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.
@article{1273002796,
author = {Barles, Guy and Mirrahimi, Sepideh and Perthame, Beno\^\i t},
title = {Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result},
journal = {Methods Appl. Anal.},
volume = {16},
number = {1},
year = {2009},
pages = { 321-340},
language = {en},
url = {http://dml.mathdoc.fr/item/1273002796}
}
Barles, Guy; Mirrahimi, Sepideh; Perthame, Benoît. Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result. Methods Appl. Anal., Tome 16 (2009) no. 1, pp. 321-340. http://gdmltest.u-ga.fr/item/1273002796/