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

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

2009-03-28

arxiv.org/abs/0903.4952v1

Mathematics

Analysis of PDEs

Scientific paper

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 by B. Perthame and G. Barles 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.

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