We investigate a new cross-diffusive prey-predator system which considers prey refuge and fear effect, where predator cannibalism is also considered. The prey and predator that partially depends on the prey are followed by Holling type-Ⅱ terms. We first establish sufficient conditions for persistence of the system, the global stability of constant steady states are also investigated. Then, we investigate the Hopf bifurcation of ordinary differential system, and Turing instability driven by self-diffusion and cross-diffusion. We have found that the d12 can suppress the formation of Turing instability, while the d21 promotes the appearance of the pattern formation. In addition, we also discuss the existence and nonexistence of nonconstant positive steady state by Leray-Schauder degree theory. Finally, we provide the following discretization reaction-diffusion equations and present some numerical simulations to illustrate analytical results, which show that the establishment of prey refuge can effectively protect the growth of prey.
Keywords: Hopf bifurcation; Turing instability; cross-diffusion; fear effect; predator cannibalism.