Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result

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 Dirac concentrations in "Lotka-Volterra parabolic PDEs", Indiana Univ. Math. J., 57:7 (2008), pp. 3275–3301, 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.