Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result
By Guy Barles, Sepideh Mirrahimi, Benoit Perthame Lmpt, Ljll. 2009
127 citations recorded by the source corpus.
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.
Subfields: Genetics, Public Health, Environmental and Occupational Health, Sociology and Political Science
View Paper: https://www.semanticscholar.org/paper/729ce273b876bcc1bfe13e06a04218b2ca87232e
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Citations in the corpus
- A Hamilton-Jacobi approach for a model of population structured by space and trait (2013, methodology): “Our objective is to generalize the methods developed recently on models structured only by a phenotypical trait [28, 5, 26] to spatial models, to be able to use the previous results in more general frameworks.”
- Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations (2010, unknown): “19) and that u satisfies, in the viscosity sense (modified as in [3, 4, 25]), the equation t u = r x − ̄ t C(x x̄ t )+ u 2 max d u t x ≤ 0 (9.”