Eco-evolutionary dynamics for finite populations and the noise-induced reversal of selection

Abstract

Theoretical studies from diverse areas of population biology have shown that demographic stochasticity can substantially impact evolutionary dynamics in finite populations, including scenarios where traits that are disfavored by natural selection can nevertheless increase in frequency through the course of evolution. Here, we analytically describe the eco-evolutionary dynamics of finite populations from demographic first principles to investigate how noise-induced effects can alter the evolutionary fate of populations in which total population size may vary stochastically over time. Starting from a generic birth-death process describing a finite population of individuals with discrete traits, we derive a set of stochastic differential equations (SDEs) that recover well-known descriptions of evolutionary dynamics such as the replicator-mutator equation, the Price equation, and Fishers fundamental theorem in the infinite population limit. For finite populations, our SDEs reveal how stochasticity can predictably bias evolutionary trajectories to favour certain traits, a phenomenon we call noise-induced biasing. We show that noise-induced biasing acts through two distinct mechanisms that we call the direct and indirect mechanisms. While the direct mechanism can be identified with classic bet-hedging theory, the indirect mechanism is a more subtle consequence of frequency and density-dependent demographic stochasticity. Our equations reveal that noise-induced biasing may lead to evolution proceeding in a direction opposite to that predicted by natural selection in the infinite population limit. By extending and generalizing some standard equations of population genetics, we thus describe how demographic stochasticity appears alongside and interacts with the more well-understood forces of natural selection and neutral drift to determine the eco-evolutionary dynamics of finite populations of non-constant size.