Stochastic Models of Evolutionary Dynamics

Abstract

Stochasticity is a fundamental component of evolution. Many essential evolutionary phenomena cannot be modeled without it. In developing and analyzing stochastic processes that model the dynamics of evolution, this dissertation applies tools from probability theory to study fundamental mathematical principles of evolution. These principles determine how the timeline of macroscopic evolution is constructed by the accumulation of many microscopic changes. At the microscopic scale, we focus on populations of reproducing individuals. Even under neutral evolution, the complex interaction between mutation and genealogy produces intricate dynamics. In this setting, we prove a very general result about equilibrium frequencies of genotypes, bound the mixing time to equilibrium, and find exact expressions for localization in genotype space for a general class of neutral evolutionary processes. Population structure is known to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. We consider a finite population under constant selection whose structure is given by a variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result and using large deviation estimates to understand the typical structure of random graphs, we prove that the fixation probability of an invading mutant in a randomly structured population is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. At the macroscopic scale, much is known about the timeline of life and evolution on Earth. However, current mathematical models say very little about evolution on these macroscopic timescales and are limited to describing microscopic evolutionary events, like fixation, that occur over relatively few generations. We describe several mathematical properties of genotype space, which provides the stage for long term evolution. These properties are then incorporated into a model of macroscopic evolution that accumulates many microscopic events. In the weak mutation and weak selection regime, we study the time evolution takes to discover novel functionality. Finally, we describe a mechanism called the regeneration process that suggests how evolution might behave like a tinkerer when innovating.