Eco-evolutionary dynamics of finite populations from first principles

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.