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We study the asymptotic behavior of solutions of the Cauchy problemassociated to a quantitative genetics model with a sexual mode of reproduction.It combines trait-dependent mortality and a nonlinear integral reproductionoperator "the infinitesimal model" with a parameter describing the standarddeviation between the offspring and the mean parental traits. We show thatunder mild assumptions upon the mortality rate m, when the deviations aresmall, the solutions stay close to a Gaussian profile with small variance,uniformly in time. Moreover we characterize accurately the dynamics of the meantrait in the population. Our study extends previous results on the existenceand uniqueness of stationary solutions for the model. It relies on perturbativeanalysis techniques together with a sharp description of the correctionmeasuring the departure from the Gaussian profile.

The Cauchy problem for the infinitesimal model in the regime of small variance