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

We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x ∈ R and a phenotypical trait θ ∈ Θ. 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 θ. The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term. Key-Words: Structured populations, Asymptotic analysis, Hamilton-Jacobi equation, Spectral problem, Front propagation AMS Class. No: 45K05, 35B25, 49L25, 92D15, 35F21.