Demographic Stochasticity in Evolutionary Biology
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Demographic Stochasticity in Evolutionary Biology by Yen Ting Lin A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2013 Doctoral Committee: Professor Charles R. Doering, Chair Post-Doc Assistant Professor Arash Fahim Assistant Professor Emanuel Gull Professor Mark E. J. Newman Professor John C. Schotland 5/1, 2013 at Navy Pier, Chicago. c Yen Ting Lin 2013 All Rights Reserved To Chia-Hui and Amber ii ACKNOWLEDGEMENTS I met Prof. Charles Doering, who prefers to be called \Charlie", in my very first class|Complex System 541|here at the UM. Immediately I was attractive to his teaching charisma|until then I had not realized the meaning of the famous saying \teaching is an art". Charlie really masters the \art". After 541 I took his Asymptotic Analysis offered by the Department of Mathematics and learned a great deal about asymptotic analysis, which turned out to be one of the main tools for my later research. If I can only thank one person in this Acknowledgement, Charlie is the one. Without him, I would probably never finish my Ph. D. study. He kindly accepted me transferring from Prof. Leonard Sander's group, and has been a wonderful mentor to me since then. He reshaped my \thinking" so that I can perform a more mathematical research, pushed me to take graduate courses in mathematics to broaden my view, encouraged me to attend to conferences (with financial support!) and supported me| financially and mentally|to work on subjects that I really enjoy. Charlie was the one who taught me everything about research|from performing theoretical analysis, comparing numerical data to theoretical conjectures, organizing literature reviews, drafting the dissertation, to refining the articles (he also carefully polished almost every sentences of my lame English writing). This dissertation would never exist without his guide. In every step I learned from him the rigor of scientific research, and he certainly set up a high-standard paradigm for me to follow in the future. Most important of all, as a well-respected Professor with a happy family, Charlie showed me iii that taking care of my family never contradicts the excellence in scientific research. Words cannot express my gratitude to Charlie. He has made this journey a won- derful experience in my life. I am very fortunate to have him as my mentor, and I sincerely hope to become the scientist he tried to shape. I would also like to thank to Prof. Gull, who joined the Department of Physics last year. He taught me a lot about computational physics, which had never been organized so well in the curriculum of the Department before he joined. He also made substantial suggestions to this dissertation; with his advice the dissertation turned out to be much more complete. In addition, I would like to thank him for the wonderful referral to his previously-affiliated Max Planck Institute for the Physics of Complex Systems. With his and certainly with Charlie's recommendations, I am very blessed to have the opportunity to work at such a world-class research institute in the near future. Starting from the beginning of this project, Dr. Hyejin Kim and I have been working closely together as a group. I would like to thank Hyejin for a wonderful experience of interdisciplinary research. Particularly, I enjoy our discussions, in which I gained a lot of knowledge in mathematics. Hyejin was always so patient to explain rigorous mathematics to me. Aside from the research, in the past few years our families have become very close friends, especially her 2-year-old son Minjoon and our 1-year-old daughter Amber. I certainly hope this friendship will last in the future. I appreciate Dr. Arash Fahim for giving me many excellent suggestions in the final revision of the dissertation. I also enjoy very much his Math 526, in which he showed me a comprehensive picture about general stochastic processes. Thanks very much to Prof. Mark Newman and Prof. John Schotland for serving as my committee members with a very short notice. Many of my friends helped me in the process of earning the Ph. D. Here I thank my friends Paul Bierdz, Tim Saucer, and Ran Lu for helping me improve the writing iv in my job applications and the dissertation. My groupmates in the Sweetland writing group, especially Ben Yee, also helped me to write a lot. I would also like to thank the Department of Physics for offering teaching positions several times to support me when the funding is short. Finally, special thanks to my dear wife Chia-Hui for being with me all the time and raising our lovely daughter Amber almost independently. Both Chia-Hui and I thank all of our family members who emotionally support us in Taiwan. At this time I would like to share the joy and the honor of earning this Ph. D. with all the people who ever helped|without any of them this would never be possible. v TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::: iii LIST OF FIGURES ::::::::::::::::::::::::::::::: ix LIST OF TABLES :::::::::::::::::::::::::::::::: xvi LIST OF APPENDICES :::::::::::::::::::::::::::: xvii ABSTRACT ::::::::::::::::::::::::::::::::::: xviii CHAPTER I. Introduction ..............................1 1.1 General introduction . .1 1.2 Literature review . .4 1.3 Outline of the dissertation . .8 II. Features of Fast Living: On the Weak Selection for Longevity in Degenerate Birth-Death Processes ............... 11 2.1 The models . 12 2.1.1 Deterministic dynamics . 12 2.1.2 Stochastic dynamics . 15 2.2 Drift and diffusion along the coexistence line . 19 2.2.1 A physically motivated asymptotic analysis . 19 2.2.2 Truly degenerate case: γ =1............. 23 2.2.3 Summary . 25 2.3 Numerical simulations and asymptotic verification . 26 2.4 Asymptotic analysis for arbitrary γ ............... 29 2.5 Summary and discussion . 32 vi III. Demographic Stochasticity and Evolution of Dispersal in Ho- mogeneous Environments ...................... 35 3.1 The two-patch model . 35 3.1.1 The model . 35 3.1.2 Asymptotic analysis . 40 3.1.3 Detailed computation of the physically motivated asymp- totic analysis . 45 3.1.4 Simulations and numerical computations . 55 3.2 The many-patch model . 58 3.2.1 The model . 58 3.2.2 Asymptotic analysis . 62 3.2.3 Detailed computation of the physically motivated asymp- totic analysis . 67 3.2.4 Simulations and numerical computations . 72 3.3 Discussion and conclusion . 75 IV. Nonlinear Dynamics of Heterogeneous Patchy Models .... 77 4.1 The deterministic two-patch model . 78 4.2 The deterministic many patch model . 85 4.3 Discussion and conclusion . 93 V. Demographic Stochasticity and Evolution of Dispersal in Het- erogeneous Environments ...................... 94 5.1 Stochastic two-patch model . 95 5.1.1 The model . 95 5.1.2 Physically motivated asymptotic analysis . 98 5.2 Stochastic many patch model . 104 5.2.1 The model . 104 5.2.2 Physically motivated asymptotic analysis . 106 5.3 Simulations and numerical computations . 108 5.3.1 Stochastic two-patch model . 108 5.3.2 Stochastic many-patch model . 114 5.4 Summary . 118 VI. Bifurcation Analysis ......................... 122 6.1 Analysis of the effective drifts in the center manifold . 124 6.2 Identifying the sets in the parameter space . 130 6.2.1 f$(qright; µx; µy; ρ, ξ) ≤ 0g and f$(qright; µx; µy; ρ, ξ) ≥ 0g ............................ 130 6.2.2 f$(qleft; µx; µy; ρ, ξ) ≥ 0g and f$(qleft; µx; µy; ρ, ξ) ≤ 0g ............................ 134 vii 6.2.3 fqmax 2 (qleft; qright)g and fqmax 2= (qleft; qright)g .... 135 6.2.4 f$max ≥ 0g and f$max ≤ 0g ............. 141 6.3 Properties of the sets in the parameter space . 145 6.4 Bifurcation of the Landscape in Parameter Space and Evolu- tionarily Stable Dispersal Rate . 150 VII. Discussion, Conclusion, and Future Work ............ 159 7.1 Discussion and conclusion . 159 Development of the physical asymptotic analysis . 159 Demographic stochasticity in passive dispersal models . 161 Demographic stochasticity and the emergence of a evolution- arily stable dispersal rate . 163 Analytical predictions . 164 7.2 Future Work . 166 Population dynamics . 166 General mathematical biology . 168 Applied mathematics . 169 APPENDICES :::::::::::::::::::::::::::::::::: 170 A.1 The problem . 172 A.2 Change of variable . 174 A.3 Asymptotic upper bound . 175 α A.3.1 Set 1 S1 := fy : y < 2/µg .............. 175 α A.3.2 Set 2 S2 := fy : y > 2/µg .............. 176 A.4 Asymptotic solution of the upper bound as µ ! 1 ...... 178 A.4.1 n > 4.......................... 180 A.4.2 n =3.......................... 181 A.5 Small µ approximation . 183 A.6 Numerical verification and discussion . 183 C.1 The model . 190 C.1.1 Asymptotic analysis when α =0........... 192 C.2 Effective dynamics, dynamical interpretation, and numerical simulations . 198 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 200 viii LIST OF FIGURES Figure 1.1 Outline of dissertation. 10 2.1 Deterministic trajectories and the (dashed) line of fixed points for Eq.(2.2) with γ = 10. 13 2.2 Schematic diagram of the stochastic processes and the corresponding per capita rate. The random populations are Xt = n and Yt = m at this time. 14 2.3 Mechanism for demographic fluctuation-induced drift and diffusion along deterministic coexistence line. From (x0; y0)|solid dot|the system fluctuates to (x0; y0)|indicated by the open circles|and sub- sequently relaxes back to corresponding (x0−ξ; y0+ξ) point|indicated by the filled squares|on the coexistence line along a deterministic trajectory. 20 2.4 Probability of domination of the Y -species over the X-species starting from position z on the coexistence line, i.e., u(z) from (2.56), for life- cycle ratios γ = :1;:5; 1; 2; 10 (solid lines bottom to top) as a function of starting position z on the coexistence line.