Moran models and Wright--Fisher diffusions with selection and mutation in a one-sided random environment

TL;DR

This paper studies a Moran model with selection and mutation in a random environment, showing convergence to a Wright-Fisher diffusion with jumps, and uses ancestral selection graphs to analyze genealogies and type distributions.

Contribution

It introduces a new model combining selection, mutation, and environmental randomness, and extends ancestral selection graphs to analyze genealogies and distributions.

Findings

Convergence of the type frequency process to a Wright-Fisher SDE with jumps.

Duality between the type process and the ancestral line-counting process.

Characterization of ancestral and type distributions via pruning of the ASG.

Abstract

Consider a two-type Moran population of size $N$ with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to $N$, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$, the type frequency process with time speed up by $N$ converges to the solution of a Wright-Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the \emph{ancestral selection graph} (ASG) to describe the model's genealogical picture. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.