Population dynamics in stochastic environments

TL;DR

This paper develops an advanced analytical framework combining asymptotic matching and WKB methods to better understand population fixation probabilities in stochastic environments, surpassing the limitations of traditional diffusion approximations.

Contribution

It introduces a wider-ranging theory that generalizes the diffusion approximation, enabling more accurate analysis of stochastic population dynamics and fixation probabilities.

Findings

Rederived diffusion approximation from a more general theory

Identified limitations of the diffusion approximation in fluctuating environments

Proposed alternative analytical and numerical methods for complex scenarios

Abstract

Populations are made up of an integer number of individuals and are subject to stochastic birth-death processes whose rates may vary in time. Useful quantities, like the chance of ultimate fixation, satisfy an appropriate difference (master) equation, but closed-form solutions of these equations are rare. Analytical insights in fields like population genetics, ecology and evolution rely, almost exclusively, on an uncontrolled application of the diffusion approximation (DA) which assumes the smoothness of the relevant quantities over the set of integers. Here we combine asymptotic matching techniques with a first-order (controlling-factor) WKB method to obtain a theory whose range of applicability is much wider. This allows us to rederive DA from a more general theory, to identify its limitations, and to suggest alternative analytical solutions and scalable numerical techniques when it fails. We carry out our analysis for the calculation of the fixation probability in a fluctuating environment, highlighting the difference between (on average) deleterious and beneficial mutant invasion and the intricate distinction between weak and strong selection.