Moran model and Wright-Fisher diffusion with selection and mutation in a one-sided random environment.

Consider a two-type Moran population of size $N$ subject to selection and mutation, which is immersed in a varying environment. The population is susceptible to exceptional changes in the environment, which accentuate the selective advantage of the fit individuals. In this setting, we show that the type-composition in the population is continuous with respect to the environment. This allows us to replace the deterministic environment by a random one, which is driven by a subordinator. Assuming that selection, mutation and the environment are weak in relation to $N$, we show that the type-frequency process, with time speed up by $N$, converges as $N\to\infty$ to a Wright--Fisher-type SDE with a jump term modeling the effect of the environment. Next, we study the asymptotic behavior of the limiting model in the far future and in the distant past, both in the annealed and in the quenched setting. Our approach builds on the genealogical picture behind the model. The latter is described by means of an extension of the ancestral selection graph (ASG). The formal relation between forward and backward objects is given in the form of a moment duality between the type-frequency process and the line-counting process of a pruned version of the ASG. This relation yields characterizations of the annealed and the quenched moments of the asymptotic type distribution. A more involved pruning of the ASG allows us to obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.