The evolutionary limit for models of populations interacting competitively with many resources

Details The evolutionary limit for models of populations interacting competitively with many resources The evolutionary limit for models of populations interacting competitively with many resources

2010-06-04

arxiv.org/abs/1006.0803v1

Journal of Differential Equations 251, 1 (2011) 179-195

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.jde.2011.03.007

We consider a integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between traits occur from competition for resources whose concentrations depend on the current state of the population. Following the formalism of\cite{DJMP}, we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the solution $\phi$ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully characterize in terms of the function $\phi$.

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