We study the asymptotic behavior of solutions of the Cauchy problem
associated to a quantitative genetics model with a sexual mode of
reproduction. It combines trait-dependent mortality and a nonlinear integral
reproduction operator,
the infinitesimal model. A parameter describes
the standard deviation between the offspring and the mean parental
traits. We show that under mild assumptions upon the mortality rate
,
when the deviations are small, the solutions stay close to a Gaussian profile with
small variance, uniformly in time. Moreover, we characterize accurately the
dynamics of the mean trait in the population. Our study extends previous results on
the existence and uniqueness of stationary solutions for the model. It relies on
perturbative analysis techniques with a sharp description of the correction from the
Gaussian profile.
Keywords
infinitesimal model, asymptotic analysis, quantitative
genetics, sexual reproduction, perturbative analysis
