Dynamique de concentration dans des équations aux dérivées partielles non locales issues de la biologie

Abstract : This thesis focuses on the dynamics of Dirac mass concentrations in non-local partial differential and integro-differential equations motivated by evolutionary biology. We consider population models structured in phenotypical traits and, taking into account adaptation and mutation phenomena, we aim to describe the selection of the fittest traits in a given environment. The mathematical modeling of these biological problems leads to nonlinear and nonlocal equations, with a small parameter that induces two time-scales. The asymptotic solutions to these equations are population distributions on the traits space and concentrate in Dirac masses located on the dominant traits. In the first part, we study the Dirac mass dynamics in a chemostat model, using a Hamilton-Jacobi formulation. The chemostat model is a system of equations describing the dynamics of consumers and nutrients in a bioreactor. In a second part, we investigate a competition model structured in age and phenotypical traits. By means of an appropriate factorization, we obtain the asymptotic limit of the solution as a decomposition into two profiles, one in age, the other in traits. When mutations are introduced, a Hamilton-Jacobi equation arises and we prove a uniqueness result of the solution to this equation in the framework of viscosity solutions. The last part is devoted to sexual population models. These models under investigation include asymmetric trait heredity or asymmetric trait-dependent fecundity between the parents: each individual inherits mostly its traits from the female.
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Cécile Taing. Dynamique de concentration dans des équations aux dérivées partielles non locales issues de la biologie. Mathématiques [math]. Sorbonne Université, UPMC University of Paris 6, Laboratoire Jacques-Louis Lions, 2018. Français. ⟨tel-01922892⟩

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