Abstract:
Several recent theoretical studies have shown that noise can have strong impacts on evolutionary dynamics in the limit of small population sizes. In this thesis, I analytically describe the evolutionary dynamics of finite fluctuating populations from first principles to capture the fundamental phenomena underlying such noise-induced effects. Starting from a density-dependent 'birth-death process' describing a population of individuals with discrete traits, I derive stochastic differential equations (SDEs) for how the relative population sizes and trait frequencies change over time. These SDEs generically reveal a directional evolutionary force, 'noise-induced selection', that is particular to finite, fluctuating populations and is present even when all types have the same fitness. The strength of noise-induced selection depends directly on the difference in turnover rates between types and inversely on the total population size. Noise-induced selection can reverse the direction of evolution predicted by infinite-population frameworks. This general derivation of evolutionary dynamics helps unify and organize several previous studies — typically performed for specific evolutionary and ecological contexts — under a single set of equations. My SDEs also recover well-known results such as the replicator-mutator equation, the Price equation, and Fisher's fundamental theorem in the infinite population limit, illustrating consistency with known formal descriptions of evolution. Finally, I extend the birth-death formalism to one-dimensional quantitative traits through a 'stochastic field theory' that yields equations such as Kimura's continuum-of-alleles and Lande's gradient dynamics in the infinite population limit and provides an alternative approach to modelling the evolution of quantitative traits that is more accessible than current measure-theoretic approaches.
