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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 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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Submitted on : Wednesday, September 20, 2017 - 10:06:41 AM
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Guy Barles, Sepideh Mirrahimi, Benoît Perthame. Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result. Methods and Applications of Analysis, 2009, 16 (3), pp.321-340. ⟨10.4310/MAA.2009.v16.n3.a4⟩. ⟨hal-00371416v2⟩

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