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

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.

In relation to v2, we have 1) added 5 pictures, 2) added Prop. 2.8 comparing fixation probabilities in a model with only genic selection and another with only environmental selection 3) added a table of notations at the end of the paper, and 4) improved the presentation

References in corpus (5)

• Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution
• Lines of descent under selection
• General selection models: Bernstein duality and minimal ancestral structures
• Lines of descent in the deterministic mutation-selection model with pairwise interaction
• The effective strength of selection in random environment