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September 2009 Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result
Guy Barles, Sepideh Mirrahimi, Benoît Perthame
Methods Appl. Anal. 16(3): 321-340 (September 2009).


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.


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Guy Barles. Sepideh Mirrahimi. Benoît Perthame. "Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result." Methods Appl. Anal. 16 (3) 321 - 340, September 2009.


Published: September 2009
First available in Project Euclid: 4 May 2010

zbMATH: 1204.35027
MathSciNet: MR2650800

Primary: 35B25 , 35K57 , 47G20 , 49L25 , 92D15

Keywords: Adaptive evolution , Dirac concentrations , Hamilton-Jacobi equation , Lotka-Volterra equation , viscosity solutions

Rights: Copyright © 2009 International Press of Boston


Vol.16 • No. 3 • September 2009
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