Turing Instability and Pattern Formation in a Strongly Coupled Diffusive Predator-Prey System

We are concerned with the Turing instability and pattern caused by cross-diffusion in a strongly coupled spatial predator-prey system. We explore how cross-diffusion destabilizes the spatially uniform steady state which is stable in reaction-diffusion systems, and explicitly describe the Turing space under certain conditions. Particularly, when the parameter values are taken in the Turing-Hopf domain, in which the spatiotemporal dynamical behavior is influenced by both Hopf and Turing instabilities, we investigate the formation of all possible patterns, including non-Turing structures such as wave pattern, competing dynamics as well as stationary Turing pattern. Furthermore, numerical simulations are illustrated to verify our theoretical findings.