Approches stochastiques et déterministes en biologie : dynamique adaptative, modélisation pour l'écologie, génétique des populations et dynamique moléculaire ; caractère bien posé d'équations différentielles ordinaires et stochastiques

摘要

This habilitation thesis gathers several contributions in Mathematics (probability theory, partial differential equations, dynamical systems) applied to Biology (population dynamics, evolutionary biology, ecology, population genetics, molecular dynamics). The larger part of my contributions concerns the mathematical modeling and analysis of adaptive dynamics, and particularly of the phenomenon of diversification called evolutionary branching. The second part of this mémoire presents contributions on individual-based modeling in Ecology and its links with macroscopic models of partial differential equations. Chapter 3 presents several results obtained in population genetics for populations growing according to general branching processes. Chapter 4 describes two contributions on existence and uniqueness for deterministic dynamical systems and stochastic differential equations with irregular coefficients. Finally, the last chapter studies the probabilistic interpretation of divergence-form operators in dimension 2 or more, and some consequences on the numerical resolution of linear elliptic partial differential equations by Monte Carlo methods.