Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result


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Abstract: We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in [8] for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

Mots Clés: Adaptive evolution; Lotka-Volterra equation; Hamilton-Jacobi equation; Viscosity solutions; Dirac concentrations