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

Nathanaël Boutillon, Luca Rossi

TL;DR

This work develops a rigorous analysis of a reaction–diffusion population model that is structured by space and phenotype in a periodic environment, incorporating nonlocal competition and mutation. The authors establish well-posedness and characterize long-time behavior through a generalized principal eigenvalue λ[r] of the linearized operator, proving extinction when λ[r]<0 and persistence when λ[r]>0. They also prove monotonicity of the eigenvalue with respect to environmental frequency L and mobility d, showing that increased habitat heterogeneity or higher dispersal makes persistence harder, and they extend the framework to a variant with heterogeneous diffusion. The results provide a solid mathematical foundation for optimizing environmental structure to control persistence, and set the stage for a companion paper on optimization problems for shaping persistence.

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.

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

TL;DR

This work develops a rigorous analysis of a reaction–diffusion population model that is structured by space and phenotype in a periodic environment, incorporating nonlocal competition and mutation. The authors establish well-posedness and characterize long-time behavior through a generalized principal eigenvalue λ[r] of the linearized operator, proving extinction when λ[r]<0 and persistence when λ[r]>0. They also prove monotonicity of the eigenvalue with respect to environmental frequency L and mobility d, showing that increased habitat heterogeneity or higher dispersal makes persistence harder, and they extend the framework to a variant with heterogeneous diffusion. The results provide a solid mathematical foundation for optimizing environmental structure to control persistence, and set the stage for a companion paper on optimization problems for shaping persistence.

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.
Paper Structure (26 sections, 7 theorems, 99 equations)

This paper contains 26 sections, 7 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Presentation of the model
  3. Related models
  4. The classical Fischer-KPP equation.
  5. Heterogeneous landscape.
  6. Structured population.
  7. Recent models.
  8. Main results
  9. Assumptions on the landscape
  10. Well-posedness of the problem
  11. Long-time behaviour of the solution
  12. Influence of the heterogeneity of the environment and of the diffusivity
  13. A variant with heterogeneous diffusion
  14. Well-posedness of the model (Theorem \ref{['thm:existence']})
  15. Step 1. Global boundedness of $u$ and $\rho$.
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $r\in L^{\infty}_{loc}(\mathds{R}^N\times\overline{\Theta})$ be bounded from above (not necessarily periodic in $x$), and let $u_0\in C^0(\mathds{R}^N\times\overline\Theta)\cap L^\infty(\mathds{R}^N;L^1(\Theta))$ be nonnegative and bounded. Then, there exists a unique solution $u$ to the Cauchy

Theorems & Definitions (17)

  • Theorem 2.1: Well-posedness
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof : Proof of Theorem \ref{['thm:existence']}
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 7 more