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

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