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 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).