  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).
