The evolutionary limit for models of populations interacting competitively via several resources

Nicolas Champagnat 1 Pierre-Emmanuel Jabin 1, 2
1 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : 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 $\varphi$ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully characterize in terms of the function $\varphi$.
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Nicolas Champagnat, Pierre-Emmanuel Jabin. The evolutionary limit for models of populations interacting competitively via several resources. Journal of Differential Equations, Elsevier, 2011, 251 (1), pp.179-195. ⟨10.1016/j.jde.2011.03.007⟩. ⟨inria-00488979v2⟩

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