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Consider a two-type Moran population of size $N$ with selection and mutation,where the selective advantage of the fit individuals is amplified at extremeenvironmental conditions. Assume selection and mutation are weak with respectto $N$, and extreme environmental conditions rarely occur. We show that, as$N\to\infty$, the type frequency process with time speed up by $N$ converges tothe solution of a Wright-Fisher-type SDE with a jump term modeling the effectof 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 typefrequency process and the line-counting process of a pruned version of the ASGsatisfy a moment duality. This relation yields a characterization of theasymptotic type distribution. We characterize the ancestral type distributionusing an alternative pruning of the ASG. Most of our results are stated inannealed and quenched form.

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