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

In this paper, we study the asymptotic (large time) behaviour 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 ε of mutations. We show that depending on α>0, the limit ε→0 with t=ε^−α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).