Turing pattern formation in a predator–prey–mutualist system

In this paper, we develop a theoretical framework about spatial patterns in a three-species predator–prey–mutualist system with cross-diffusion. We concentrate on three aspects of Turing pattern formation: (1) what conditions enable the occurrence of Turing patterns? (2) what are the underlying mechanisms? (3) what are the corresponding configurations? For the first two questions, by use of the stability analysis for the positive uniform solution and the Leray–Schauder degree theory, we prove that under some conditions, the system admits at least a nonhomogeneous stationary solution. For the third question, we carry out numerical simulations for a Turing pattern, and we show that the configurations of Turing pattern are stable spotted patterns, which resemble a real ecosystem.