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We provide an asymptotic analysis of a nonlinear integro-differentialequation describing the evolutionary dynamics of a population which reproducessexually and which is subject to selection and competition. The sexualreproduction is modeled via a nonlinear integral term, known as the''infinitesimal model''. We consider a regime of small segregational variance,where a parameter in the infinitesimal operator, which measures the deviationbetween the trait of the offspring and the mean parental trait, is small. Weprove that, in this regime, the phenotypic distribution remains close to aGaussian profile with a fixed small variance and we characterize the dynamicsof the mean phenotypic trait via an ordinary differential equation. Whilesimilar properties were already proved for a closely related model using aHopf-Cole transformation and perturbative analysis techniques, we provide analternative proof which relies on a direct study of the dynamics of the momentsof the phenotypic distribution and a contraction property of the Wassersteindistance.

A moment-based approach for the analysis of the infinitesimal model in the regime of small variance