Stationary distribution of a 2-island 2-allele Wright-Fisher diffusion model with slow mutation and migration rates

TL;DR

This paper derives the stationary distribution of a two-island, two-allele Wright-Fisher diffusion model with slow mutation and migration, providing explicit approximations for different genetic states.

Contribution

It introduces a novel approximation method for the stationary distribution in a complex population genetics model with small mutation and migration rates.

Findings

Explicit line densities on the edges of the sample space.

Point masses at the corners of the sample space.

Analytic verification using backward generator and coalescent methods.

Abstract

The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent.