Evolutionary dynamics on multi-dimensional fitness landscapes

Evolution is the common theme linking everything in biology from individual alleles to languages. Darwin believed that those who were mathematically inclined had a di erent insight and he regretted not having it. He probably would feel grati ed knowing that now evolution has gained a solid mathematical foundation. The general principles of evolution can be represented by precise mathematical equations. Simplicity is invoked by making use of the minimum factors that matter. But we cannot even imagine how many factors a single honeybee takes into account to vouch for a particular ower. How can we take this complexity into account if we aim at retrieving simple tractable explanations of biological principles? This thesis addresses this problem particularly in two scenarios: Static and dynamic tness landscapes. A tness landscape is a tool for visualising the the tness of a population in a space in which each dimension is a trait a ecting the tness. The population is ever searching for tness maxima on this landscape. This is the process of adaptation. In a static tness landscape the tness is xed, determined by the trait combination. Here we present results pertaining to the time required for a population to move from one point to another on this landscape if the paths consists of broad valleys or narrow ridges. In dynamic tness landscapes the tness is a function of the population composition. Hence as the population moves over the landscape the landscape changes shape and the tness maxima can be eternally moving. To analyse frequency dependence we employ evolutionary game theory. Traditional evolutionary game theory deals with two player games with two strategies. This thesis invokes higher dimensions and non-linearities by studying multiple players and strategies. Important results from the two player two strategy case are generalised to multiple players. Finally we employ this theoretical development to analyse a possible evolutionary application in genetic pest management.Kurzfassung v Abstract vi 1 Introduction 1 1.1 Evolution of Evolutionary Theory . . . . . . . . . . . . . . . . . 1 1.2 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Speed of evolution 13 2.1 Pace of evolution across tness valleys . . . . . . . . . . . . . . 13 2.2 Fitter but slower . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Evolutionary Game Theory 37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Evolutionarily Stable Strategies . . . . . . . . . . . . . . . . . . 42 3.3 Evolutionary Game Dynamics . . . . . . . . . . . . . . . . . . . 45 4 Evolution in the multiverse 64 4.1 Evolutionary games in the multiverse . . . . . . . . . . . . . . . 64 4.2 The assumption of \small" mutation rates . . . . . . . . . . . . 78 4.3 Mutation selection equilibrium in evolutionary games. . . . . . . 91 4.4 Evolutionary games and Medea allele dynamics . . . . . . . . . 106 5 Summary and Outlook 122 References 125 Acknowledgements cxl Declaration cxli Curriculum vitae cxli