On a predator–prey system with cross-diffusion representing the tendency of prey to keep away from its predators

In this work, we study a predator–prey system with cross-diffusion, representing the tendency of prey to keep away from its predators, under the homogeneous Dirichlet boundary condition. Using fixed point index theory, we provide some sufficient conditions for the existence of positive steady-state solutions. Furthermore, we investigate the non-existence of positive solutions.