c 2016 by Meghan Galiardi. All rights reserved. MATHEMATICAL MODELS IN EVOLUTIONARY DYNAMICS BY MEGHAN GALIARDI DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2016 Urbana, Illinois Doctoral Committee: Professor Zoi Rapti, Chair Professor Lee DeVille, Director of Research Professor Renming Song Professor Vadim Zharnitsky Abstract We consider two mathematical models in evolutionary dynamics. The first model is an extension of an evolutionary game theory model proposed by Martin Nowak [50]. We consider both a mean field deterministic approach and a weak noise stochastic approach, but the focus is on latter which is an uncommon approach for this type of model. The second model is an extension of a competitive exclusion model studied by DeVille et. al. [7]. We again consider both a mean field deterministic approach and a weak noise stochastic approach, this time with the focus on the former where we are able to prove numerous global stability results. ii To Bumpa, who gave me my math brain. iii Acknowledgments This project would not have been possible without the support of many people. First I thank my advisor Lee DeVille who provided me with much guidance and many helpful insights without which I would not have gotten this far. I thank my committee members Zoi Rapti, Renming Song and Vadim Zharnitsky. I also thank many staff members at Sandia National Laboratories for all their support and guidance, especially my mentors Eric Vugrin and Steve Verzi. I am very appreciative of all the financial support I have received over the years: National Science Foundation grant DMS 08-38434 \EMSW21-MCTP: Research Experience for Graduate Students" for funding my summers during my first few years, National Science Foundation grant DMS 1345032 \MCTP: PI4: Program for Interdisciplinary and Industrial Internships at Illinois" for providing travel support for my trips to Albuquerque, and Sandia National Laboratories for funding my last two summers and last three semesters which enabled me to focus more on my research. Lastly, I would like to thank my cat Solo for always sitting by my side though all the research and writing. iv Table of Contents List of Figures . viii List of Abbreviations . x List of Symbols . xi Chapter 1 Introduction . 1 1.1 Early History . .1 1.2 Evolutionary Game Theory . .3 1.3 Competitive Exclusion . .6 1.4 Layout of Dissertation . .9 Chapter 2 Finite{Size Effects and Switching Times for Moran Dynamics with Mutation 10 2.1 Introduction . 10 2.2 Model . 11 2.2.1 Definition . 11 2.2.2 Parameter Regimes . 13 2.2.3 Simulation Results . 14 2.2.4 Connection to Prisoner's Dilemma Dynamics . 14 2.3 Deterministic Bifurcation Analysis . 17 2.3.1 Case 1.1 | a + b = c + d with a > d and b < c ...................... 17 2.3.2 Case 2 | a + b > c + d ................................... 19 2.4 Stochastic Analysis . 20 2.4.1 Diffusion Approximation (See Appendix E.2.1) . 22 2.4.2 WKB Approximation (See Appendix E.3) . 26 2.4.3 Comparison of Quasipotentials . 26 2.5 Simulation Results . 28 2.5.1 Comparison Between van Kampen Approximation and Deterministic System . 28 2.5.2 Comparison of MFPT . 29 2.6 Discussion . 30 Chapter 3 Lowering the Cooperation Threshold Using Tit{for{Tat . 33 3.1 Introduction . 33 3.2 Model . 33 3.2.1 Three Strategies . 33 3.2.2 Four Strategies . 34 3.3 Deterministic Equation . 35 3.3.1 Three Strategies . 35 3.3.2 Four Strategies . 36 3.4 Quasistationary Distribution and Mean First{Passage Times (See Appendix D.5.3) . 36 3.5 Analysis of Examples with AllC, AllD, TFT, and WSLS . 38 3.5.1 Three{Strategy Game between AllC, AllD, and TFT . 40 v 3.5.2 Three{Strategy Game between AllC, AllD, and WSLS . 40 3.5.3 Four{Strategy Game between AllC, AllD, TFT, and WSLS . 44 3.6 Discussion . 46 Chapter 4 Global Stability of a Trait{Based Competitive Exclusion Model with Mutation 49 4.1 Introduction . 49 4.2 Model . 50 4.3 Deterministic Analysis . 51 4.3.1 Spectrum of Hamming Matrices . 52 4.3.2 Linear Stability Analysis . 54 4.3.3 Global Stability Analysis . 55 4.4 Stochastic Analysis . 66 4.5 Simulation Results . 69 4.6 Discussion . 69 Chapter 5 Algorithms . 74 5.1 Numerical Solutions of Differential Equations . 74 5.1.1 Adaptive Time Step for Competitive Exclusion . 74 5.1.2 Parallelization Performance Model . 76 5.2 Stochastic Simulations of Markov Processes . 82 5.3 Simulations of Stochastic Differential Equations . 83 5.4 Quasipotential in Multidimensions . 83 Chapter 6 Concluding Remarks . 86 Appendix A Stability of Differential Equations . 87 A.1 Definitions . 87 A.2 Stability of Linear Equations . 87 A.3 Stability of Non-Linear Equations . 88 Appendix B Background on Markov Processes . 90 B.1 Stochastic Process . 90 B.2 Markov Process . 91 B.3 Chapman{Kolmogorov Equation . 91 B.4 Differential Chapman{Kolmogorov Equation . 92 B.4.1 Interpretation of Conditions (1)-(3) . 96 B.4.2 Diffusion Process . 101 B.5 Example | Ornstein{Uhlenbeck Process . 102 Appendix C Stochastic Differential Equations . 105 C.1 It^oStochastic Integral . 105 C.1.1 dW n(t)............................................ 106 C.1.2 Mean-Value and Correlation . 107 C.2 Stochastic Differential Equations . 108 C.2.1 Existence and Uniqueness . 109 C.2.2 It^o'sChange of Variables Formula . 109 C.2.3 Relationship between SDEs and the Fokker{Planck Equation . 110 C.3 Example | Ornstein{Uhlenbeck Process . 111 C.4 Small Noise Expansions for SDEs . 113 C.5 Multidimensional SDEs . 122 C.5.1 It^o'sChange of Variables Formula . 122 C.5.2 Relationship between SDEs and the Fokker{Planck Equation . 122 vi Appendix D The Fokker{Planck Equation . 123 D.1 Forward Fokker{Planck Equation . 123 D.1.1 Boundary Conditions . ..