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

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.