We provide an asymptotic analysis of a nonlinear integro-differential equation which describes 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. We also briefly discuss the extension of the method to the study of steady solutions and their stability.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]

What would be the impact of an environment change on the persistence and the genetic/phenotypic distribution of a population? We present some integro-differential models describing the evolutionary adaptation of asexual phenotypically structured populations subject to mutation and selection in changing environments. Using an approach based on Hamilton-Jacobi equations, we provide an asymptotic analysis of such equations in the regime of small mutational variance. This analysis allows us to characterize different evolutionary outcomes depending on the type of the environmental change.[-]