Asymptotic analysis of a quantitative genetics model withnonlinear integral operator

Abstract: We study the asymptotic behavior of stationary solutions to a quantitative genetics model with trait-dependent mortality and sexual reproduction. The infinitesimal model accounts for the mixing of parental phenotypes at birth.Our asymptotic analysis encompasses the case when deviations between the offspring and the mean parental trait are typically small. Under suitable regularity and growth conditions on the mortality rate, we prove existence and local uniqueness of a stationary profile that get concentrated around a local optimum of mortality, with a Gaussian shape having small variance. Our approach is based on perturbative analysis techniques that require to describe accurately the correction to the Gaussian leading order profile. Our result extends previous results obtained with an asexual mode of reproduction, but using an alternative methodology.