The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

TL;DR

This paper analyzes a one-dimensional spatial population model with local heterogeneities, showing that these heterogeneities slow gene diffusion and increase local population density, converging to a stochastic heat equation with Wright-Fisher noise.

Contribution

It introduces a model incorporating local heterogeneities into the stepping stone framework and demonstrates their impact on gene diffusion and population density at large scales.

Findings

Local heterogeneity slows effective gene diffusion.

Heterogeneity increases effective local population density.

Model converges to a stochastic heat equation with Wright-Fisher noise.

Abstract

We study a one-dimensional spatial population model where the population sizes at each site are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.