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

TL;DR

This paper analyzes a reaction-diffusion system modeling prey and predator interactions, introducing a simplified model with fast reaction limits, and proves convergence results while examining stability and pattern formation.

Contribution

It develops a simplified reaction-diffusion model with fast reaction limits for predator-prey interactions, including cross-diffusion effects, and proves convergence in 1D and 2D.

Findings

No Turing instability in the new system

Convergence of the simplified model to the original in the fast reaction limit

Modified diffusive terms with cross-diffusion effects

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)