Turing instability and stationary patterns in a predator-prey systems with nonlinear cross-diffusions

In this paper, we study a strongly coupled reaction-diffusion system which describes two interacting species in prey-predator ecosystem with nonlinear cross-diffusions and Holling type-II functional response. By a linear stability analysis, we establish some stability conditions of constant positive equilibrium for the ODE and PDE systems. In particular, it is shown that Turing instability can be induced by the presence of cross-diffusion. Furthermore, based on Leray-Schauder degree theory, the existence of non-constant positive steady state is investigated. Our results indicate that the model has no non-constant positive steady state with no cross-diffusion, while large cross-diffusion effect of the first species is helpful to the appearance of Turing instability as well as non-constant positive steady state (stationary patterns).