Asymptotic analysis of a selection model with space

Selection of a phenotypical trait can be described in mathematical terms by 'stage structured' equations which are usually written under the form of integral equations so as to express competition for resource between individuals whatever is their trait. The solutions exhibit a concentration effect (selection of the fittest); when a small parameter is introduced they converge to a Dirac mass. An additional space variable can be considered in order to take into account local environmental conditions. Here we assume this environment is a single nutrient which diffuses in the domain. In this framework, we prove that the solution converges to a Dirac mass in the physiological trait which depends on time and on the location in space with Lipschitz continuity. The major mathematical difficulties come from the lack of compactness in time, space and trait variables. Usual Bounded Variation estimates in time are not available and we recover strong convergence in space time, from uniqueness in the limiting constrained Hamilton-Jacobi equation after Hopf-Cole change of unknown. For this reason, we are forced to work in a concavity framework for the trait variable, where enough compactness allows us to derive this constrained Hamilton-Jacobi equation. Our analysis is motivated by a model of tumor growth introduced in [15] in order to explain emergence of resistance to therapy. (C) 2015 Elsevier Masson SAS. All rights reserved.