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

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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Cited by in corpus (9)

• A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach
• Adaptation and migration of a population between patches
• Dynamics of concentration in a population model structured by age and a phenotypical trait
• A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources
• Clamping and Synchronization in the strongly coupled FitzHugh-Nagumo model
• Uniqueness in a class of Hamilton-Jacobi equations with constraints
• Dynamics of concentration in a population structured by age and a phenotypic trait with mutations. Convergence of the corrector
• Small populations corrections for selection-mutation models
• Convergence to equilibrium in competitive Lotka-Volterra equations