A Hamilton–Jacobi approach for a model of population structured by space and trait

Abstract

We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x R d and a phenotypical trait . Considering diffusion, mutations, and space-local competition between the individuals, we analyze the asymptotic (long-time/long-range in the x variable) exponential behavior of the solutions. Using some kind of real phase WKB ansatz, we prove that the propagation of the population in space can be described by a Hamilton-Jacobi equation with obstacle which is independent of . The effective Hamiltonian is derived from an eigenvalue problem.