Stationary Patterns Caused by Cross-Diffusion for a Three-Species Prey-Predator Model

We study a system arising from a three-component prey-predator model with prey- dependent and ratio-dependent functional responses, where cross-diffusion is included in such a way that predator chases the prey and the prey runs away from the predator. We prove that the cross- diffusion can generate the stationary patterns (nonconstant positive steady states).