth rate, a, and the development rate, b, of the carrying capacity. Oscillations occur in the growth curves if the carrying capacity development rate is more than a quarter of the population growth rate.

In the next chapter, we extend this model to consider two-species populations using a similar approach in which the environmental carrying capacity is represented as a biotic resource. It is expected that more dynamical properties will be found and more interesting features will be observed.

Chapter 5

Two-species population-growth models with variable carrying capacities

In this chapter, we study a food-chain model in which the predator and prey share the same base resource. The limited biotic resource is modelled through varying carrying capacities of the two populations; their presence has the eﬀect of increasing and/or decreasing the carrying capacity by consuming the resource. With the analogy used in the previous chapter, the enrichment parameter of the resource sets the primary bifurcation parameter in all models in this chapter. Changing the values of this parameter generates diﬀerent kinds of dynamics in the system’s behaviour. We investigate the steady-state solutions and their corresponding eigenvalues. The results can be displayed as bifurcation diagrams, which are useful in visualising the steady-state solution branches that exist in the system. Some of the results are compared to those found in the literature. Part of the analysis presented in this chapter appears in Safuan et al. [2013b] and Safuan et al. [2014b].

5.1 Predator-prey models

Single-population dynamics provide insight in the inter-relationship that coexists between the population and its carrying capacity. The classic Lotka-Volterra (LV) model is written as [Murray, 1989, Berryman, 1992]

dN = aN − bNP, (5.1a) dt dP = cNP − dP, (5.1b) dt where N is the number of prey and P is the number of predator. In the absence of predator, the prey grows exponentially, but with the absence of prey, the predator

57 will not survive. To overcome the unlimited growth of prey, a ﬁxed carrying capacity, K, is introduced to Equation (5.1a) [Vandermeer and Goldberg, 2013] to give

dN N = aN 1 − − bNP, (5.2a) dt K dP = cNP − dP. (5.2b) dt

Oscillations in predator and prey populations are common in these types of models. This behaviour is caused by the nonlinear interaction between the predator and its prey. However, the ﬁxed carrying capacities used in these models do not allow for varying environmental factors. Thus, many studies have been done to provide more realistic predator-prey models. These models are non-autonomous since they incorporated time-varying carrying capacities [Ahmad, 1987, 1993, Peng and Chen, 1994, Oca and Zeeman, 1996]. Besides non-autonomous predator-prey models, autonomous systems have also been studied to investigate the inter-relation between these populations. The next section will discuss some modiﬁed predator-prey models studied by others.

5.2 Modiﬁed predator-prey models

Consider a general type of predator-prey model (Figure 5.1a) in which the prey,

X, and the predator, Y , are assumed to grow logistically with growth rates r1 and r2, and have ﬁxed carrying capacities K1 and K2, respectively [Hoyle and Bowers, 2007],

dX X = r1X 1 − − aF (X)Y, (5.3a) dt K1 dY Y = r2Y 1 − + bF (X)Y. (5.3b) dt K2

The functional response term F (X) describes the per capita intake of population X by population Y . The commonly used response functions are the Holling Types I, II, and III [Collings, 1997, Seo and Kot, 2008]. Type I is linear in X; Type II and Type III correspond to the Monod and sigmoid functional response, respectively. Generally, Types II and III have saturation of the number of prey consumed by the predator.

58 predator predator

prey

prey resource

(a) (b)

Figure 5.1: (a) Predator-prey system; (b) intraguild predation [Polis et al., 1989].

Another predator-prey model commonly studied is the Leslie-Gower (LG) model with a linear response function [Collings, 1997],

dX X = r X 1 − − aXY, (5.4a) dt 1 K dY Y = r Y 1 − . (5.4b) dt 2 bX

Introduced by Leslie [1958] and Leslie and Gower [1960], the model assumed that the carrying capacity of the predator Y is proportional to the resource (prey X), and that the carrying capacity of the prey was limited by a ﬁxed value K. The parameter b in Equations (5.3b) and (5.4b) is not the same, and so it has diﬀerent units.

This model is a type of ratio-dependent predator-prey model whose positive equilibrium solution is globally stable [Korobeinikov, 2001, Wang et al., 2003, Seo and Kot, 2008]. On substituting a Holling Type II response function, F (X) = X/(a + X), into system (5.4), it becomes a ratio-dependent Holling-Tanner model and possesses rich dynamical behaviour, such as limit cycles [Tanner, 1975, Hsu and Huang, 1995, Saez and Olivares, 1999, Gasull et al., 1997]. These models can be extended beyond the two-species food-chain model [Korobeinikov and Lee, 2009, Gakkhar and Naji, 2003]. Collings [1997] applied several functional forms to system (5.4) to investigate changes in the dynamical behaviour of spider-mite (predator-prey) populations.

Seo and Kot [2008] compared a LG-type model with another predator-prey model, and found both models exhibit limit cycle solutions. A LG model with the eﬀect of prey refuge has been considered by Chen et al. [2009]. They found that, if the amount of prey refuge is increased, it can increase or decrease predator and

59 prey densities. Chen and Chen [2009] investigated the LG model by incorporating feedback controls, and found that the feedback control variables have no eﬀects on the global stability of the system, but only alter the location of the coexistence equilibrium.

Another type of model, similar to the LG model, is the Basener-Ross (BR) model, that uses a simple response function, F (X) = 1 [Basener and Ross, 2005]. The model is applied to the human population on Easter Island, in which the carrying capacity was represented as a resource such as food or trees. The term aY in their model represented a harvesting term proportional to Y . Bologna and Flores [2008] also applied the LG model to the same demographic on Easter Island. In the 2 absence of the crowding term r1X /K in Equation (5.4a), the model reduces to the models studied by Korobeinikov [2001] and Safuan et al. [2012].

Intraguild predation, in which a predator and prey share/exploit the same environmental resource, consists of three diﬀerential equations to describe the tritrophic interactions [Polis et al., 1989, Polis and Holt, 1992, Holt and Polis, 1997]. A theoretical review of intraguild predation is presented by Janssen et al. [2006]. For example, a system of three populations with intraguild predation consists of a prey/intermediate consumer, a predator and a resource (see Figure 5.1b; speciﬁc examples are shown in Figure 5.2).

(a) (b)

Figure 5.2: Examples of intraguild predation; (a) ﬁsh populations [Mylius et al., 2001]; (b) spider-mite populations [Janssen et al., 2006].

60 The abundance of resources in intraguild models is known to have a signiﬁcant impact on the predator and prey dynamics [Diehl and Feissel, 2001, Mylius et al., 2001]. For two microorganisms feeding on the same resource (a bacterium), a simple linear response function was used by Diehl and Feissel [2001]. Using the carrying capacity as the enrichment parameter, they found that at a low level of enrichment, only the intraguild prey persisted in the system. Increased enrichment to an intermediate level, the three species coexisted. However, at a high enrichment level, the system only permitted the survival of the predator; the prey suﬀered and became extinct.

The main objective of this chapter is to use a type of ratio-dependent model to investigate the dynamics of a predator and prey that share the same biotic resource. The carrying capacities of both predator and prey depend on the amount of the resources. The eﬀects of resource enrichment at low, intermediate and high levels are investigated. Some dynamics of the model are predicted to have a similar type of dynamics as the intraguild predation model discussed above, in which the three populations persist at an intermediate resource level, but the prey population becomes extinct at a high level.

5.3 Environmental carrying capacity as biotic resource enrichment

To study the predator-prey system, a model based on system (5.3) with a linear response function F (X) = X is proposed. The proposed model is an extension of the models by Leslie [1958], Basener and Ross [2005] and Safuan et al. [2012],

dX X = r X 1 − − aXY, (5.5a) dt 1 pK dY Y = r Y 1 − + bXY, (5.5b) dt 2 qK dK = K (c − dX − eY ) . (5.5c) dt

The two populations, X (prey) and Y (predator), grow logistically with growth rates r1 and r2, respectively, and their growth is limited by the availability of a biotic resource, K. We introduce Equation (5.5c) to represent the biotic resource. The terms pK and qK set the environmental carrying capacity for populations X and Y , respectively. If p + q = 1, then pK + qK is the total carrying capacity. If p > q, it means that species X has access to a higher proportion of the biotic resource and can bear to be more crowded than species Y . Equal carrying capacity (p = q = 1/2) was discussed by Lacitignola and Tebaldi [2003]. In Equation (5.5c), the biotic resource grows with developmental rate, c. The rate of uptake of

61 the resource by X is dKX and by Y is eKY , where d and e are constants. All parameters are greater than or equal to zero.

This scenario may be thought of as a three-species ratio-dependent food-chain model [Wang and Pang, 2008, Gakkhar and Naji, 2003]. Since we may think of K as the instantaneous carrying capacity of a replenishing environment that is also being depleted by the presence of X and Y , increases or decreases in K could aﬀect one or both of X and Y . Our aim is to understand the dynamical behaviour that system (5.5) is capable of displaying.

5.3.1 Equilibrium and stability analysis

In this section, three possible cases are discussed. The first case is fixing the carrying capacity as a constant. This means K0(t) = 0 in Equation (5.5c). The second case is the absence of inter-specific interactions, a = b = 0 in Equations (5.5a) and (5.5b). The third case is the analysis of the full model (5.5).

5.3.2 Constant carrying capacity: c = d = e = 0

First, a simple case where the carrying capacity K is assumed to be constant. Thus, only Equations (5.5a) and (5.5b) are considered. This model is similar to those in Hoyle and Bowers [2007], but for completeness its equilibria and the corresponding stability are presented.

Using the transformations x = X/pK, y = Y/qK and τ = r1t, the non-dimensionalised system is

x0 = x (1 − x) − αxy,ˆ (5.6a)

y0 = βyˆ (1 − y) +γxy, ˆ (5.6b) where aqK r bpK αˆ = , βˆ = 2 , γˆ = , r1 r1 r1 and the derivatives of x and y are with respect to τ. There are four equilibria in ˆ ˆ the (x, y) plane that satisfy system (5.6). Two are P1 = (0, 0) and P2 = (1, 0), with eigenvalues

E = {1, βˆ}, Pˆ1 E = {−1, βˆ +γ ˆ}, Pˆ2 respectively. From the expressions for E and E , it is obvious that equilibrium Pˆ1 Pˆ2 ˆ ˆ P1 (the origin) and equilibrium P2 (prey only) are unstable. The remaining two

62 equilibria are ˆ ˆ ! ˆ ˆ β(1 − αˆ) β +γ ˆ P3 = (0, 1) and P4 = , . (5.7) βˆ +α ˆγˆ βˆ +α ˆγˆ ˆ The equilibrium P3 (predator only) has eigenvalues

E = {1 − α,ˆ −βˆ}, Pˆ3

ˆ and the threshold stability condition for equilibrium P3 isα ˆ = 1. Ifα ˆ > 1, then ˆ ˆ P3 is stable and ifα ˆ < 1, then the non-trivial equilibrium P4 (predator and prey coexist) becomes stable, as discussed next. ˆ The coexistence equilibrium P4 has the characteristic equation

2 λ − a1 λ + a2 = 0, (5.8) where λ denotes an eigenvalue, and βˆ αˆ − (1 + βˆ +γ ˆ) a1 = , βˆ +α ˆγˆ βˆ(βˆ +γ ˆ)(1 − αˆ) a2 = . βˆ +α ˆγˆ

The positive quadrant, x > 0 and y > 0, of system (5.6) is invariant. Using Dulac’s Test, we deﬁne a function ρ(x, y) = 1/xy which admits continuous ﬁrst partial derivatives [Jordan and Smith, 1987]. Denoting

f(x, y) = x(1 − x) − αxy,ˆ

g(x, y) = βyˆ (1 − y) +γxy, ˆ and then applying Dulac’s Test to system (5.6) gives

1 ρ(x, y)f(x, y) = (1 − x − αyˆ ), y 1 ρ(x, y)g(x, y) = (βˆ − βyˆ +γx ˆ ). x

63 Therefore,

∂(ρ(x, y)f(x, y)) 1 = − , ∂x y ∂(ρ(x, y)g(x, y)) βˆ = − , ∂y x which yields ! ∂(ρ(x, y)f(x, y)) ∂(ρ(x, y)g(x, y)) 1 βˆ + = − + < 0. ∂x ∂y y x

Hence, there are no periodic solutions in the positive quadrant of system (5.6).

5.3.3 Absence of inter-speciﬁc interactions: a = b = 0

Consider the model (5.5) in the absence of inter-speciﬁc interactions, a = b = 0. A similar model was considered by Sprott [2011] in his work on the extended BR model. Using the non-dimensional variables x = eX/r1, y = fY/r1, k = epK/r1 and τ = r1t, system (5.5) can be reduced to x x0 = x 1 − , (5.9a) k ! βy˜ y0 =αy ˜ 1 − , (5.9b) k

k0 = k (˜γ − x − y) , (5.9c) where r dp c α˜ = 2 , β˜ = , γ˜ = . r1 eq r1

There are three equilibria in the (x, y, k) phase-space of system (5.9). The ﬁrst two ˜ ˜ ˜ are P1 = (0, γ,˜ βγ˜) and P2 = (˜γ, 0, γ˜), with eigenvalues

E = {1, −α/˜ 2 ± pα˜(˜α − 4˜γ)}, P˜1 E = {α,˜ −1/2 ± p1 − 4˜γ}, P˜2 respectively. Thus, both equilibria are always unstable. The coexistence equilibrium

˜ ˜ ! ˜ βγ˜ γ˜ βγ˜ P3 = , , , (5.10) 1 + β˜ 1 + β˜ 1 + β˜

64 has the characteristic equation

3 2 λ + b1λ + b2λ + b3 = 0, (5.11) where

b1 = 1 +α, ˜

γ˜(˜α + β˜) b2 =α ˜ + , 1 + β˜

b3 =α ˜γ.˜

Using the Routh-Hurwitz criterion [Walter and Peterson, 2010], all roots have negative real parts if and only if

b1 > 0, b3 > 0, b1b2 > b3. (5.12)

The ﬁrst two criteria are obviously satisﬁed, and on substitution of b1, b2, b3 into the third, ! γ˜(˜α + β˜) (1 +α ˜) α˜ + > α˜γ,˜ 1 + β˜

˜ it is satisﬁed as well. Hence, the equilibrium P3 is asymptotically stable.

5.3.4 Full model: a, b, c, d, e 6= 0

In this section, the full model (5.5) is studied. Introducing nondimensional variables x = bX/r1, y = aY/r1, k = bpK/r1, τ = r1t, system (5.5) becomes [Safuan et al., 2013b]

x x0 = x 1 − − xy, (5.13a) k βy y0 = αy 1 − + xy, (5.13b) k

k0 = k (γ − δx − y) , (5.13c) where r bp c d e α = 2 , β = , γ = , δ = , = . r1 aq r1 b a

65 In system (5.13):

α is the ratio of the growth rates of y to x; β is the ratio of the feeding rate of y to the death rate of x in proportion to their carrying capacities; γ is the ratio of the developmental rate of the carrying capacity to the growth rate of x; δ is the ratio of the saturation rate of the carrying capacity to the feeding rate of y; and is the ratio of the consumption rate of the resource by y to the death rate of x by y.

The stability is determined from the eigenvalues of the Jacobian matrix at an equilibrium solution (x∗, y∗, k∗)

 2x∗ x∗2  1 − − y∗ −x∗  k∗ k∗2        J(x∗, y∗, k∗) =  2αβy∗ αβy∗2  .  y∗ α − + x∗   ∗ ∗2   k k      −k∗ −φk∗ γ − δx∗ − y∗

Equating each equation of system (5.13) to zero leads to the equilibria of the system.

One of the equilibria is P1 = (γ/δ, 0, γ/δ), with eigenvalues ( ) γ 1 p(1 − 4γ) E = α + , − ± . P1 δ 2 2

From the above expression, equilibrium P1 is unstable, and will no longer be considered in the analysis. The other equilibria are

ˆ ˆ ˆ ˆ ! γ βγ αk(β − k) k(α + k) ˆ P2 = 0, , and P3 = , , k , αβ + kˆ2 αβ + kˆ2 where kˆ is a root of the quadratic equation

ˆ2 ˆ c1k + c2k + c3 = 0, (5.14)

66 with

c1 = (γ − )/α + δ,

c2 = −(βδ + ),

c3 = βγ.

As the parameter values vary, there exists one equilibrium point for each of P1 and

P2, whereas there can be either one or two equilibria for P3, based on the quadratic equation (5.14). Ecologically, the equilibrium point P1 represents a steady-state that is free from predator and P2 is free from prey. The coexistence of predator and prey is represented by the steady-state solution branch P3.

Since c3 > 0, Equation (5.14) will have strictly one positive root if c1 = 0, one positive root and one negative root if c1 < 0, two positive roots if c1 > 0, or two 2 complex roots if c2 − 4c1c3 < 0.

Equilibrium P2 (predator and resource only) has the characteristic equation

γ α(γ − − γ) αγ(γ − ) λ3 + α − 1 + λ2 + λ + = 0, (5.15) with eigenvalues ( ) γ α pα(α − 4γ) E = 1 − , − ± . P2 2 2

The threshold condition from the ﬁrst eigenvalue, λ1 = 1 − γ/ is

γ = . (5.16)

The condition (5.16) results in a transcritical bifurcation (tb), and is mentioned in

Chapter 2. The transition of the real part of eigenvalue λ1 through zero will result in diﬀerent stability behaviour for equilibrium P2. The equilibrium P2 is stable if the eigenvalue λ1 is negative (γ > ). If γ < , however, equilibrium P2 becomes unstable. For this case (γ < ), the non-trivial steady-state P3 (coexistence) becomes stable. This “switch” or change results in a bifurcation, and γ (enrichment parameter) of the biotic resource (carrying capacity) is treated as the primary bifurcation parameter for the model (5.13), and all other parameters are candidates as secondary bifurcation parameters. Using this parameter for system (5.13) results in changing the ratio of c and r1 in the original model (5.5). However, changing r1 also results in changing γ in model (5.13). The actual bifurcation parameter is the developmental rate c of the unscaled model (5.5), which results in changing γ.

67 In our investigation, by adjusting the enrichment of the resource, the size of x (prey) and y (predator) can be manipulated. If an ecosystem requires the persistence of all populations, controlling the biological factor γ is crucial for future conservation. If population x, however, is considered as being damaging to the environment (e.g. a pest population), it is important to establish a threshold level for γ beyond which the deleterious eﬀects of x on the environment is reduced or eliminated.

5.3.5 Hopf bifurcation analysis

Recall that a Hopf bifurcation occurs when a pair of complex eigenvalues crosses the imaginary axis. It is a local bifurcation that results in a periodic solution branch emanating from an equilibrium point. The eigenvalue EP2 corresponding to the steady-state P2 (prey extinction) can not be purely imaginary. This implies that there is no occurrence of Hopf bifurcation in the steady-state P2. However, a Hopf bifurcation can exist in the coexistence steady-state P3, which will be discussed later in this section.

The numerical software packages XPPAUT [Ermentrout, 2010] and MATLAB R [MathWorks, 2008] are used to obtain the steady-state curves. Setting parameters β = 0.5, δ = 2 and = 2.5, a bifurcation diagram for varying α and γ is obtained, as shown in Figure 5.3. The ﬁgure shows the curve of Hopf points, which is the locus of origin of periodic solutions.

In our numerical example, the maximum point of the Hopf locus (sometimes referred to as the h1 degenerate Hopf bifurcation) is found at αdh ≈ 0.0052. It corresponds to the locus at which two Hopf bifurcation points are created (or destroyed) as they separate (or merge) from (with) each other. Hence, for values of α less than αdh, periodic solutions are possible. Otherwise, by choosing α > αdh, no limit cycle behaviour is possible in the system.

First, a case in which no periodic solution appears in the system is chosen. Fixing α = 0.3, for example, physically represents a growth rate of species y that is 0.3 times than of population x. Steady-state diagrams of the system variables (5.13) against the enrichment parameter of the resource γ are shown in Figure 5.4. The following representations are utilised: solid lines are stable steady-states; dashed lines are unstable steady-state solutions. The two steady-state branches denoted by P2 and P3 are obtained from the stability analysis in Section 5.3.4.

As γ increases in value, the equilibrium values of x, y and k also increase, as shown in Figure 5.4. The equilibrium densities of x and y increase, since the biotic resource k increases, thus making it more conducive for both species to grow. This is represented by the solid curves P3 (coexistence) in Figure 5.4a and Figure 5.4b.

68 0.01 Hopf locus

0.008 No periodic solutions

0.006 α DH α

0.004 Periodic solutions 0.002

0 1.6 1.8 2 2.2 2.4 2.6 γ

Figure 5.3: Bifurcation diagram showing the Hopf locus with β = 0.5, δ = 2, and = 2.5 (αdh – degenerate Hopf point).

At an intermediate level of γ, both k and y continue to increase, but the equilibrium density of x reaches a maximum, before decreasing due to the predation by y.

From condition (5.16), a transcritical bifurcation occurs at γtb = 2.5. As γ increases in magnitude, the branch P3 is stable below the threshold value, γ < γtb, whereas branch P2 becomes stable for γ > γtb. At a high value of γ, both k and y continue to grow linearly, following the steady-state branch P2; species x, however, dies out, leaving only the dominant population y to persist in the rich biotic environment. The predator y switches to a specialist predator that persists solely on the abundance of the biotic resource k [Diehl and Feissel, 2000].

These results show that the survival of prey population x is determined by the bifurcation parameter γ. If x is considered as a harmful (or not desirable) population, a rich resource environment will guarantee the elimination of it. In contrast, adjusting γ to an intermediate or low level ensures the persistence of all species.

Some of these behaviours, shown in Figure 5.4, can be linked to an intraguild model, discussed in Section 5.2. Most intraguild models assume that due to enrichment, the equilibrium density of the resource remains at the same level but at the same time allowing the predator population to keep increasing [Mylius et al., 2001, Hin et al., 2011]. The model (5.13) predicts that by increasing the enrichment parameter (γ → ∞), the equilibrium density of the predator population y also increases, following the increasing resource k. Further, from the system (5.13), if γ exceeds δ and , it means that the available resource is excessive compared to the amount that the prey and predator could possibly consume.

69 0.1 P 0.08 3

0.06

x 0.04

0.02 P TB→ P 2 2 0 P −0.02 3 0 1 2 3 γ (a)

1.2 P 3 P

1 → P TB 0.8 2

y 0.6 P 0.4 3

0.2

0 0 1 2 3 γ (b)

1.5

P 1 3 k

0.5 P P 2 → 2 TB

P 3 0 0 1 2 3 γ (c)

Figure 5.4: Steady-state diagrams of the system (5.13) against the enrichment parameter, γ, with α = 0.3, β = 0.5, δ = 2, = 2.5 for: (a) x; (b) y; (c) k. P2 and P3 – steady-state solution branches, tb – transcritical bifurcation. Solid line – stable steady-state; dashed line – unstable steady-state.

70 0.2 1.4

1.2

1 y

x 0.1 0.8

0.6

0 0.4 0 10 20 30 0 10 20 30 τ τ (a) (b)

1.5

1 1 k k 0.5 0.5

0 0.2 0 1.5 1 0.1 0 10 20 30 y 0.5 0 x τ 0 (c) (d)

Figure 5.5: Time series plots of system (5.13) with parameters as in Figure 5.4 with γ = 2.12 for: (a) x; (b) y; (c) k. Initial conditions (x, y, k) = (0.1, 0.4, 0.5). (d) Corresponding phase-plot.

71 0.3 1.4

1.2 0.2 1 y x 0.8 0.1 0.6

0 0.4 0 100 200 300 400 500 0 100 200 300 400 500 τ τ (a) (b)

2.5

2.5 2 2 1.5 1.5 k k 1 1

0.5 0.5 0 0.4 0 1.5 1 0.2 0 100 200 300 400 500 y 0.5 0 x τ 0 (c) (d)

Figure 5.6: Time series plots of system (5.13) with parameters as in Figure 5.5 with α = 0.013, γ = 2.12 for: (a) x; (b) y; (c) k. Initial conditions (x, y, k) = (0.1, 0.4, 0.5). (d) Corresponding phase-plot.

72 To illustrate the coexistence equilibrium P3, Figure 5.5 shows the time series solution of the system (5.13) with the parameters used in Figure 5.4 and at a speciﬁc value of γ = 2.12. The phase-space plot is shown in Figure 5.5d. Suﬃcient enrichment of the resource enables the predator and prey to increase, exhibiting (damped) oscillations until the populations reach their stable steady-state values. Keeping all parameters the same and decreasing α (α = 0.013), more pronounced damped oscillations can be seen, as illustrated in Figure 5.6 and in the phase-plot in Figure 5.6d.

Now if we choose α = 0.003 < αdh, two Hopf points are obtained (hb: γ ≈ 1.93 and hb: γ ≈ 2.39). Physically, decreasing α has the eﬀect of decreasing the growth rate of population y with respect to population x; α = 0.003 implies the growth rate of population y is 0.003 times that of population x. Again the steady- state diagrams of x, y and k against γ are generated, and these are presented in Figure 5.7. The behaviour of the steady-states is relatively similar to Figure 5.4, except there are now regions where periodic behaviour occurs between hb and hb. Closed circles represent stable periodic solutions and open circles represent unstable periodic solutions. A subcritical Hopf bifurcation exists for the steady- state P3, which exhibits unstable periodic solutions emanating from the Hopf point hb; stable periodic solutions are found to originate from the Hopf point hb.

From these figures, low resource enrichment enables x, y and k to persist. At an intermediate level, however, destabilisation of the system occurs, which allows the populations to switch to a limit cycle. Destabilisation effects can occur in predator- prey models with increasing enrichment of the resources. This behaviour is known as the “paradox of enrichment’ [Rosenzweig, 1971]. The model (5.13) shows that varying the carrying capacity through resource enrichment at an intermediate level gives rise to this destabilisation effect. The effect of further enrichment causes the system to switch back to a steady-coexistence equilibrium, P3, for γhb < γ < γtb, and a prey-extinction equilibrium, P2, for γ > γtb.

Figure 5.8 illustrates the time series solution of the periodic behaviour for γ = 2.12, that is within the periodic behaviour region (hb < γ < hb). Persistent oscillations can be seen in x, y and k. These oscillations are associated with a limit cycle that emanates from the supercritical Hopf point, hb. Compared to the damped oscillation plots in Figure 5.5 and Figure 5.6, these oscillations are self-generated with speciﬁc amplitudes. The amplitude of k is larger than those of y and x for a suﬃciently rich resource. The corresponding phase-space plot is shown in Figure 5.8d. Appendix A.1 explains the numerical computation of the steady-states for this example.

73 0.12

0.1

0.08

x 0.06

0.04 HB1 HB2 P TB 0.02 3 P P 0 2 2 0 1 2 3 γ (a)

1.5 HB2

P 1 2 HB1

y TB P 2 0.5 P 3

0 0 1 2 3 γ (b)

P 4 3

3 k

2 HB2 HB1

1 P

2 P

P 3

2 ← TB 0 0 1 2 3 γ (c)

Figure 5.7: Steady-state diagrams of the system (5.13) against the enrichment parameter, γ, with Hopf bifurcation with α = 0.003, β = 0.5, δ = 2, = 2.5 for: (a) x; (b) y; (c) k. hb and hb – Hopf points. Closed circle – stable periodic solutions; open circle – unstable periodic solutions.

74 Figure 5.9 shows the h1 degeneracy in the (, α)–parameter plane. There is a critical value of (resource uptake rate by the predator). If > 1.21, there is an h1 degeneracy; if α is below the h1 degeneracy, the steady-state diagram contains two Hopf points like those shown in Figure 5.7. However if < 1.21, there is no h1 degeneracy, and consequently there is no possibility of Hopf bifurcation points on the steady-state diagram.

Previously the enrichment parameter γ, was varied. The bifurcation diagrams for varying consumption parameter using the condition (5.16) are shown in Figure 5.10. A stability behaviour the reverse of the previous can be seen in the plots, in which P2 is stable for < γ and P3 becomes stable for > γ. In the long run, overconsumption exhausts the resource, with the populations then driven to extinction. The equilibrium densities of x, y and k decay:

lim x = lim y = lim k = 0. →∞ →∞ →∞

The system (5.13) predicts the eﬀects of biotic resource enrichment on the top predator and the prey/consumer populations. Although destabilisation is found in the system, the possible parameter range where it occurred is determined and explained in the analysis. In the next section, for completeness of the model, we discuss the case in which the resource enrichment is logistically limited.

75 0.3 1.4

1.2 0.2 1 y x 0.8 0.1 0.6

0 0.4 0 100 200 300 400 500 0 100 200 300 400 500 τ τ (a) (b)

6 8 5 6 4 k

3 k 4

2 2 1 0 0.4 0 1.5 1 0.2 0 100 200 300 400 500 0.5 0 x τ y 0 (c) (d)

Figure 5.8: Time series solutions of the system (5.13) with α = 0.003, β = 0.5, γ = 2.12, δ = 2, = 2.5 for: (a) x; (b) y; (c) k. Initial conditions (x, y, k) = (0.1, 0.4, 0.5). (d) Corresponding phase-plot.

0.01 Degenerate Hopf locus

0.008

0.006 No periodic solutions α 0.004 Periodic solutions 0.002

0 1 1.5 2 2.5 3 ε

Figure 5.9: Bifurcation diagram showing the degenerate Hopf locus with β = 0.5, δ = 2.

76 0.1 P 3

0.05 x P P 2 2 0

P 3 −0.05 0 4 8 12 16 ε (a)

P 1.5 2

P 3

y 1

P P 2 0.5 3

0 0 1 2 3 4 5 6 7 ε (b)

P 1.5 3

k P 1 2

P 0.5 2 P 3 0 0 1 2 3 4 5 6 7 ε (c)

Figure 5.10: Steady-state diagrams of the system (5.13) against the consumption parameter, , with α = 0.3, β = 0.5, γ = 2.5, δ = 2 for: (a) x; (b) y; (c) k.

77 5.4 Environmental carrying capacity with logistically limited biotic resource enrichment

In this section, the model (5.5) is modiﬁed to investigate the case in which the resource is a limited quantity in the absence of consumption by the prey or predator or both. The modiﬁed model is [Safuan et al., 2014b]

dX X = r X 1 − − aXY, (5.17a) dt 1 pK dY Y = r Y 1 − + bXY, (5.17b) dt 2 qK dK = K (c − dK − eX − fY ) . (5.17c) dt

The diﬀerence between Equations (5.5c) and (5.17c) is that there is now an extra term dK2 which gives the resource a carrying capacity c/d.

5.4.1 Equilibrium and stability analysis

Using the same dimensionless variables as in model (5.13), the system (5.17) becomes

x x0 = x 1 − − xy, (5.18a) k βy y0 = αy 1 − + xy, (5.18b) k k0 = k (γ − θk − ξx − φy) , (5.18c) where r bp c d e f α = 2 , β = , γ = , θ = , ξ = , φ = . r1 aq r1 bp b a

The ﬁrst three parameters are the same as those used in system (5.13), except for θ, ξ and φ.

In system (5.18):

θ is the ratio of the saturation rate of the carrying capacity to the feeding rate of y in proportion to their carrying capacities; ξ is the ratio of the consumption rate of the resource by x to the feeding rate of y on x; and φ is the ratio of the consumption rate of the resource by y to the removal rate of x due to y.

78 The Jacobian matrix at an equilibrium solution (x∗, y∗, k∗) is

 2x∗ x∗2  1 − − y∗ −x∗  k∗ k∗2        J(x∗, y∗, k∗) =  2αβy∗ αβy∗2  .  y∗ α − + x∗   ∗ ∗2   k k      −ξk∗ −φk∗ γ − 2θk∗ − ξx∗ − φy∗

Equating system (5.18) to zero, there are up to six equilibria. Two of the equilibria,

γ γ γ Q = 0, 0, and Q = , 0, , 1 θ 2 θ + ξ θ + ξ have eigenvalues

EQ1 = {α, −γ, 1} , ( s ) γ 1 γθ 1 γθ 2 E = α + , − + 1 ± + 1 − 4γ , Q2 θ + ξ 2 θ + ξ 2 θ + ξ

respectively. It is clear that Q1 and Q2 are unstable, as at least one of the eigenvalues is positive; for this reason they will no longer be considered in the analysis. The other equilibria are

ˆ ˆ ˆ ˆ ! γ βγ αk(β − k) k(α + k) ˆ Q3 = 0, , and Q4 = , , k , (5.19) βθ + φ βθ + φ αβ + kˆ2 αβ + kˆ2 where kˆ is the root of the cubic polynomial

δkˆ3 + (φ − γ − αξ)kˆ2 + α(βξ + βθ + φ)kˆ − αβγ = 0. (5.20)

As the parameter values are varied, there exists one equilibrium point for each of

Q1,Q2 and Q3, whereas there can be either one or three equilibria for Q4, based on the cubic equation (5.20). The equilibrium point Q1 represents a steady-state that is free from both predator and prey, Q2 is free of predator only, while Q3 is free of prey only. The coexistence of predator and prey is represented by the steady-state ˆ solution branch Q4. The latter is physically realistic (x > 0) when β > k, which holds on some ﬁnite interval of γ.

79 For Q3, the eigenvalues are

( s ) γ 1 βγθ 1 βγθ 2 E = 1 − , − + α ± + α − 4αγ . Q3 βθ + φ 2 βθ + φ 2 βθ + φ

From the structure of the eigenvalues of Q3, the condition

γ > βθ + φ, (5.21)

ensures that it is stable. Further, Q3 is a stable spiral if the expression under the square root sign is negative, a stable node otherwise.

For the equilibrium points on the solution branch Q4, general expressions for the eigenvalues could not be obtained explicitly. However, Appendix A.2 provides a discussion of the special case presented in the next section. Further, the condition for stability of Q4 is more complicated as it exhibits a hysteresis phenomenon which is discussed in more detail in the next section. A necessary (but not suﬃcient) condition for stability is γ < βθ + φ.

5.4.2 Cusp singularity analysis

Singularity theory is applied to ﬁnd regions which show diﬀerent steady-state behaviour. As discussed in Chapter 2, the steady-state solution of the system (5.18) can be written in the form of a reduced singularity function,

G(k, γ, p) = 0, (5.22) where k is the state-variable, γ is the primary bifurcation parameter and p represents all other parameters (the secondary bifurcation parameters) – see Section 2.1.2 for further details. There are several important singularities, including the cusp and isola, among others. In all cases, the primary singularity conditions G = Gk = 0 are assumed to be satisﬁed. Additional conditions are invoked to distinguish between the diﬀerent types of singularities.

For the equilibrium point Q3, no singularity function G(k, γ, p) exists for which the primary singularity conditions are satisﬁed. On the other hand, for Q4, using the cubic equation (5.20) the singularity function is written as

G = δk3 + (φ − γ − αξ)k2 + α(βξ + βθ + φ)k − αβγ = 0. (5.23)

80 It is now quite easy to show, for example, that the isola does not exist: Gγ = −(k2 + αβ) 6= 0. We focus on the existence of the cusp singularity, as this type of singularity may indicate the existence of bistability (hysteresis phenomenon).

The cusp singularity is found when G satisﬁes the additional conditions

Gkk = 0 with Gγ 6= 0 and Gkkk 6= 0. (5.24)

Combining the primary singularity conditions with those of (5.24) leads to

G = δk3 + (φ − γ − αξ)k2 + α(βξ + βθ + φ)k − αβγ = 0, (5.25a)

2 Gk = 3δk + 2(φ − γ − αξ)k + α(βξ + βθ + φ) = 0, (5.25b)

Gkk = 6δk + 2(φ − γ − αξ) = 0, (5.25c)

2 where Gγ = −(k + αβ) 6= 0, and Gkkk = 6θ 6= 0.

Until now the analysis has been general. To make further progress, we make the simpliﬁcation φ = 0. The ecological consequence is that the resource is not consumed by the predator Equation (5.18c). Solving Equations (5.25) gives

β(8θ − ξ)(θ + ξ)2 γ = , (5.26a) 27θξ β(8θ − ξ)(θ + ξ) k = , (5.26b) 9θξ β(8θ − ξ)2(θ + ξ) α = . (5.26c) 27θξ2

To ensure that the state-variable k and the parameters γ and α are nonnegative requires the condition ξ < 8θ be satisﬁed. As system (5.18) has a large number of parameters, the dimensionality of p is reduced by exploring a two-parameter bifurcation diagram by ﬁxing φ = 0, β = θ = 1.

Figure 5.11 shows the (ξ, α)–parameter space obtained from Equation (5.26c). The curve denoting the cusp locus divides the parameter space into two regions. Region

A is where the steady-state curve corresponding to Q4 does not possess limit points; region B, is where limit points exist.

If φ > 0, then y also consumes the biotic resource, and so it too degrades the environment. The dashed line in Figure 5.11 represents the cusp locus for φ = 2. Noticeably, Region A becomes larger as φ increases. In the next example, we discuss the case φ = 0.

81 Figure 5.11: Two-parameter bifurcation diagram when φ = 0 (solid line) and φ = 2 (dashed line) with β = θ = 1. Region A – no bistability; Region B – bistability exists.

First, consider the case in Region A when ξ = 3 and α = 0.1. XPPAUT and R MATLAB generated two steady-state branches, denoted by Q3 and Q4 in Fig- ure 5.12. A transcritical bifurcation (tb) occurs at γtb = 1. As γ increases in magnitude, the branch Q4 is stable below the threshold value, γ < γtb, whereas branch Q3 becomes stable for γ > γtb.

For increasing values of γ, the equilibrium value of the carrying capacity k also increases, as shown in Figure 5.12c. The sizes of y and x also increase, as the environment becomes more conducive to growth, as represented by the curves Q4 in Figure 5.12b and Figure 5.12a. As γ continues to increase, so do k and y, the equilibrium population size of x reaches a maximum, before declining due to predation by y. Passing through the threshold, γtb, k and y continue to increase following branch Q3. However, population x dies out, leaving only the dominant population y to survive in the rich biotic environment. These results show that the survival of population x is determined by the bifurcation parameter γ.

To illustrate the behaviour in Region B, ξ remains the same, ξ = 3, but α is increased from 0.1 to 1.5 . Physically, increasing α has the eﬀect of increasing the growth rate of y relative to x . The steady-state curves are shown in Figure 5.13. Moving from Region A to B, two limit points (lp and lp) exist on the S-shaped steady-state curve Q4 . Only the ﬁrst limit point (lp) is physically meaningful, since the second limit point (lp) occurs for x < 0 (Figure 5.13a). These latter solutions are unstable and will not be discussed further.

82 Within Region B, bistability can occur for a certain range of γ. From Figure 5.13, bistability occurs for 1 < γ < 1.184. In other words, for values of γ between the transcritical bifurcation (tb) and the limit point lp, depending on the choice of initial conditions, the solution can approach the stable equilibria on the steady- state solution branch Q3 or Q4. This bistable behaviour determines the transition between a favourable and an unfavourable environment for the two populations.

To illustrate bistability, we choose two diﬀerent initial conditions for the prey in the vicinity of lp. Figure 5.13 shows the time series solutions of system (5.18) with the parameters used in Figure 5.14 at γ = 1.1. For the initial condition (0.01, 0.1, 0.2), both k and y increase until they reach their steady-state values, whereas the prey population x asymptotically decays to zero. However, for a slight change in the initial conditions (0.02, 0.1, 0.2), the result is quite diﬀerent as shown in Figure 5.14b. The prey x also increases until it reaches its steady-state value.

At this stage it is worth contrasting model (5.18) with that of a model with constant carrying capacity, k. The equilibrium points of a model with constant carrying capacity are k Q˜ = (0, 0), Q˜ = (k, 0), Q˜ = 0, , 1 2 3 β and αk(β − k) (α + k)k Q˜ = , . 4 k2 + αβ k2 + αβ ˜ ˜ ˜ ˜ Here Q1 and Q2 are unstable. Q3 is stable if k > β, while Q4 is unstable. For k < β, ˜ ˜ Q3 is unstable, while Q4 is stable. There exists a transcritical bifurcation when k = β. This means that for k < β the predator and prey coexist; however, for a suﬃciently large carrying capacity (k > β), the prey dies out. This scenario is qualitatively similar to the results in Region A of model (5.18). Importantly, a model with constant k does not exhibit bistability.

This feature of the model (5.18), that the prey population can become extinct when the enrichment parameter exceeds some threshold value, is again reminiscent of the “paradox of enrichment” [Rosenzweig, 1971]. In essence, the paradox states that in a predator-prey system, increasing the nutrition to the prey may lead to an extinction of both prey and predator. The extinction of both the prey and predator, due to an oscillatory instability following an enrichment in the carrying capacity, does not arise in the proposed model. A change of stability occurs, but the predator population remains stable.

83 0.15

0.1 Q 4 0.05 x Q ←TB Q 0 3 3

−0.05 Q 4

−0.1 0 0.5 1 1.5 2 γ (a)

Q 3 1.5

Q 4 ← y 1 TB

Q 0.5 3

Q 4 0 0 0.5 1 1.5 2 γ (b)

2 Q 4 Q 3 1.5

k 1 ← TB

Q 3 0.5

Q 4 0 0 0.5 1 1.5 2 γ (c)

Figure 5.12: Steady-state diagrams of the system (5.18) against the enrichment parameter, γ, with φ = 0, β = δ = 1 and = 3, α = 0.1 for: (a) x; (b) y; (c) k.

84 ← 0.2 Q LP1 4 Q Q ←TB 0 3 3 −0.2 −0.4 x −0.6 ← LP2 −0.8

−1 Q 4

0 0.5 1 1.5 2 γ (a)

Q 3 1.5 LP2 → Q 4 ← y 1 TB ← LP1 Q 0.5 3 Q 4

0 0 0.5 1 1.5 2 γ (b)

5 Q 4

k ← LP2 Q 2 3

← 1 Q TB 3 Q ← LP1 4 0 0 0.5 1 1.5 2 γ (c)

Figure 5.13: Steady-state diagrams of the system (5.18) against the enrichment parameter, γ, with φ = 0, β = δ = 1 and = 3, α = 1.5 for: (a) x; (b) y; (c) k.

85 1.2

1 x y 0.8 k 0.6

0.4

0.2

0 0 20 40 60 80 100 τ (a)

1.2 x 1 y k 0.8

0.6

0.4

0.2

0 0 20 40 60 80 100 τ (b)

Figure 5.14: Time plots of two initial conditions (x0, y0, k0) in the bistability region, γ = 1.1, for: (a) (0.01, 0.1, 0.2); (b) (0.02, 0.1, 0.2).

86 5.5 Conclusion

This chapter investigates a two-species population model with competition in an environment enriched by a biotic resource. Changes in the environmental carrying capacity, via changes in the abundance of the resource, provide an understanding of the dynamics of a resource shared by the two populations. In the models, the biotic resource only aﬀects the carrying capacity of each species through the intra-species interaction terms; however, the species can directly alter the amount of resource available by interacting with it. In addition, the inter-species interaction between predator and prey also contributes to the overall dynamics.

Bifurcation and stability analyses, with the aid of singularity theory, provide the tools in understanding the eﬀects of the environmental carrying capacity on the two populations. The environmental development rate (the enrichment parameter), γ, is a critical parameter in the systems (5.13) and (5.18). It determines whether or not population x becomes extinct.

The resource enrichment model (5.13) discussed in Section 5.3 predicts three types of behaviours: coexistence; extinction; and limit cycles. The persistence of each species can be controlled at low enrichment levels of the resource. Under these conditions, both predator and prey populations have the chance to compete for food. We found that the predator always has the eﬀect of inhibiting the prey in highly enriched environments, ultimately leading to extinction of the prey population. Similar behaviour has been reported in most intraguild models [Mylius et al., 2001, Diehl and Feissel, 2001, Hin et al., 2011]. For a certain parameter range, Hopf bifurcations are found to exist at which enrichment destabilises the system and results in periodic behaviour. The regions in which limit-cycle behaviour exist are presented in the bifurcation diagrams.

Section 5.4 discussed bistability that was found to exist in a model with logistically limited resource enrichment system (5.18). To maintain both populations in the system, γ has to be less than the threshold value at the transcritical bifurcation point. If the eﬀect of population y is neutral (φ = 0) on the environment, increasing γ will guarantee its survival at the expense of the (harmful) population x.

So far in this thesis, only ODE models have been discussed to describe populations in changing environments. Although these models provide useful information on the population dynamics, they do not allow any population motion or diﬀusion. The next chapter extends the ODE models to systems with spatial dependence (PDE formulation) in order to investigate the local dispersal or spread and competition that will aﬀect population distribution.

Chapter 6

One-dimensional spatial-temporal population-growth models

This chapter investigates the extension of the ODE models discussed in Chapter 5 to include spatial effects. In other words, in this chapter we formulate the models as systems of PDEs. Although ODE models help biologists and ecologists to understand the dynamics that evolve between populations temporally, they do not allow for any spatial effects. Spatial models make an important contribution to the study of population dispersal by simulating the distribution of populations over time and space domains. In this chapter, travelling waves in homogeneous diffusive population models are investigated. Population movement and how the spatial dimension affects the distribution of the environmental resources are discussed. The wave speed and the effects of diffusivity are also discussed. A linear stability analysis is performed on the system moving with speed equaling the speed of the wave. In addition to the aforementioned stability analysis, full numerical simulations of the model was also undertaken. Different initial conditions are used in simulating the PDE systems to ensure that the travelling waves are stable. For the two-species model with diffusion, travelling wave that possess pulsating behaviours are found, and the non-constant wave speeds are tracked and plotted. Complex pulsation behaviour has also been detected in certain parameter regions. Parts of work in this chapter appear in Safuan et al. [2014c] (Manuscript submitted for publication) and Safuan et al. [2014a] (Manuscript in preparation).

6.1 Travelling waves

The wavefront solution provides information on how populations disperse over space. It is a standard approach used to investigate travelling wave solutions in reaction-diffusion systems. For example, Dunbar [1983] investigated a diffusive Lotka-Volterra (LV) model that gave rise to a travelling wave solution, and simulated the PDE using the method of lines (MOL). Huang and Weng [2013] applied several numerical methods to study travelling waves in a diffusive predator-prey system with a general functional response. Our aim is to apply an approach similar

89 to that used by Dunbar [1983] to determine and analyse travelling wave solutions for diﬀusive population models in one spatial dimensional. This chapter investigates the diﬀusive Leslie-Gower (LG) model for one-species and two-species population models with variable carrying capacities, as discussed in Chapter 5.

6.2 Single-species population growth model with biotic resource enrichment

Before investigating the diﬀusive two-species model with variable carrying capacity, we will discuss a simpler model, that is the LG model with diﬀusion, which may be written as

∂U ∂2U bU 2 = D + aU − , (6.1a) ∂t 1 ∂x2 K ∂K ∂2K = D + cK − dK2 − eUK, (6.1b) ∂t 2 ∂x2 where U and K represent the predator population and environmental carrying capacity (a biotic resource), respectively. Parameters D1 and D2 are the diffusion coefficients for U and K, respectively. U grows logistically with growth rate, a, and is limited by the availability of the resourcem with proportionality constant b. The resource also grows logistically with growth rate ,c, and carrying capacity c/d. The term eUK represents the effect of predation, which reduces the resource’s per capita growth rate.

There have been several investigations into the diffusive LG model (6.1). Du and Hsu [2004] investigated the behaviour of steady-state solutions in environments that are homogenous and heterogeneous (in which the constant parameter b in Equation (6.1a) is replaced by a spatially dependent function b(x)). They found that the system has no non-constant positive solutions in a homogeneous environment, whereas in a heterogeneous environment, a non-constant solution can be obtained. Ko and Ryu [2007] investigated non-constant positive steady-states of system (6.1) with a general functional response, and found in some situations, there may be more than one non-constant positive steady-state. A diffusive LG model with a protection zone for the prey was studied by Du et al. [2009], and cross-diffusion LG models were considered by Li and Zhang [2012] and Zhang and Fu [2013].

6.2.1 Equilibrium and stability analysis

The transformations

eU eK r a u = , k = , τ = at, ξ = x, c c D2

90 change the PDE system (6.1) to its dimensionless form

∂u ∂2u αu = δ + u 1 − , (6.2a) ∂τ ∂ξ2 k ∂k ∂2k = + βk(1 − γk − u), (6.2b) ∂τ ∂ξ2 where b c d D α = , β = , γ = , δ = 1 . a a e D2

Let the travelling wave solution take the form u(ξ, τ) = u(ζ), and v(ξ, τ) = v(ζ), where ζ = ξ −sτ is a moving frame with speed s. Substituting these into the system (6.2) gives a system of second order ODEs,

αu δu00 + su0 + u 1 − = 0, (6.3a) k k00 + sk0 + βk(1 − γk − u) = 0, (6.3b) where 0 ≡ d/dζ. Next, the eﬀects of the diﬀusivity parameter δ on the system (6.3) are analysed.

Sedentary population and diﬀusing resource (δ = 0)

Most predators are fast-moving, which is a crucial survival skill for some animals. For example, tigers have to be able to run fast to catch a deer; a large fish has to swim faster than a school of small fish if it wants to feed. However, there are other cases where predators move slower than the resource. For example, a praying mantis remains still while waiting for butterflies, moths, bees and beetles to move towards it. Similarly, the pacman frogs, which have a wide mouth that enable them to swallow prey that crosses their path, use a ‘sit-and-wait’ strategy. Another group that applies the same strategy are web-building spiders [Riechert, 2009]. This type of spiders stay and wait for the presence of any small insects on sheet webs until they can make their move. Thus it is a relevant case to consider. In this model, the predator is assumed to move very slowly relative to the resource, d1 d2, so the ratio of the diffusion coefficients can be neglected, δ = 0. The system (6.3) then becomes

αu su0 + u 1 − = 0, (6.4a) k k00 + sk0 + βk(1 − γk − u) = 0. (6.4b)

91 From system (6.4) there is a pair of critical points. The ﬁrst critical point is C1 = ∗ ∗ (0, 1/γ) (extinction of predator), and the second critical point is C2 = (u , k ) (coexistence of predator and resource), where

1 α u∗ = , k∗ = . αγ + 1 αγ + 1

The system (6.4) is written as a system of ﬁrst order equations,

u αu u0 = − 1 , (6.5a) s k k0 = w, (6.5b)

w0 = βk(γk + u − 1) − sw. (6.5c)

Our interest is to study system (6.5) which corresponds to orbits in the phase space connecting one critical point to the other. C1 and C2 correspond to the critical points E1 = (0, 1/γ, 0) and E2 = 1/(αγ + 1), α/(αγ + 1), 0 . The Jacobian matrix of system (6.5) takes the form

2αu − k αu2  − 0  sk sk2        J =  0 0 1  , (6.6)       βk β(2γk + u − 1) −s and at the equilibrium E1 = (0, 1/γ, 0), the Jacobian becomes

 1  − 0 0  s        JE1 =  0 0 1  . (6.7)        β  β −s γ

The characteristic equation of (6.7) is

s2 + 1 β P (λ) = λ3 + λ2 − (1 − β) λ − = 0, 1 s s which gives the eigenvalues at E1 as

1 s ps2 + 4β s ps2 + 4β λ = − , λ = − + , λ = − − . (6.8) 1 s 2 2 2 3 2 2

92 λ1 and λ3 are always negative, while λ2 is always positive. Thus, there is a one- dimensional unstable manifold based at equilibrium E1. The requirement that u and k do not oscillate below zero is met since s2 + 4β > 0. From the expressions for the eigenvalues, the wave speed can take any positive value, s > 0. At equilibrium E2 = 1/(αγ + 1), α/(αγ + 1), 0 , the Jacobian becomes

 1 1  − 0  s αs        JE2 =  0 0 1  . (6.9)        αβ αβγ  −s αγ + 1 αγ + 1

The characteristic equation at E2 is

s2 − 1 αβγ β P (λ) = λ3 + λ2 − 1 + λ + = 0. 2 s αγ + 1 s

From P2(λ), there is exactly one negative root and two positive roots, or a pair of complex roots with positive real parts. Thus there is a two-dimensional unstable manifold based at equilibrium E2.

Population and resource diﬀuse (δ > 0)

Generally, populations move at different rates which implies that the ratio of the diffusion coefficients δ > 0. Where both species move, one could be moving faster than the other or both move at a similar rate. For example, equal diffusivity of both species (δ = 1) may be due to spatial mixing from turbulence in a lake or in the sea. For such a case, one can assume that the magnitude of the diffusivity for the predator (zooplankton) and the resource (phytoplankton) is the same [Petrovskii and Li, 2006]. Here we discuss a general case δ > 0. Again the system (6.3) is written as a system of first order equations,

u0 = w, (6.10a)

k0 = z, (6.10b) u αu sw w0 = − 1 − , (6.10c) δ k δ z0 = βk(γk + u − 1) − sz. (6.10d)

93 The critical points are F1 = (0, 1/γ, 0, 0) and F2 = 1/(αγ + 1), α/(αγ + 1), 0, 0 .

At equilibrium F1, the characteristic equation is

s(1 + δ) 1 s β Q (λ) = λ4 + λ3 + s2 + 1 − βδ λ2 + (1 − β) λ − = 0, 1 δ δ δ δ with eigenvalues

s ps2 + 4β s ps2 + 4β λ = − + , λ = − − , (6.11) 1 2 2 2 2 2 √ √ s s2 − 4δ s s2 − 4δ λ = − + , λ = − − . (6.12) 3 2δ 2δ 4 2δ 2δ

From Equation (6.11), the eigenvalue λ is always positive,, while λ is always √ 1 2 negative. From (6.12), if s ≥ 2 δ, then λ and λ are a pair of real negative √ 3 4 eigenvalues. If 0 < s < 2 δ, then λ3 and λ4 are a pair of complex conjugate eigenvalues with negative real part. There is a one-dimensional unstable manifold √ and a two-dimensional stable manifold based at equilibrium F1. For 0 < s < 2 δ, the equilibrium F1 is a spiral point on the stable manifold. Therefore, if a travelling wave solution exists, the minimum possible speed of system (6.10) for a biologically relevant solution with non-negative u and k is √ s ≥ 2 δ. (6.13)

With s satisfying condition (6.13), a realistic solution may exist with limits u = 0 and k = 1/γ as ζ → +∞.

At equilibrium F2, the corresponding characteristic equation is

s(1 + δ) 1 αβγδ s αβγ β Q (λ) = λ4 + λ3 + s2 − 1 − λ2 − 1 + λ + = 0. 2 δ δ αγ + 1 δ αγ + 1 δ

From Q2(λ), there are exactly two negative roots or a pair of complex roots with negative real parts, and two positive roots or a pair of complex roots with positive real parts. Thus there is a two-dimensional unstable manifold based at equilibrium

F2.

94 6.2.2 Numerical results

In this section, we describe how a two-point boundary-value problem given by system (6.3) is solved, and the PDE problem (6.2) for both δ = 0 and δ = 2 are simulated.

Travelling wave proﬁles

A relaxation or collocation method as discussed in Chapter 2 is used to solve the travelling wave system (6.3), together with the boundary conditions

1 u(−∞) = , u(+∞) = 0, (6.14a) αγ + 1 α 1 v(−∞) = , v(+∞) = . (6.14b) αγ + 1 γ

In the process of ﬁnding travelling wave proﬁles, the following initial trial functions are used

 1  α , ζ < ζ ,  , ζ < ζ0, αγ + 1 0 αγ + 1 u(ζ) = v(ζ) = (6.15)   1 0, ζ > ζ ,  , ζ > ζ0. 0 γ

For a sedentary population and a moving resource (δ = 0), the parameters are set to be α = 0.5, β = 0.9, γ = 1, δ = 0, s = 2 in system (6.4) with the trial functions (6.15). The parameters are chosen to approximate the shape of the travelling wave solutions. The plot of the travelling wave solution for u and k is shown in Figure 6.1. The travelling wave solutions join equilibrium (1/(αγ + 1), α/(αγ + 1)) to (0, 1/γ).

From the ﬁgure, initially at the right boundary, the resource exists and the predator is present at a low density level (asymptotically approaching zero). The sedentary predator consumes abundant moving resource. The predator numbers increase after consuming the resource, and as a consequence, resource numbers decrease due the predation activity. After passing the interaction regime, both populations stabilise at the coexistence equilibrium. Figure 6.1 also shows the solutions of the system (6.4) for various wave speeds of the resource. As found in Equations (6.8), s can be any value greater than zero. As s increases, the peak population of predator u decreases slightly. Note that the resource passes through the front faster, so the height reduces but the interaction region increases in length as the populations stabilise at their asymptotic levels.

95 In system (6.4), each parameter α, β, γ has a different effect on the properties of the travelling wave solution. Different values of α and γ affect the location of the coexistence equilibrium 1/(αγ + 1), α/(αγ + 1) and the predator-extinction equilibrium (0, 1/γ). The variable β does not have any significant impact on the travelling wave of the resource, but the density of the predator decreases as β increases. This is a result of the predator’s carrying capacity being dependent on the availability of the resource. Thus, any slight change in the system will affect the numbers of the predator. This behaviour is caused by the nature of the system (6.4), where β is the growth term factor in Equation (6.4b).

Figure 6.1: Numerical solution of the travelling wave obtained by employing the relaxation method with α = 0.5, β = 0.9, γ = 1, δ = 0, ζ0 = 50, and speeds s = 0.5, 1, 1.5, 2.

The case in which both species diffuse (δ > 0) is shown in Figure 6.2. The region of existence of the travelling waves is given by the speed condition (6.13). The √ minimum speed curve s = 2 δ = s∗ divides the parameter space into two regions. The shaded region represents the parameter space in which s ≥ s∗, the unshaded region s < s∗. We set the ratio of diffusion coefficients δ = 2 and other parameters remain the same as before. In this example, the predator’s diffusion rate is twice √ that of the resource’s, with a minimum wave speed s∗ = 2 2. The travelling wave profile is shown in Figure 6.3a, using this set of parameters (in the shaded region). However, if a parameter set in the unshaded region is used, the wave profile will have negative population density, which is not biologically sensible. Figure 6.3b shows the predator’s profile for different values of the diffusivity parameter δ. The character of the travelling wave remains the same, but the width of solution expands as δ increases.

96 4

s 2

0 0 1 2 3 δ √ Figure 6.2: The curve s = 2 δ = s∗ (minimum speed from (6.13)) separates the shaded and the unshaded regions in (s, δ) parameter space. Shaded region represents parameter space of existence of the travelling waves, the unshaded region represents parameter space of the extinction of travelling waves.

97 (a)

1 δ = 0.5 δ = 1 0.8 δ = 2 δ = 10 δ = 20 0.6 u

0.4

0.2

0 30 60 90 120 ζ (b)

Figure 6.3: (a) Wave profile obtained from√ a relaxation method with α = 0.5, β = ∗ 0.9, γ = 1, δ = 2, ζ0 = 100, and s = s = 2 2; (b) u profile with different values of diffusivity δ.

98 Simulations

The PDE system (6.2) is numerically simulated via MOL (refer to Chapter 2 for details) with the initial conditions obtained from the relaxation method for δ = 0 with speed s = 2. The results of the simulation are shown in Figure 6.4. The distribution of each species is shown across the domain for time intervals of size 100 and we discretise the one-dimensional space into equal sizes of 2000 with space interval [0, 200]. In all simulations, a Neumann boundary condition ∂u/∂n = ∂v/∂n = 0 is used, which reﬂects that no population can escape across the boundary.

Figure 6.4 shows the fixed wave profiles for both u and k with constant speed s. Both equilibria are connected by the travelling wave solution – similar results to those from the relaxation method in the previous section. Once the predator starts consuming the resource, the density of the resource reduces. The resource solution show to have a tanh-shaped kink, whereas the predator solution show to have a one-hump travelling wave as a result of consumption. Both populations stabilise asymptotically to their equilibrium at the left boundary. The peak in the predator solution and a slight dip in the resource solution are induced by the nonlinear interaction terms in system (6.4). √ The solution for δ = 2 with minimum speed s∗ = 2 2 is shown in Figure 6.5. Compared to the sedentary case δ = 0 in Figure 6.4, the waves travel further for the case with δ = 2. This shift is driven by the diffusion parameter δ. The greater value of δ, the further the populations move in space. In this case, the predator moves twice as fast as the resource.

99 (a)

(b)

Figure 6.4: Solutions of system (6.2) via MOL with initial conditions obtained from the relaxation method. Solutions are plotted on the same axes for τ = 5.7, 12.7, 19.6, 26.7, 33.7, 40.7. Parameters are α = 0.5, β = 0.9, γ = 1, δ = 0 and ﬁxed speed s = 2.

100 (a)

(b)

Figure 6.5: Solutions of system (6.2) via MOL with initial conditions obtained from the relaxation method. Solutions are plotted on the same axes for τ = 5.7, 12.7, 19.7, 26.7,√33.7. Parameters are α = 0.5, β = 0.9, γ = 1, δ = 2 and minimum speed s∗ = 2 2.

101 After analysing the eﬀects of δ on the travelling wave solution of the LG model, we are now interested in the dynamics of population distribution for diﬀerent types of initial conditions. For example, Medvinsky et al. [2002] studied a system that depends on the choice of initial conditions. They found that a small perturbation in an otherwise homogeneous distribution gave rise to a heterogeneous spatial distribution of species. However, they did not discuss the travelling wave solutions. Dunbar (1983) found a travelling wave solution in a predator-prey system using a particular step-function form of the initial conditions.

The PDE system (6.2) is numerically simulated with several initial conditions: constant-gradient (IC1); step-function (IC2); and a sinusoidal function (IC3). For all simulations, the PDE is solved with a Neumann boundary condition and δ = 2. √ The minimum wave speed according to condition (6.13) is s∗ = 2 2. We begin with a simple step function for predator u and constant distribution for resource k,

 1 , ξ < ξ , αγ + 1 0 α IC1: u(ξ, 0) = k(ξ, 0) = ,  αγ + 1 0, ξ > ξ0, then consider step functions for both species

 1  α , ξ < ξ ,  , ξ < ξ0, αγ + 1 0 αγ + 1 IC2: u(ξ, 0) = k(x, 0) =   1 0, ξ > ξ ,  , ξ > ξ0, 0 γ and ﬁnally investigate a sinusoidal distribution in resource k

 1  , x < x0, αγ + 1 α 2π(ξ − ξo) IC3: u(ξ, 0) = v(ξ, 0) = +A sin ,  αγ + 1 S 0, ξ > ξ0,

α where A and S are constants with A < . αγ + 1

The simulations of the travelling waves with diﬀerent initial conditions shown in Figure 6.6. For all three types of initial conditions, after some initial transients (Figure 6.6a(ii), 6.6b(ii), and 6.6c(ii)), the travelling wave proﬁles settle to those found by the relaxation method. These simulations show that a wide range of initial conditions for the PDE system will relax to a travelling wave. Hence, it demonstrates that the travelling wave solutions found are attractors in the PDE system and are nonlinearly stable.

102 (a)

(b)

(c)

Figure 6.6: Simulations of system (6.2) with initial conditions (a) IC1; (b) IC2; (c) IC3; with ξ0 = 50,A = 0.1,S = 60.

103 6.3 Two-species population growth model with biotic resource enrichment

In the diffusive LG model (6.2), the effects of the diffusivity parameter, δ, on the properties of the travelling wave solutions and the population distribution were studied. We extend the model (5.5) in Section 5.3 to investigate the effects of the enrichment parameter, γ, on the population distribution. It is important to note that periodic solutions were found in the non-diffusive model, in which the periodic solution branch originated from the steady-state solution via Hopf bifurcations, i.e. oscillating behaviour was seen for hb < γ < hb (refer Figure 5.7).

In this section, we proceed to further investigate the dynamics of the system incorporating spatial eﬀects. The two-species model with diﬀusion is

∂U ∂2U U = D + r U 1 − − aUV, (6.16a) ∂t 1 ∂x2 1 pK

∂V ∂2V V = D + r V 1 − + bUV, (6.16b) ∂t 2 ∂x2 2 qK

∂K ∂2K = D + K (c − dU − eV ) , (6.16c) ∂t 3 ∂x2 where U is prey, V is predator and K is the biotic resource. Here D1, D2 and

D3 represent diﬀusion coeﬃcients for U, V and K, respectively. The remaining parameters are the same as used in Section 5.3.

6.3.1 Equilibrium and stability analysis

The transformations r bU aV bpK r1 u = , v = , k = , τ = r1t, ξ = x, r1 r1 r1 D1 convert the system of PDEs (6.16) to the dimensionless version

∂u ∂2u u = + u 1 − − uv, (6.17a) ∂τ ∂ξ2 k

∂v ∂2v βv = d + αv 1 − + uv, (6.17b) ∂τ 1 ∂ξ2 k

∂k ∂2k = d + γk − δuk − vk, (6.17c) ∂τ 2 ∂ξ2

104 where

D2 D3 r2 bp c d e d1 = , d2 = , α = , β = , γ = , δ = , = . D1 D1 r1 aq r1 b a

The parameters α, β, γ and δ are diﬀerent from the model (6.2). Let the travelling wave solution take the form u(ξ, τ) = u(ζ), v(ξ, τ) = v(ζ) and k(ξ, τ) = k(ζ), where ζ = ξ −sτ, with speed s. Substituting these expressions into the system (6.17) gives

u u00 + su0 + u 1 − − uv = 0, (6.18a) k

βv d v00 + sv0 + αv 1 − + uv = 0, (6.18b) 1 k

00 0 d2k + sk + γk − δuk − vk = 0. (6.18c)

Next, system (6.18) is written as a system of six equations

0 u = u1, (6.19a)

0 v = v1, (6.19b)

0 k = k1, (6.19c) u u0 = uv − u 1 − − su , (6.19d) 1 k 1 0 αv βv uv sv1 v1 = − 1 − − , (6.19e) d1 k d1 d1

0 1 k1 = (δuk + vk − γk − sk1) . (6.19f) d2

The equilibria for the non-diﬀusive system are found in Section 5.3, and the corresponding equilibria in the diﬀusive system are

γ γ H = , 0, , 0, 0, 0 , 1 δ δ

γ βγ H = 0, , , 0, 0, 0 , 2

ˆ ˆ ˆ ˆ ! αk(β − k) αk(α + k) ˆ H3 = , , k, 0, 0, 0 , αβ + kˆ2 αβ + kˆ2

105 where kˆ is a root of the quadratic equation

ˆ2 ˆ c1k + c2k + c3 = 0, with

c1 = (γ − )/α + δ,

c2 = −(βδ + ),

c3 = βγ.

From equilibria H1,H2 and H3, there are pairs of connections that correspond to two diﬀerent types of solutions. For biologically relevant solutions, the travelling wave solutions u, v and k are non-negative, and satisfy the boundary conditions

γ u(−∞) =u, ˆ u(+∞) = , (6.20a) δ v(−∞) =v, ˆ v(+∞) = 0, (6.20b) γ k(−∞) = k,ˆ k(+∞) = , (6.20c) δ which we refer to as a type I wave, and

u(−∞) =u, ˆ u(+∞) = 0, (6.21a) γ v(−∞) =v, ˆ v(+∞) = , (6.21b) βγ k(−∞) = k,ˆ k(+∞) = , (6.21c) which is denoted as a type II wave, where u(±∞) means lim u(ζ). ζ→±∞

At equilibrium H1, the Jacobian leads to the eigenvalues

p 2 p 2 −s s − 4d1(γ + αδ)/δ −s s − 4d1(γ + αδ)/δ λ1 = + , λ2 = − , (6.22) 2d1 2δd1 2d1 2δd1 and the remaining eigenvalues are roots of the quartic polynomial

4 3 2 2 d2λ + s(1 + d2)λ + (s − d2)λ − sλ + γ = 0. (6.23)

It turns out that only the negative root λ of this quartic equation gives rise to travelling type I wave. From Equations (6.22), there exists a minimum speed for

106 type I wave r d (γ + αδ) s∗ = 2 1 . (6.24) 1 δ

At equilibrium H2, the Jacobian leads to the eigenvalues

−s ps2 − 4( − γ)/ −s ps2 − 4( − γ)/ λ = + , λ = − , (6.25) 1 2 2 2 2 2 with the remain eigenvalues the roots of the quartic polynomial

4 3 2 2 d1d2 λ + s(d1 + d2)λ + (s − αd2)λ − αsλ + αγ = 0. (6.26)

Once again only the negative root λ of this quartic equation gives rise to travelling type II wave. From Equations (6.25), for ≥ γ there exists a minimum speed for type II wave r − γ s∗ = 2 . (6.27) 2

These waves will be biologically observable only if they are stable [Dunbar, 1983]. Stability analysis of travelling wave solutions to the reaction-diﬀusion equation is generally complex. Thus, our approach is to demonstrate the stability of these waves numerically.

6.3.2 Numerical results

In this section, we describe how the full PDE problem (6.17) is simulated to observe the behaviour of the wave.

Simulations

The full PDE model (6.17) is solved using MOL in MATLAB R and FlexPDETM to simulate the wave proﬁles. A Neumann boundary conditions is applied in all simulations. The initial conditions used in the simulations are    ˆ u,ˆ ξ < ξ , v,ˆ ξ < ξ , k, ξ < ξ0,  0  0  u(ξ, 0) = v(ξ, 0) = k(ξ, 0) = γ γ  , ξ > ξ , 0, ξ > ξ ,  , ξ > ξ0,  δ 0 0  δ (6.28) for type I wave, and

107    k,ˆ ξ < ξ , u,ˆ ξ < ξ , v,ˆ ξ < ξ ,  0  0  0  u(ξ, 0) = v(ξ, 0) = k(ξ, 0) = γ βγ 0, ξ > ξ ,  , ξ > ξ ,  , ξ > ξ0, 0  0  (6.29) for type II wave. These are basically step functions that connect the corresponding boundary conditions.

ODE15s is used in MOL with relative tolerance 1e-5 and absolute tolerance 1e-8, and errlim 1e-5 in FlexPDETM . In FlexPDETM , the average wave speed calculation is straightforward, since it is a space- and time-adaptive finite-element package which minimises errors to a relative error tolerance level, whereas in MOL, an algorithm has to be developed to trace the speeds. In our simulations, we apply the algorithm developed by Fornberg [1988, 1998] to generate and compute finite- difference weights at each grid point. In Fornberg’s work, the algorithm can be used for equispaced grids and also for arbitrarily spaced grids. In our algorithm, we use adaptive time stepping so the solution is found at arbitrarily spaced time values. The distribution of each species is calculated for time intervals 3000, and we discretise the one-dimensional space into equal sizes of 15000 with space interval [0,4000].

The travelling wave profiles of type I and type II for predator, prey and the resource are shown in Figure 6.7. Most parameters are set to be the same to those used in Section 5.3, in which the Hopf bifurcation was observed for the ODE model. The parameters used for these examples are α = 0.003, β = 0.5, γ = 1.5, δ = 2, = ˆ 2.5, d1 = 0.1, d2 = 2. The travelling wave solutions join equilibrium (û, v,ˆ k) to γ/δ, 0, γ/δ for type I wave and (û, v,ˆ kˆ) to 0, γ/, βγ/ for type II wave.

For type I wave in Figure 6.7a, initially at the right boundary, prey u, and the resource k are in equilibrium, and a low density of predator v, is present. For type II wave in Figure 6.7b, initially at the right boundary, predator v, resource k, and a low density of prey u, are present. For both waves, the prey consumes the resource; at the same time, the predator consumes the resource and the prey. After passing the interaction regime, all population stabilise at the coexistence equilibrium (ˆu, v,ˆ kˆ). The predator stabilises at a higher number than u and k, as it is a dominant population that feeds on both species.

108 1.2 u v 1 k

0.8

0.6

0.4

0.2

0 0 200 400 600 800 1000 ξ (a)

0.8

0.6 u v k 0.4

0.2

0 0 200 400 600 800 1000 ξ (b)

Figure 6.7: Wave profiles of system (6.17) with parameters α = 0.003, β = 0.5, γ = 1.5, δ = 2, = 2.5, d1 = 0.1, d2 = 2, ξ0 = 50 for: (a) type I wave starting from ∗ initial condition (6.28) with speed of profile s1 = 0.549 at τ = 240; (b) type II wave ∗ starting from initial condition (6.29) with speed of profile s2 = 1.265 at τ = 240.

109 In Figure 6.8 we plot the speed s against the enrichment parameter, γ using the wave speed relationships (6.24) and (6.27). For type I wave, as γ increases, the speed of the travelling wave increases. In contrast with type II wave in which the speed of the travelling wave decreases as γ increases. Here we have also plotted the wave speed obtained via MOL (denoted by 4). For example in type II wave, ∗ we see good agreement between the numerically obtained speed and that of s2. We observe three regions, denoted by A, B and C. Results are compared with the minimum speed formulation in Equation (6.27). Overall, the results obtained by the two solvers are relatively similar. For γ = 0.5, 1.5 and 2.45, the speeds are found to be constant and are consistent with the minimum speeds, as shown in Table 6.1.

TM ∗ Table 6.1: Relative errors of wave speeds from FlexPDE and MOL relative to s2 (type II wave) for the case of constant wave speeds.

γ FlexPDETM MOL 0.5 3.4e-3 7.3e-3 1.5 7.9e-4 1.6e-3 2.45 9.9e-3 6.4e-3

In Region A, we observe the existence of two stable travelling waves — solutions pertaining to type I and II waves. In this bistability region, the choice of the initial condition determines to which wave the solution converges, either type I or type II. If the prey and the resource start at the same equilibrium, and predator is present at a low density, then the solution approaches type I wave with minimum speed ∗ s1. If the predator and the resource start at diﬀerent values with low density of ∗ predator, then the solution approaches type II wave with minimum speed s2. The wave speeds of type I wave are lower relative to the wave speeds of type II wave for

γ < γhb. For the convenience of the reader, we will mainly present results for type II wave solutions here, and discuss signiﬁcant diﬀerences between these two types of solutions..

We begin by choosing the enrichment parameter γ = 1.5 in ‘steady speed’ in Region A, where the wave speed is found to be constant s = 1.265 (see Figure 6.9a). This speed implies that steady propagating waves with unique speed can be observed in the ξ-direction (see Figure 6.10), and the proﬁle looks similar to those in Figure 6.7b. All equilibria are connected by the travelling wave. Initially, the predator consumes the resource at the right boundary. However, when the prey is present, the predator now has the option to feed on both food sources. Thus, the predator’s numbers increase, before stabilising at equilibriumv ˆ at the left boundary. The prey also consumes the available resource, resulting in an increase in its number, and as

110 result of the predation by u and v, the resource decreases in number, before both stabilising atu ˆ and kˆ, respectively. We note that for the same value γ = 1.5, a type I wave solution also exist with speed s = 0.549.

2.5

2 A B A C

1.5 s

1 HB1

0.5 HB2 0 0 0.5 1 1.5 2 2.5 3 γ

p Figure 6.8: The wave speed s = 2 d1(γ + αδ)/δ (blue curve) for type I wave and p s = 2 ( − γ)/ (red curve) for type II wave with oscillations occur at hb < γ < hb where hb ≈ 1.92 and hb ≈ 2.39. Parameters are α = 0.003, β = 0.5, δ = 2, = 2.5, d1 = 0.1, d2 = 2. 4 denotes the speeds obtained via MOL, and the locations hb and hb of the Hopf bifurcation points for the ODE model. Region# A: ‘steady speed’; Region B: ‘oscillating and irregular speeds’; Region C: travelling wave extinction.

111 (a) (b)

Figure 6.9: Wave speed s derived using FlexPDETM and MOL in MATLAB R for type II wave with: (a) γ = 1.5; (b) γ = 1.925 (close to hb); (c) γ = 2.385 (close to hb); (d) γ = 2.45.

Next, we investigate the case when the environment parameter, γ = 1.925 (which is close to the location of the bifurcation point hb) in Region B, where ‘oscillating speeds’ can be seen in Figure 6.9b. The origin of the oscillating behaviour is very close to the location of Hopf bifurcation found in the ODE model in Chapter 5 (see locations of hb and hb in Figure 6.8). These speeds imply that propagating waves exhibit a pulsating behaviour in the ξ-direction (see Figure 6.11 for wave profiles). Compared to the constant-speed case, the wave speeds in Figure 6.11 possess pulsating or oscillatory behaviour, characterised by a constant amplitude and period of the wave speed oscillations. Consequently, the travelling wave fronts exhibit regular pulsating behaviour along the ξ-axis. The waves travel at different speeds than the steady propagation of waves in Figure 6.10. From Figure 6.11, there are regions in which the travelling wave profiles alternate to move between fast and slow speeds.

112 This behaviour could be analogous to a population-search tactic called a “pause- travel” mode [Andersson, 1981, Getty and Pulliam, 1991, Sonerud, 1992, Barbosa, 1995]. This strategy is used by a predator to search for food from a sedentary search position, followed by moving in a stop-and-go pattern. This pattern of movement enables the predator to alternate between rest and movement at “minimum power speed”, rather than moving continuously at a lower speed. The predator then would spend less energy in searching for food. Larval ﬁsh, hawk owls (Surnia ulula), and shorebirds are among populations that practise this strategy. To the best of our knowledge, these pulsating travelling waves have not been observed in this type of population models.

Similar behaviour can be seen for γ = 2.385 (close to the location of the bifurcation point hb), where the speed exhibits oscillations as shown in Figure 6.9c. After passing through hb, the wave speed switches back to a ‘steady speed’ behaviour in Region A (see Figure 6.9d for the case γ = 2.45). Here we note that bistability exists in this region also, and that the wave speeds for type I is now greater 2 than those of type II. Beyond the critical value γc = (1 − s /4) (and for a set of parameter value γc = 2.5) there is no travelling wave solution (s = 0). We de- note this region as C. For γ > γc, the behaviour of the system follows the solution found in the ODE model. The solution approaches the equilibrium (0, γ/, βγ/) discussed in Section 5.3.4. This behaviour indicates that the spatial dependence does not have any effect on the distribution of the population for γ > γc. We believe that the rich environment enables the predator to consume the abundant resource without the effort to travel far for food. Note that in the ODE model, the transcritical bifurcation occurred between the equilibrium points (0, γ/, βγ/) and (û, v,ˆ kˆ), whereas equilibrium (γ/δ, 0, γ/δ) was never stable. We conjecture that the travelling wave extinction in the type I wave was due to this unstable equilibrium point. It is suggested that further investigations need to be undertaken to establish the non-existence of this wave for large γ, and the reasons behind the loss of stability of the type I solution branch.

In our numerical simulation, besides obtaining speeds that are constant and periodic in nature, we also observe wave speeds that pulsate in a complex manner (complex oscillations) within Region B (see Figure 6.12 for the time series plot for the speed). As an illustration, we choose γ = 2.12 and γ = 2.3. In these simulations, the oscillations in the speeds are not due to numerical artifact, as two diﬀerent solvers independently verify the results. The corresponding travelling wave solution proﬁles are shown in Figure 6.13, with γ = 2.12. It can be observed that the pulsation of the propagating waves are irregular in pattern. Further investigations need to be carried out to determine the route to this complex behaviour and to characterise if

113 this is indeed chaotic pulsating behaviour. Preliminary numerical simulations appear to suggest that depending on initial conditions, diﬀerent regular and irregular pulsations occur as they correspond to solutions from type I and type II waves.

6.4 Conclusion

We have studied a spatio-temporal reaction-diffusion system as a Leslie-Gower (LG) population model with diffusion, and showed that system (6.3) gives rise to travelling wave solutions that connect one critical point to the other. Using linear stability analysis, a minimum wave speed condition was derived for the case with diffusion parameter δ > 0. The relaxation method was used to find the travelling wave profile for both δ = 0 and δ > 0. The PDE system was simulated numerically using Method of Lines (MOL). The PDE system was also solved numerically for several types of initial conditions using the same method, and have demonstrated that travelling waves are nonlinearly stable in the system.

Our investigations uncovered the existence of regions of bistability — co-existence of stable travelling waves corresponding to diﬀerent speeds. This occurrence of bistability has numerous implications, since changes in conditions, may be suﬃcient to perturb the solution to move from one solution branch to the other. This should be investigated further in future work.

We found three diﬀerent types of wave speed behaviour present in the system as the enrichment parameter γ was varied. First, constant wave speeds were found for a range of γ values, implying a steady propagating wave in the population with a unique and constant speed. Second, periodic wave speeds were found to emanate when γ was chosen to be very close to the location of the Hopf bifurcation points found in the ODE model discussed in Chapter 5. As a result, the propagation wave exhibits regular pulsation behaviour to describe that the population, which alternates its movement between stop and travel patterns. Third, a complex wave speed with irregular oscillating patterns arose in the system when γ moves away from Hopf points. It is suggested that further work be undertaken to explain this behaviour, and if it is indeed chaotic pulsating behaviour. On a further increase in γ results in the wave speeds returning to constant speeds and ﬁnally, beyond the critical value, γc, no travelling wave solution was observed, and the solution converge to the equilibrium 0, γ/, βγ/ found in the ODE model discussed in Chapter 5.

114 0.035

0.03

0.025 → s

0.02 u 0.015

0.01

0.005

0 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 ξ (a)

0.75

0.7

0.65 → s v

0.6

0.55

0.5 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 ξ (b)

0.35

0.3

0.25

0.2 → s k 0.15

0.1

0.05

0 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 ξ (c)

Figure 6.10: Numerical solution of PDE system (6.17) with τ = 946, 949, 952, 955, ..., 1109, starting from initial condition (6.29) for: (a) u; (b) v; (c) k. Parame- ters are α = 0.003, β = 0.5, γ = 1.5, δ = 2, = 2.5, d1 = 0.1, d2 = 2, ξ = 50.

115 0.02

0.016

0.012 → s u

0.008

0.004

0 1250 1300 1350 1400 1450 1500 ξ (a)

0.86

0.84

0.82

0.8 → s v 0.78

0.76

0.74

0.72 1250 1300 1350 1400 1450 1500 ξ (b)

0.4

0.35

0.3

0.25 → s

k 0.2

0.15

0.1

0.05

0 1250 1300 1350 1400 1450 1500 ξ (c)

Figure 6.11: Numerical solution of PDE system (6.17) with τ = 1360, 1362, 1364, ..., 1468 starting from initial condition (6.29) for: (a) u; (b) v; (c) k. Pa- rameters are α = 0.003, β = 0.5, γ = 1.925, δ = 2, = 2.5, d1 = 0.1, d2 = 2, ξ = 50.

116 (a)

(b)

Figure 6.12: Wave speed, s, derived using FlexPDETM and MOL in MATLAB R for: (a) γ = 2.12; (b) γ = 2.3.

117 0.016

0.012 → s

u 0.008

0.004

0 1050 1100 1150 1200 1250 1300 ξ (a)

0.9

0.89

0.88 → s

0.87

v 0.86

0.85

0.84

0.83

0.82 1050 1100 1150 1200 1250 1300 ξ (b)

0.45

0.4

0.35 → s

0.3

k 0.25

0.2

0.15

0.1

0.05 1050 1100 1150 1200 1250 1300 ξ (c)

Figure 6.13: Numerical solution of PDE system (6.17) with τ = 1385, 1387, 1389, ..., 1600 starting from initial condition (6.29) for: (a) u; (b) v; (c) k. Pa- rameters are α = 0.003, β = 0.5, γ = 2.12, δ = 2, = 2.5, d1 = 0.1, d2 = 2, ξ = 50.

118 Chapter 7

Concluding remarks and suggestions for future work

7.1 Conclusions

We undertook a detailed investigation of several mathematical models of population growth subjected to environmental change. We started with a fundamental model, which was then extended to include both spatially independent and dependent models. The main aims of this thesis were to:

The investigations in this thesis provide an understanding of a changing environment – represented by a variable carrying capacity in both time-dependent and space-dependent models. Here, we highlight the main conclusion of each chapter and suggest possible extensions for future work.

7.2 Thesis highlights

This section is structured in three parts: background and method; analyses of spatially independent models; and analyses of spatially dependent models. For each part we provide a list of highlights and a summary of the main ﬁndings.

119 Background and methods

The important concept of a carrying capacity used in the literature was explained in Chapter 1. The notion of carrying capacity has been used in many applications in ecology, and generally it represents a feature of an environment. Changes in the environment correspond to changes in the carrying capacity that could turn out to be beneﬁcial or detrimental to a population. Thus, maintaining the environment is important to population’s conservation. The carrying capacity in the logistic equation became the main framework in this thesis. Using the logistic equation, we investigated a variable carrying capacity and how it changed the population density, compared to the case when carrying capacity is assumed to be constant. Varying the carrying capacity in the logistic equation have been successfully applied to describe resource over-exploitation, enrichment, toxicants and seasonal environments. It was our aim to develop mathematical models for populations subjected to environmental change. Comparing and analysing these previous studies, new models are proposed, developed and analysed with the aid of mathematical methods and tools discussed in Chapter 2.

Chapter 2 introduced the mathematical methods and software that were used in this thesis. Stability and bifurcation analyses were the main analytical techniques used to investigate most models. In order to simulate and plot the solutions, the software packages MATLAB R , MAPLE, G3DATA, XPPAUT and FlexPDETM were used. G3DATA was used to extract data in Chapter 3; MAPLE was used to analyse the models using its interactive symbolic computation, especially in Chapters 5 and 6. XPPAUT was used extensively in Chapter 5 to obtain the steady-state plots as well as the periodic solution branches. MATLAB R was used to plot all ﬁgures in this thesis. Travelling wave simulations in Chapter 6 were numerically and independently validated using MATLAB R and FlexPDETM .

Analyses of spatially independent models

Modelling carrying capacity as a time-dependent form was our ﬁrst approach in investigating a changing environment (Chapter 3). A time-dependent carrying capacity was incorporated into the logistic equation, which led to a non-autonomous model. This model was applied to bacterial growth on the surface of skin that was occluded. The derivation of the exact solution of the model was discussed at the end of Chapter 3. The solution was found via a power series resulting from a straightforward algebraic method. For practical applications, the power series may be truncated; a simple criterion was established that led to a good approximate solution. The approximate solution was consistent with the numerical simulations, even though only a small number of terms was used.

120 The next approach was to treat the carrying capacity as a state-variable. The carrying capacity was included in a coupled model with a growth equation (the logistic equation). Chapter 4 explained in detail this approach, in which the environmental carrying capacity was considered as a biotic resource. The results showed that an increase in the carrying capacity encouraged the population to grow. Controlling the developmental rate of the carrying capacity can aﬀect the population’s growth. This model provided an understanding of the inter-relationship that exists between the environment and its population. This simple model can be extended to a more complex.

With the analogy used in Chapter 4, we extended the model in Chapter 5 to a two-species population with variable carrying capacities. The models treat the environmental carrying capacity as a biotic resource. In these models, a predator and a prey are dependent on a shared resource, which is linked to previous intraguild models. We investigated the eﬀects of the enrichment parameter of the resource γ as the primary bifurcation parameter throughout the analyses.

In the analysis, we found that in:

• low enriched environment, the predator, prey and resource persisted. Hence, the predator and prey were able to compete for the resource; • intermediate enriched environment, Hopf bifurcation and cusp singularity were found to exist. In this stage, populations exhibited limit-cycle (persistent oscillations) and bistability behaviours caused by the changes in the environment; • high enriched environment, the predator became dominant as it switched to a specialist predator that persisted on the abundance of the resource. The prey, on the other hand, was driven to extinction.

Analyses of spatially dependent models

Chapter 6 extended the models in Chapter 5 to include spatial dependence in one dimension. The purpose of incorporating diﬀusive terms was to investigate the dis- persion of population in space domains. To achieve this, travelling wave solutions were considered to describe the spatial propagation of the population. In the stability analysis, the wave speed relationships were derived analytically and tracked numerically. The travelling wave systems were solved using the relaxation method for the case in which the asymptotic solution can be found. For a more complex system, the full PDE systems were numerically solved and simulated using two independent solvers – Method of Lines (MOL) in MATLAB R and FlexPDETM . These

121 results were in good agreement with the minimum speeds derived in the analysis. In the simulations, we found that in:

• low enriched environment, steady speeds were found; the population dispersal were steady with constant speed; • intermediate enriched environment, oscillating and complex speeds were found; the population dispersal exhibited regular and irregular pulsating behaviours. We believe that these pulsating behaviour originated close to the Hopf bifurcations that were found in the non-diffusive system. Further increase in the enrichment parameter resulted in steady speeds being observed beyond the second Hopf bifurcation point. • high enriched environment, travelling wave solutions cease to exist as the predator does not have to travel far to find food. As observed in the non- diffusive model, the prey was also driven to extinction in the PDE system.

7.3 Future work

The analyses presented here have helped us gained a better understanding of the growth of population subjected to environmental change. The models developed exhibit a rich dynamics in population behaviour. However, there are some aspects of these models that could be studied further.

7.3.1 Spatially independent models

A variable carrying capacity, K(t), discussed in this thesis was used to describe a changing environment via a non-autonomous equation. For other applications, if the changes in the environment has the eﬀect on the population’s growth rate, it is suggested that time-dependent growth rate, r(t), could be incorporated to describe cycles in the environment

dN(t) N(t) = r(t)N(t) 1 − . (7.1) dt K(t)

For example, time-dependent growth rate has been studied to model sea-surface temperature [Carson et al., 2009], seasonality [Rogovchenko and Rogovchenko, 2009] and soil moisture [Wallach and Gutman, 1976] in the logistic equation. Using variable growth rate and carrying capacity, the model can be investigated taking into account these changes in the environment.

A type of ratio-dependent model with a simple interaction term was used in Chapter 4 and 5, but it served as a valuable approach to describe predator-prey populations in a varying environment. Other response functional forms could be considered as an

122 extension of our model, such as the Holling types ii, iii, iv, Beddington-DeAngelis and Ivlev function, among others. Collings [1997] has considered Holling-type functional responses in his study and found population oscillations and outbreaks. Other studies [Sugie et al., 1997, Ruan and Xiao, 2001, Liang and Pan, 2007] have also incorporated functional responses in their work. We believe that more dynamical behaviour may be possible if these functional responses are formulated in our models in future studies.

In this thesis, we have studied predator-prey type models with an environment enriched by a biotic resource. Competition between predator and prey for the same resource led to interesting dynamics. Future studies could incorporate inter-speciﬁc competition and mutualism between two populations toward the same resource. Utilising such models, one can undertake a similar analysis as presented here, and use the enrichment parameter as the primary bifurcation parameter to investigate the behaviour of the systems.

Models discussed in this thesis, however, do not suﬃciently describe realistic ecosystem. The population has a single age-structure; and its environment changes in- stantaneously to external perturbations. To model a realistic scenario, these models must take into account the feedback the state of the environment (available resources) reaches with a delay due to various factors by incorporating (multiple) time delays τ1, τ2 such that dX X(t − τ1) = r1X 1 − − aXY, (7.2a) dt pK(t − τ2) dY Y (t − τ1) = r2Y 1 − + bXY, (7.2b) dt qK(t − τ2) dK = K (c − dX − eY ) . (7.2c) dt

Possible factors that could generate delay eﬀects are for example, generation and maturation periods, diﬀerential resource consumption with respect to age-structure and hunger threshold levels. The work in Nindjin et al. [2006], Nindjin and Aziz- Alaoui [2008], Liu et al. [2014] have studied delays in the LG type growth models and models such as system (7.2) can be considered as an extension of our models for future work. In other contexts, it is known that delays can cause instability in the system giving rise to oscillatory or even chaotic dynamics.

123 7.3.2 Spatially dependent models

For future investigation of the current work, it is suggested that a proper stability analysis of the travelling wave solution be undertaken. The linear stability analysis of the travelling waves normally results in ﬁnding points in the discrete spectrum for some diﬀerential operator that is obtained via the linearisation of the governing PDEs around the travelling wave solution. Past researchers have been able to apply the Evans function method [Evans, 1973, 1975, Sandstede, 2002] to the stability analysis of the travelling waves for various PDE systems such as in biology [Gardner, 1991], optics [Kapitula, 1996], chemistry [Balmforth et al., 1999] and combustion [Gubernov et al., 2003, Towers et al., 2013]. This approach reduces the stability problem to the corresponding problem of locating the zeros of Evans function. In Chapter 6, the locations of the Hopf points at which regular and irregular pulsating travelling wave solution emerged, were found numerically by solving the governing PDEs. The locations of these Hopf points were found to be close to the Hopf points from the ODE model discussed in Chapter 5. Utilising the Evans function, the Hopf bifurcation points in the spatially dependent model can be determined accurately. The route to complex oscillations observed in Chapter 6 should also be investigated in detail.

In the current model, bistability regions were found to exist. Depending on the choice of initial conditions, the PDE solution converged to one of the stable travelling waves — type I or type II. Further studies should be undertaken to investigate the speeds that correspond to the regular and irregular pulsations in the two waves, and to explore the reasons behind the loss at stability of the type I wave solution branch beyond the critical value determined in Chapter 6.

In this thesis, travelling waves in homogeneous diﬀusive population models were investigated. Another possible extension is to consider heterogeneous environment by studying the eﬀects of temporal and spatial variations in the carrying capacity, K(x, t), such that

" # ∂u(x, t) ∂2u(x, t) u(x, t) = D + ru(x, t) 1 − . (7.3) ∂t ∂x2 K(x, t)

Study by Rotenberg [1982] discussed exact solutions can be found for a number of diﬀerent cases of carrying capacities in the diﬀusive logistic Equation (7.3). The carrying capacities involved in Rotenberg’s work were deterministic and stochastic (random functions of time and space). These types of carrying capacities could be incorporated in the current work to include heterogeneity of habitats such as a

124 moving habitat, straight-sided or periodic. The travelling wave solutions could be investigated for future work for these type of environments.

The diffusive models discussed in Chapter 6 were formulated only in one-dimension. An obvious extension is to include a two-dimensional space and investigate the dispersal of population and patterns that could be generated from such a system. The study of pattern formation was pioneered by Turing [1952] to study two kinds of reacting chemicals in a reaction-diffusion model. Since then, different kinds of stationary and non-stationary spatio-temporal patterns have been observed, such as spiral, spot, target and tip-splitting patterns [Upadhyay et al., 2012, Camara and Aziz-Alaoui, 2009]. The steady, pulsating and complex wave speeds found in our work could generate further interesting behaviours and patterns if two-dimensional spatial effects are incorporated.

125

Appendix A

Numerical calculation of steady-states

A.1 Steady-states for two-species population model with resource enrichment

In this appendix, numerical calculations are performed to determine the eigenvalues of the corresponding equilibria that lie on the P2 and P3 steady-state solution branches for system (5.13) using the parameter set in the text (α = 0.003, β =

0.5, δ = 2, = 2.5). P1 is unstable as discussed in the text. The trivial steady-state is P2 = (0, γ/, βγ/), and the non-trivial steady-state solution branch is

ˆ ˆ ˆ ˆ ! αk(β − k) k(α + k) ˆ P3 = , , k , αβ + kˆ2 αβ + kˆ2 where kˆ is a root of the quadratic equation

γ − + δ kˆ2 − (βδ + )kˆ + βγ = 0. (A.1) α

According to Figure 5.7, with γ in the Hopf region, 1.925 ≤ γ ≤ 2.385, the quadratic polynomial (A.1) has two positive roots, both residing on a diﬀerent branch of P3.

There are four equilibria: P1, P2 and two lying on the P3 solution curve, which are denoted as P31 and P32 . A Hopf bifurcation occurs when a pair of complex eigenvalues crosses the imaginary axis. To observe this, two cases are considered. In the ﬁrst case, the value of γ is chosen in the region where there is no Hopf occuring (γ = 1.92); in the second case γ in the Hopf region (γ = 1.93). The transition between these two values exhibits the occurrence of a Hopf bifurcation.

127 If γ is chosen to be in the region where there is no Hopf bifurcation (γ = 1.92), then the corresponding steady-states are

P2 = (0, 0.786, 0.384),

P31 = (0.015, 0.756, 0.062),

P32 = (−0.018, 0.782, −0.081), and for γ = 1.93 in Hopf region, the steady-states are

P2 = (0, 0.772, 0.386),

P31 = (0.015, 0.760, 0.063),

P32 = (−0.018, 0.786, −0.082).

Table A.1 shows the eigenvalues of the Jacobian matrix J for P2, P31 and P32 . From Table A.1, it can be observed that a pair of complex eigenvalues crosses the imaginary axis from negative to positive for steady-state P31 at Hopf point hb1. This change indicates an occurrence of a Hopf bifurcation which is associated with the formation of a limit cycle. The limit-cycle behaviour ceases to exist beyond the other Hopf point hb2, at which there is another eigenvalues sign change.

Table A.1: Eigenvalues of Jacobian matrices for steady-states P2 and P3 of system (5.13). γ = 1.92 γ = 1.93

P2 λ1 = 0.232 λ1 = 0.228 λ2,3 = −0.0015 ± 0.076i λ2,3 = −0.0015 ± 0.076i

P31 λ1 = −0.262 λ1 = −0.258 λ2,3 = −0.000095 ± 0.24i λ2,3 = 0.000095 ± 0.24i

P32 λ1 = 0.245 λ1 = 0.245 λ2,3 = −0.224 ± 0.0817i λ2,3 = −0.222 ± 0.082i

128 A.2 Steady-states for two-species population model with logistically limited resource enrichment

In this appendix, numerical calculations are performed to determine the eigenvalues of the equilibria that lie on the Q3 and Q4 steady-state solution branches of system (5.17) using the parameter set in the text (α = 1.5, β = θ = 1, ξ = 3, φ = 0).

Q1 and Q2 are unstable as discussed in the text. The trivial steady-state Q3 = (0, γ/βθ, γ/θ), and the non-trivial steady-state solution branch is

ˆ ˆ ˆ ˆ ! αk(β − k) k(α + k) ˆ Q4 = , , k , αβ + kˆ2 αβ + kˆ2 where kˆ is a root of the cubic polynomial

δkˆ3 − (γ + αξ)kˆ2 + α(βξ + βθ)kˆ − αβγ = 0. (A.2)

According to Figure 5.13c, with γ in the bistability region, 1 < γ < 1.184, the cubic polynomial (A.2) has three positive roots, all residing on a diﬀerent branch of Q4. ˆ ˆ ˆ If γ = 1.1, the approximate roots of (5.20) are k1 = 0.45, k2 = 0.86 and k3 = 4.29, with the corresponding points on Q3 and Q4 at

Q3 = (0, 1.1, 1.1),

Q41 = (0.22, 0.51, 0.45),

Q42 = (0.08, 0.91, 0.86),

Q43 = (−1.06, 1.25, 4.29),

respectively. There are six equilibria: Q1, Q2, Q3 and three lying on the Q4 solution curve which are denoted as Q41 , Q42 and Q43 . Q1 and Q2 are always unstable; we show the eigenvalues of the Jacobian matrix J for Q3, Q41 ,Q42 and Q43 in Table A.2.

From Table A.2, Q3 is a stable node, Q41 is a stable spiral with negative real eigenvalues, whereas both Q42 and Q43 are unstable. Thus, for these parameter values, bistability can only occur between steady-states Q3 and Q41 , depending on the choice of initial conditions.

129 Table A.2: Eigenvalues of Jacobian matrices for steady-states Q3 and Q4 of system (5.17).

Q3 Q41 Q42 Q43

λ1 = −0.10 λ1 = −0.24 λ1 = 0.06 λ1 = 0.86

λ2 = −0.86 λ2,3 = −1.21 ± 0.47i λ2,3 = −1.30 ± 0.40i λ2 = −1.13

λ3 = −1.74 λ3 = −4.21

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