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 $α> 0$, the limit $\varepsilon \to 0$ with $t = \varepsilon^{-α}$ can lead to population number densities which are either Gaussian-like (when $α$ is small) or Cauchy-like (when $α$ is large).

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