EVOLUTIONARYDYNAMICS INCHANGING ENVIRONMENTS Dissertation to acquire the doctoral degree in mathematics and natural science ’Doctor rerum naturalium’ at the Georg-August-Universität Göttingen in the doctoral degree programme ’Physics of Biological and Complex Systems’ at the Georg-August University School of Science (GAUSS) Submitted by Frank Stollmeier from Lippstadt, Germany Göttingen, 2018 Thesis Committee: • Prof. Dr. Marc Timme (First Referee) Institute for Theoretical Physics, Technical University of Dres- den • Prof. Dr. Theo Geisel (Second Referee) Max Planck Institute for Dynamics and Self-Organization • Dr. Jan Nagler (Supervisor) Computational Physics for Engineering Materials, IfB, ETH Zurich • Prof. Dr. Ulrich Parlitz Max Planck Institute for Dynamics and Self-Organization Further members of the examination board: • Prof. Dr. Stefan Klumpp Institute for Nonlinear Dynamics, University of Göttingen • Prof. Dr. Reiner Kree Institute for Theoretical Physics, University of Göttingen Date of the oral examination: April 19, 2018 RELATEDPUBLICATIONS A F. Stollmeier and J. Nagler. Unfair and Anomalous Evolutionary Dynamics from Fluctuating Payoffs. In: Physical Review Letters 120.5 (2018), p. 058101. B F. Stollmeier. [Re] A simple rule for the evolution of cooperation on graphs and social networks. In: ReScience 3.1 (2017). C F. Stollmeier and J. Nagler. Phenospace and mutational stability of shared antibiotic resistance. Submitted to Physical Review X. Chapter 3 is based on publication A, chapter 4.3 is based on publica- tion B, and chapter 5 is based on submitted paper C. Each chapter also includes some additional content and the texts, the equations and the figures in these chapters are modified and partly rearranged compared to the publications. iii CONTENTS 1 introduction1 1.1 Motivation . 1 1.2 Outline . 3 2 fundamentals of evolutionary game theory5 2.1 Unexpected results of Darwinian evolution . 5 2.2 Evolutionarily stable state . 6 2.3 Replicator equation . 8 2.4 Frequency-dependent Moran process . 9 2.5 Evolutionary games . 10 3 evolutionary game theory with payoff fluctua- tions 15 3.1 Introduction . 15 3.2 Anomalous evolutionarily stable states . 16 3.3 Deterministic payoff fluctuations . 20 3.4 Classification of games with payoff fluctuations . 22 3.5 Discussion . 27 3.6 Proofs and methods . 28 3.6.1 Approximation of the geometric mean . 28 3.6.2 Stochastic payoff fluctuations . 28 3.6.3 Correspondence of payoff rank criteria and gen- eralized criteria with constant payoffs . 33 4 evolutionary games on networks 37 4.1 Introduction . 37 4.2 Existing results on structured populations indicate ef- fects of inherent noise . 37 4.3 Implementation of evolutionary games on networks . 38 4.3.1 Motivation . 38 4.3.2 Short review of Ohtsuki et al.: A simple rule for the evolution of cooperation on graphs and social net- works (Nature, 2006)................ 39 4.3.3 Description of algorithms . 40 4.3.4 Comparison of own implementation with re- sults in the paper by Ohtsuki et al. 41 4.4 The effect of inherent payoff noise on evolutionary dy- namics . 45 4.4.1 Methods . 45 4.4.2 Results . 45 4.5 Discussion . 47 5 shared antibiotic resistance 49 5.1 Motivation . 49 5.2 Introduction . 50 v vi contents 5.3 The phenospace of antibiotic resistance . 51 5.4 Cell model . 52 5.5 Colony model . 53 5.6 Fitness landscape . 55 5.7 Evolutionary dynamics . 55 5.7.1 Strain A alone . 55 5.7.2 Strain A invaded by B . 55 5.7.3 Strain A and B, invaded by C . 57 5.8 Mutational stability and the direction of evolution . 58 5.9 Conclusion . 60 5.10 Outlook . 61 6 adaptation to fluctuating temperatures 63 6.1 Introduction . 63 6.2 Outline . 64 6.3 Theoretical examples . 64 6.3.1 Periodically changing temperature . 64 6.3.2 Randomly fluctuating temperature . 67 6.4 Modeling the growth of nematodes . 70 6.4.1 Development . 70 6.4.2 Reproduction . 72 6.4.3 Population growth . 75 6.4.4 Growth rates at constant temperatures . 76 6.4.5 Growth rates with switching temperatures . 77 6.5 Weather data . 79 6.6 Estimation of soil temperatures . 80 6.7 Results . 82 6.8 Discussion . 89 7 conclusion 91 7.1 Summary and Discussion . 91 7.1.1 Overview . 91 7.1.2 Chapter 3: Evolutionary game theory with pay- off fluctuations . 92 7.1.3 Chapter 4: Evolutionary games on networks . 93 7.1.4 Chapter 5: Shared antibiotic resistance . 93 7.1.5 Chapter 6: Adaptation to fluctuating temperatures 94 7.2 Outlook: Towards applied evolutionary dynamics . 95 a appendix 97 a.1 Parameters for figure 5.4 .................. 97 a.2 Metadata and results of weather data analysis . 99 bibliography 103 acknowledgments 113 1 INTRODUCTION 1.1 motivation The aim of evolutionary dynamics is to describe how species change. It emerged after two fundamental insights. First, that species change at all, and second, that this happens due to variation, heredity and natural selection. Both have been controversially debated, but since Charles Darwin presented a line of arguments with a large number of supporting observations these insights became the established the- ory of evolution. Which open questions remain after these underlying mechanisms of evolution are discovered? Can we calculate the course of evolution using a set of basic laws for variation, heredity and nat- ural selection? The starting point of such calculations is the fitness function, which is defined as the reproductive success of an organism. The fitness de- pends first of all on the phenotype of the organism. This part of the fitness function, often metaphorically referred to as the “fitness land- scape”, can be considered as the potential of evolutionary dynamics and we can obtain the direction of evolution from the gradient of the fitness with respect to the phenotype. While the species evolves, the fitness landscape changes dynamically, because the fitness also depends on the environmental conditions, on the interactions with other individuals of the same species and on the interactions with other individuals of other evolving species. At the same time, the in- dividuals of the species also have an impact on the environment and on the fitness of individuals of other species. Taken together, these coupled fitness functions constitute a complex, nonlinear, high dimensional, heterogeneous, stochastic system. Even if we knew all relevant biological mechanisms, seemingly simple ques- tions remain challenging problems. For example, which conditions lead to extinction or speciation? Or how many species can coexist and how stable is such a system of coexisting species? The literature contains already a great number of answers, but each relies on a very specific set of assumptions. For example, let us assume a homogeneous habitat with species whose fitness depends only on their own phenotype and on the environmen- tal conditions. We further assume that the environmental conditions 1 2 introduction do not change and that the populations are infinitely large. Since the species with the highest fitness outcompetes all species with a lower fitness regardless how small the advantage is, only one species can survive under these assumptions. This reasoning is known as the competitive exclusion principle. However, the relevance for natural systems is very limited, because the assumptions are almost never sat- isfied. A famous case is the high diversity of phytoplankton species in ocean water. Due to the supposed homogeneity of ocean water and the limited number of resources, the surprisingly high diversity of phytoplankton species in ocean water is called the “paradox of the plankton”. Today, it is known that none of the aforementioned as- sumptions is satisfied for phytoplankton in ocean water, so the com- petitive exclusion principle does not apply to this case [1]. We can extend the example by assuming that the fitness of a species also depends on the phenotype of other species. To keep it simple, we further assume that the individuals interact with equal probabil- ity with every other individual, and that no new mutations occur except for those that are already present at the beginning. The re- sulting coupled evolutionary dynamics is the subject of evolutionary game theory. Imagine two species which each have a high fitness if their population is small and a low fitness if their population is large compared to the population of the other species. These two species can coexist, because natural selection always favors the species that is rare compared to the other species, which leads to an evolutionarily stable state at the ratio of populations for which both species have an equal fitness. These typical assumptions for evolutionary game theory are still far from being realistic. Changing only one of them can lead to com- pletely different results, but can also make it difficult or even impos- sible to find exact solutions. Hence, almost any attempt to relax one assumption has resulted in a new direction of research, e. g. evolu- tionary game theory with finite instead of infinite populations [2–11], with populations on networks or spatially distributed populations in- stead of well-mixed populations [12–19] or with mutations occurring during the process [7, 20, 21]. The attempts to replace the assump- tion of a constant environment with variable environments headed in different directions because the environment can affect several pa- rameters in the system. For example, the environment may affect the selection strength [8], the reproduction rate [22–25], or the population size [10, 11, 26–28] for each species independently. Typical properties of interest in these studies are the probability of extinction or fixation of a mutation and the mean time to extinction or fixation of a muta- tion [3, 5, 8, 9, 27, 29–43]. In this thesis, we aim to study the effects of changing environments on evolutionary dynamics in a different way.