Cross-diffusion induced Turing instability in two-prey one-predator system

TL;DR

This paper investigates how cross-diffusion effects can induce Turing instability in a two-prey one-predator system, leading to spatial pattern formation despite stability in classical diffusion models.

Contribution

It demonstrates that cross-diffusion can cause Turing instability in a coupled prey-predator system, which is not possible under classical diffusion assumptions, and provides numerical pattern simulations.

Findings

Positive equilibrium is globally stable without diffusion.

Cross-diffusion induces linear instability and pattern formation.

Numerical simulations confirm spatial pattern emergence.

Abstract

In this paper, we study a strongly coupled two-prey one-predator system. We first prove the unique positive equilibrium solution is globally asymptotically stable for the corresponding kinetic system (the system without diffusion) and remains locally linearly stable for the reaction-diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. Moreover we prove that the positive equilibrium solution is globally asymptotically stable for the reaction-diffusion system without cross-diffusion. But it becomes linear unstable only when cross-diffusion also plays a role in the reaction-diffusion system, thus it is a cross-diffusion induced instability. Finally, the corresponding numerical simulations are also demonstrated and we obtain the spatial patterns.