Méthode de Hamilton–Jacobi pour décrire des équilibres évolutifs dans les environnements hétérogènes avec des mutations non évanescentes

Abstract

In this note, we characterize the solution to a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems arisen from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows us to go further than the Gaussian approximation commonly used by biologists, and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.