Taming the diffusion approximation through a controlling-factor WKB method

TL;DR

This paper introduces a controlled WKB-based method to improve the diffusion approximation in stochastic population dynamics, enabling more accurate analysis in regimes where traditional methods fail, especially when selection switches sign.

Contribution

It develops a novel WKB-based approach combined with asymptotic matching to derive controlled diffusion approximations and introduces a scalable numerical technique for population genetics models.

Findings

Enhanced approximation accuracy in regimes with sign-changing selection

A scalable WKB-based numerical method for large populations

Application to fixation probability in two-type competition

Abstract

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (large-deviations) method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.