Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result
نویسندگان
چکیده
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. Key-Words: Adaptive evolution, Lotka-Volterra equation, Hamilton-Jacobi equation, viscosity solutions, Dirac concentrations. AMS Class. No: 35B25, 35K57, 47G20, 49L25, 92D15
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