Turing instability of the periodic solutions for a vegetation-water model with cross-diffusion

Abstract

In this paper, we focus on a vegetation-water model with cross-diffusion and investigate Turing instability of its periodic solutions. Firstly, we discuss the qualitative properties of the corresponding ODE system without diffusion. By employing center manifold theory and normal form method, we deal with the stability of periodic solutions of the perturbed ODE system. Based on Floquet theory and the change of coefficients of self-diffusion and cross-diffusion, we derive conditions that the stable periodic solution from Hopf bifurcation can become Turing unstable.