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

We provide an asymptotic analysis of a nonlinear integro-differential equation describing the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction 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 deviation between the trait of the offspring and the mean parental trait, is small. We prove that, in this regime, the phenotypic distribution remains close to a Gaussian profile with a fixed small variance and we characterize the dynamics of the mean phenotypic trait via an ordinary differential equation. While similar properties were already proved for a closely related model using a Hopf-Cole transformation and perturbative analysis techniques, we provide an alternative proof which simplifies considerably the analysis. Our method relies on a direct study of the dynamics of the moments of the phenotypic distribution and a Lipschitz property of the Wasserstein distance. Such a simplification is mandatory to tackle analytically more comprehensive models with for instance a spatial structure.