  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.
