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Open Research Online The Open University’s repository of research publications and other research outputs An investigation into the relationship between the stability of a biological community and both community complexity and environmental heterogeneity. Thesis How to cite: Larter, Richard John (1984). An investigation into the relationship between the stability of a biological community and both community complexity and environmental heterogeneity. The Open University. For guidance on citations see FAQs. c 1983 The Author https://creativecommons.org/licenses/by-nc-nd/4.0/ Version: Version of Record Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.21954/ou.ro.0000f94a Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk UfVRE5TR^CTE:D An investigation into the relationship between the stability of a biological community and both community complexity and environmental heterogeneity. Richard John barter BSc Dissertation for the degree of Bachelor of Philosophy (Mathematics) This dissertation may be made available to readers and photocopied subject to the discretion of the Librarian. ProQ uest Number: 27919407 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. in the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 27919407 Published by ProQuest LLC (2020). Copyright of the Dissertation is held by the Author. Ail Rights Reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 Abstract The aim of this dissertation Is to study two aspects of the stability properties of a biological community. These are firs tly their relationship with the structural complexity of the community and secondly the effect of spatial heterogeneity of the environment in which the community lives. Chapter 1 introduces the subject of population dynamics and discusses the reasons for using mathematical models to study it. The firs t part of Chapter 2 defines the terminology, especially the meaning of stability. The second part considers the biological evidence for a relationship between stability and complexity while the third part investigates the stability properties of mathematical models of communities of varying complexities. It is concluded that there is no general relationship, but that the more complex a community the more unlikely it is to be stable. The firs t part of Chapter 3 discusses the bioligical evidence for the importance of the effect of spatial heterogeneity on stability and proposes a definition of this term. The second part describes different ways of modelling commwnitypopulation dynamics in spatially heterogeneous environments. It is concluded that spatial heterogeneity is not likely to make a community less stable. Contents Chapter 1 Introduction 1 Section 1.1 Aim of the dissertation 1 1.2 Reasons for studying population dynamics 1 1.3 The use of mathematics 2 1.4 Types of mathematical models 3 1.5 Why study stability? 6 1.6 The importance of definitions 7 Chapter 2 The relationship between the stability of a natural community and its structural complexity. 8 Section 2.1 introduction 8 2.2 Definition of a population 8 2.3 Definition of a community 10 2.4 Why does the size of a population change through time? 10 2.5 How can we model population dynamics? 12 2.6 Definition of stab i l ity 17 2.6.1 Mathematical stability 19 2.6.2 How can we relate thismathematical definition of stability to a natural population? 22 2.6.3 Conclusions 26 2.7 Community structure 26 2.8 Community complexity 28 2.9 Evidence that stability increases as the community becomes more complex 30 2.9.1 Introduction 30 2.9.2 R.H. MacArthur (1955) 30 2.9.3 C.S. Elton 34 2.9.3.1 C.S. Elton (1958) 34 2.9.3.2 Discussion of Elton's ideas 35 2.9.4 G.E. Hutchinson 37 2.9.4.1 G.E. Hutchinson (1959) 37 2.9.4.2 Discussion 37 2.9.5 Pimentel 38 2.9.5.1 D. Pimentel (1961) 38 2.9.5.2 Discussion of Pimentel's paper ' 40 2.10 Evidence that the relationship between stability and complexity is not straightforward 4l 2.10.1 Introduction 4l 2.10.2 K.E.F. Watt 4l 2.10.2.1 Watt (1965) 4l 2.10.2.2 Discussion of Watt's ideas 46 2.10.3 R.T. Paine 47 2.10.3.1 Paine (1966) 47 2.10.3.2 Discussion of Paine's paper 49 2.11 The use of mathematical models to investigate relationships between community stability and complexity. 50 2.11.1 Introduction 50 2.11.2 May's argument ' 5 0 2.11.3 Models of communities consisting of only 1 species. 58 2.11.3.1 Simple exponential growth 58 2.11.3.2 Self-interact ion 59 2.11.4 Models of communities consisting of two species 61 2. 11.5 Models of communities consisting of three species 66 2.11.6 The general case 71 2.11.6.1 Introduction 71 2.11.6.2 M.R. Gardner and W.R. Ashby 75 2.11.6.3 May 76 2.12 Conclusions 77 Chapter 3 Environmental heterogeneity and community stability 79 Section 3.1 Introduction 79 3.2 Biological evidence of the importance of environmental 79 heterogeneity in determining community stability 3.2.1 Introduction 79 3.2.2 C.B. Huffaker 80 3.2.3 J.N.M. Smith and R. Dawkins 83 3.2.4 I.J. Polyakov 85 3.2.5 L. Hansson 86 3.2.6 J.R. Beddington, C.A. Free and J.H. Lawton 90 Definition of environmental heterogentity 93 3.3.1 Introduction 93 3.3.2 Definition 94 3.4 Approcahes to modelling population dynamics in a heterogeneous environment. 95 3.4.1 Introduction 95 3.4.2 Temporal environmental heterogeneity 97 3.4.2.1 Deterministic temporal heterogeneity 98 3.4.2.2 Random temporal heterogeneity 98 3.4.3 Approaches to modelling the effect of spatial environmental heterogeneity by implicit methods.101 3.4.4 Physical spatial heterogeneity 104 3.4 .4 .1 Cover and refuges (hiding places) - type A heterogeneity 104 3.4.4.2 Continuous heterogeneity - type B heterogeneity 105 3.4.5 Non-random spatial distributions of populations - type C heterogeneity 107 3.4 .5 .1 Implicit refuges (May (1978)) 107 3.4.5.2 Hassell and May (1973) 112 3.4 .5 .3 Discussion of Hassell and May's model 121 3.4.6 Migration - type D heterogeneity 122 3.4 .6.1 A. Hastings 122 3.4.6.2 J. Maynard Smith 127 3.4.6.3 B.P. Zeigler 129 3.4.6.4 Discussion 134 3.5 Conclusions 135 References Figure 2.1 Example of exponential growth 13 2.2 Example of logistic growth 14 2.3 Prey oscillations predicted by the Lotka-Volterra 15 equations 2.4 Abundance of tawny owls near Oxford 19 (p.265 Krebs (1972)) 2.5 Estimated abundance of Pacific sardines along Pacific Coast of N. America (p.357 Krebs (1972)). 23 2.6 Fluctuations in population density of the azuki bean weevil (p.254 Krebs (1972)). 24 2.7 Two hypothetical populations (see text) 25 2.8 Two food webs (a) and b)) with equal stab ility. 32 See text for explanation (from MacArthur (1955)). 2.9 Effect of number of tree hosts species on abundance of forest Macrolepidoptera 43 2.10 Effect of number of tree host species on stability of forest Macrolepidoptera. 2^^ 2.11 Effect of number of insect species eating same host plants on abundance of a particular species of forest Macrolepidoptera. 45 2.12 Effect of number of insect species eating same host plants on stability of a particular species of forest Macrolepidoptera. 45 2.13 The Pisaster dominated subweb at Mukkaw Bay (from Paine (1966)) 48 2.14 Variation of stability with connectance where n is the size of the system (from Figure 1 of Gardner and Ashby (1970)). 75 3.1 The importance of landscape heterogeneity as related to seasonality 87 3.2 The importance of landscape heterogeneity as related to anjmal mobility and environmental grain size. 88 3.3 The importance of landscape heterogeneity as related to types of breeding habitat and of disturbance. 88 3.4 The importance of landscape heterogeneity as related to distance to and permanence of favourable disturbances. 89 3.5 Predator and prey isoclines for Rosenzweig and MacArthur's model (see text) 102 3.6 The general shapes of the functions f and 0 considered by Rosenzweig and MacArthur. 103 3.7 Stability boundaries from a host-parasite model in which parasites aggregate in the area of highest host density (yw-><») 118 3.8 Stability boundaries from a host-parasite model showing the effect of parasite aggregation (see text) 120 3.9 Stability properties of Hastings' (1977) model. 126 3.10 Results from simulation runs of Zeigler's (1977) discrete event model. 131 3.11 Pictorial representation of a cycle of predator and prey population sizes. 132 Table 2.1 Equilibrium and stability properties of simple exponential growth. 59 2.2 Equilibrium and stability properties of a one-species self-interaction model.