Asymptotic analysis of a selection model with space

TL;DR

This paper analyzes how a phenotypical trait selection model behaves when considering spatial environmental factors, proving convergence to a localized trait distribution that varies with space and time, relevant to tumor growth modeling.

Contribution

It extends existing trait selection models by incorporating spatial diffusion of nutrients and proves convergence to a space-dependent Dirac mass, addressing compactness issues.

Findings

Solution converges to a Dirac mass in trait space depending on space and time.

Lipschitz continuity of the limiting distribution in space.

Strong convergence established via uniqueness after Hopf-Cole transformation.

Abstract

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 main difficulties come from the lack of compactness in time and trait variables. Strong convergence can be recovered from uniqueness in the limiting constrained equation after Hopf-Cole change of unknown. Our analysis is motivated by a model of tumor growth introduced by Lorz et al. (preprint) in order to explain emergence of resistance to therapy.