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Ecology and Evolution of Dispersal in Metapopulations Western University Scholarship@Western Electronic Thesis and Dissertation Repository 10-18-2018 10:00 AM Ecology and Evolution of Dispersal in Metapopulations Jingjing Xu The University of Western Ontario Supervisor Wild, Geoff The University of Western Ontario Graduate Program in Applied Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Jingjing Xu 2018 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Ordinary Differential Equations and Applied Dynamics Commons Recommended Citation Xu, Jingjing, "Ecology and Evolution of Dispersal in Metapopulations" (2018). Electronic Thesis and Dissertation Repository. 5765. https://ir.lib.uwo.ca/etd/5765 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract Dispersal plays a key role in the persistence of metapopulations, as the balance between local extinction and colonization is affected by dispersal. Dispersal is also the primary means by which a species’ range changes, as well as an important mechanism for reducing compe- tition and breeding among relatives. In this thesis, I present three pieces of work related to dispersal. The first two are devoted to the ecological aspect of delayed dispersal in metapop- ulations. The first one focuses on how dispersal may disrupt the social structure on patches from which dispersers depart. Examinations of bifurcation diagrams of the dynamical system show that a metapopulation will, in general, be either in the state of global extinction or per- sistence. The key finding is that dispersal, and the state changes associated with dispersal, have significant qualitative and quantitative effects on long-term dynamics only in a narrow range of parameter space, so life-history features other than dispersal (e.g., mortality rate) have a greater influence over metapopulation persistence. The second one asks whether the effort intending to enrich the metapopulation could always promote the persistence of metapopula- tions, incorporating time delays into the ODE models by Levins and Hanski. Investigations of critical delays and the absolute stability of equilibrium in DDE models show that: 1) delays associated with dispersal only cannot destabilize the population; 2) reducing local extinction in metapopulations with delays associated with available territories or establishment may lead to oscillations; 3) metapopulations with a structure based on the quality of occupied patches suffer less from the problem described in 2) caused by establishment delays. The third work stud- ies the evolution of conditional/unconditional dispersal in environments with temporal global- scale fluctuations. Methods from theoretical evolutionary biology are applied, and perturbation methods, numerical procedures and individual-based simulations show that difference between conditional dispersal probabilities for poor and good environment states increases as fluctua- tion frequency and disparity of dispersal cost increase. At last, conclusions and discussion of the implication of the above studies to cooperative breeding are presented, as well as a future direction to construct a kin selection model and investigate the evolution of helping. i Keywords: Population Dynamics, Differential Equation, Inclusive-Fitness Theory, Reac- tion Norm, Ecological Constraint, Fragmented Habitat ii Co-Authorship Statement The dissertation herein is written by Jingjing Xu under the supervision of Dr. Geoff Wild. Chapter 2 is based on a paper published in Journal of Biological Dynamics co-authored with G. Wild. Chapter 3 is based on a paper in preparation for publication co-authored with G. Wild. Chapter 4 is based on a paper under review for publication in The American Naturalist co-authored with G. Wild. Chapter 5 is based on an on-going work co-authored with G. Wild. All authors contributed to the writing of the respective manuscripts. Since Chapters 2, 3, and 4 are – or will be – papers co-authored by J. Xu and G. Wild, first-person plural pronouns are used therein. Chapters 1 and 5, by contrast, will not be sub- mitted as co-authored papers, and so first-person singular pronouns are used there. iii Acknowlegements First and foremost, I would like to express my sincere gratitude to my supervisor Geoff Wild for the continuous support of my PhD, and for his knowledge, time commitment, infinite patience and encouragement. His insightful perspective in both mathematics and biology inspired me in our countless discussions. His guidance leaded me through my research and writing of the thesis. None of this would be possible without his advice. Second, I am grateful to the Department of Applied Mathematics, in the University of Western Ontario, for offering me such an incredible opportunity to work with numerous knowl- edgeable professionals and intelligent fellow students for five years. I would also like to thank all the professors in the department, especially my supervisory committee, Lindi Wahl and Rob Corless, and professors in the mathematical biology group, Xingfu Zou and Pei Yu, for their valuable suggestions and enlightenment. In addition, I would like to express my gratitude to the administrative staff, especially Audrey Kager and Cinthia MacLean, as they have been always kind and helped me promptly ever since the beginning of my application of PhD program. Third, I would like to thank fellow students in the department and my friends, in particular, Ao Li and Yang Wang, for their patience, feedback, discussions, and for all the fun we have had in the last few years. Last but not least, I would like to thank my family for supporting and encouraging me throughout my study. In addition to all the help I received from the professionals when navi- gating through the ocean of doctoral research, it is the priceless company with my family and friends that made my days delightful at Western, so that I am strong enough to overcome all the obstacles and accomplish this dissertation eventually. iv Contents Abstract i Co-Authorship Statement iii Acknowledgements iv List of Figures x List of Tables xvii List of Appendices xviii 1 Introduction 1 1.0 Summary . 1 1.1 Metapopulation . 1 1.1.1 Metapopulations, Human Activities and Conservation Biology . 2 1.2 Metapopulation Dynamics . 4 1.2.1 Mathematical Models of Metapopulation Dynamics . 4 1.3 Evolution of Dispersal in Metapopulations . 7 1.3.1 Mechanisms for the Evolution of Dispersal . 8 1.3.2 Kin Selection: a Methodology for Modelling the Evolution of Dispersal 9 1.4 Thesis Motivation and Outline . 12 Bibliography . 14 v 2 Dispersal Altering Local States Has a Limited Effect on Persistence of a Metapop- ulation 24 2.1 Introduction . 24 2.2 Model . 26 2.2.1 Preamble . 26 2.2.2 Main Assumptions . 26 2.2.3 Demographic Processes . 27 2.2.4 Dynamics . 29 2.3 Results . 31 2.3.1 Extinction Equilibrium . 31 2.3.2 Positive Equilibria . 32 2.4 Discussion . 42 Bibliography . 44 3 “Paradox of Enrichment” in Metapopulations with Delays 48 3.1 Introduction . 48 3.2 A Gallery of Models . 51 3.2.1 Three DDE versions of Levins’ Model . 51 3.2.2 Two DDE versions of Hanski’s Model . 56 3.3 The Paradox of Enrichment . 59 3.3.1 The Paradox of Enrichment in DDE versions of Levins’ model . 59 3.3.2 The Paradox of Enrichment in DDE versions of Hanski’s model . 62 3.4 Discussion . 67 Bibliography . 71 4 Evolution of Dispersal with Temporal, Global-Scale Fluctuations in its Cost 76 4.1 Introduction . 76 4.2 Models . 78 vi 4.2.1 Life Cycle . 78 4.2.2 The Inclusive-Fitness Effect of Dispersal . 81 4.2.3 Mathematical Analysis . 84 4.2.4 Numerical Procedures, Simulations, and Model Validation . 84 4.3 Results . 85 4.3.1 Unconditional Dispersal . 85 4.3.2 Dispersal Conditional on Environmental State . 87 4.4 Discussion . 91 4.4.1 General Issues . 91 4.4.2 Implications for Cooperative Breeding . 93 4.4.3 Limitations and Technical Considerations . 94 Bibliography . 96 5 Conclusions, Discussion and a Future Direction 102 Bibliography . 107 A Appendix for Chapter 3 113 A.1 Analysis of Eq. (3.2) . 113 A.2 Characteristic Equation . 113 A.3 For Proposition 6: Analysis of Critical Delay in Eq. (3.6) . 115 A.4 For Proposition 7: Absolute Stability . 117 A.5 For Proposition 8, Part I: Absolute Stability . 119 A.6 For Proposition 8, Part II: Calculating Critical Delay τ0 . 121 Bibliography . 122 B Appendix for Chapter 4 124 B.1 Environmental Stochasticity . 124 B.2 Deriving the Inclusive-Fitness Effect . 126 B.3 Genetic Relatedness . 130 vii B.3.1 The Relationship Between Pairs of Alleles . 130 B.3.2 Relatedness Calculations . 133 B.4 Perturbation Methods . 136 B.4.1 Dispersal Before Mating (DM) . 136 B.4.2 Dispersal After Mating (MD) . 143 B.5 Computational Methods . 144 B.6 Supplementary Figures . 147 Bibliography . 147 C Code 156 C.1 Code for Chapter 2 . 156 C.1.1 myode dm.m . 156 C.1.2 fig2.m . 156 C.1.3 fig3.m . 158 C.1.4 fig4a.m . 160 C.1.5 fig4b.m . 161 C.1.6 fig5.m . 162 C.1.7 figure 6 . 165 ode0913.h . 165 fig6a.cpp . 167 fig6a plot.p . 172 fig6b.cpp . 174 fig6b plot.p . 179 C.2 Code for Chapter 3 . 181 C.2.1 fig2 . 181 main function for figure 2 . ..
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