Reaction-diffusion model for a population structured in phenotype and space I -- Criterion for persistence

TL;DR

This paper develops a reaction-diffusion model for a phenotypically structured population in a heterogeneous environment, establishing criteria for population persistence based on a generalized eigenvalue, and analyzing how environmental heterogeneity and mobility affect survival.

Contribution

It introduces a mathematical criterion for population persistence using a generalized eigenvalue, accounting for environmental heterogeneity, mobility, and mutation effects.

Findings

Higher environmental heterogeneity reduces persistence likelihood.

Increased mobility makes persistence more difficult.

The model provides a foundation for environmental optimization studies.

Abstract

We consider a reaction-diffusion model for a population structured in phenotype. We assume that the population lives in a heterogeneous periodic environment, so that a given phenotypic trait may be more or less fit according to the spatial location. The model features spatial mobility of individuals as well as mutation. We first prove the well-posedness of the model. Next, we derive a criterion for the persistence of the population which involves the generalised principal eigenvalue associated with the linearised elliptic operator. This notion allows us to handle the possible lack of coercivity of the operator. We then obtain a monotonicity result for the generalised principal eigenvalue, in terms of the frequency of spatial fluctuations of the environment and in terms of the spatial diffusivity. We deduce that the more heterogeneous is the environment, or the higher is the mobility of individuals, the harder is the persistence for the species. This work lays the mathematical foundation to investigate some other optimisation problems for the environment to make persistence as hard or as easy as possible, which will be addressed in the forthcoming companion paper.