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

• Autores: Guy Barles, Sepideh Mirrahimi, Benoît Perthame
• Localización: Methods and applications of analysis, ISSN 1073-2772, Vol. 16, Nº 3, 2009, págs. 321-340
• Idioma: inglés
• DOI: 10.4310/maa.2009.v16.n3.a4
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• Resumen

  • 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.