Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions

In this paper, we study the asymptotic (large time) behavior of a selection-mutation-competition model for a population structured with respect to a phenotypic trait, when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable $t$ and the rate $\varepsilon$ of mutations. We show that depending on $\alpha > 0$, the limit $\varepsilon \to 0$ with $t = \varepsilon^{-\alpha}$ can lead to population number densities which are either Gaussian-like (when $\alpha$ is small) or Cauchy-like (when $\alpha$ is large).