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Fast reaction limit for a Leslie-Gower model including preys, meso-predators and top-predators

Desvillettes Laurent, Fiorentino Ludovica, Mautone Teresa

Abstract

We consider a system of three reaction-diffusion equations modeling the interaction between a prey species and two predators species including functional responses of Holly type-II and Leslie-Gower type. We propose a reaction-diffusion model with five equations with simpler functional responses which, in the fast reaction limit, allows to recover the zero-th order terms of the initially considered system. The diffusive part of the initial equations is however modified and cross diffusion terms pop up. We first study the equilibria of this new system and show that no Turing instability appears. We then rigorously prove a partial result of convergence for the fast reaction limit (in 1D and 2D)

Fast reaction limit for a Leslie-Gower model including preys, meso-predators and top-predators

Abstract

We consider a system of three reaction-diffusion equations modeling the interaction between a prey species and two predators species including functional responses of Holly type-II and Leslie-Gower type. We propose a reaction-diffusion model with five equations with simpler functional responses which, in the fast reaction limit, allows to recover the zero-th order terms of the initially considered system. The diffusive part of the initial equations is however modified and cross diffusion terms pop up. We first study the equilibria of this new system and show that no Turing instability appears. We then rigorously prove a partial result of convergence for the fast reaction limit (in 1D and 2D)

Paper Structure

This paper contains 9 sections, 1 theorem, 113 equations.

Table of Contents

  1. Introduction
  2. Presentation
  3. Turing instability analysis
  4. Adimensionalization
  5. Biological meaningful equilibria
  6. Preliminaries to stability
  7. Linear instability in absence of diffusion
  8. Turing Instability when linear and cross diffusions are added
  9. Rigorous results for the passage to the limit in microscopic models

Key Result

Theorem 3.1

Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$ for $N=1,2$, let $d_1 , d_2^{(1)}, d_2^{(2)}, d_3^{(1)}, d_3^{(2)} > 0$ be diffusion rates, $\Tilde{\alpha}, \Tilde{\beta}, \Tilde{\gamma}, \Gamma, \Tilde{\eta}, \Tilde{\mu}, \Tilde{c}, K, \Tilde{m}, \Tilde{r}, \Tilde{s}>0, \delta > 0$ b together with the homogeneous Neumann boundary conditions for all $t \in \mathbb{R}_+$, $\textbf{x

Theorems & Definitions (2)

  • Theorem 3.1
  • proof