Effect of prey-taxis and diffusion on positive steady states for a predator-prey system

In this paper, we consider a generalized predator-prey system with prey-taxis under the Neumann boundary condition. We investigate the local and global asymptotical stability of constant steady states (including trivial, semitrivial, and interior constant steady states). On the basis of a priori estimate and the fixed-point index theory, several sufficient conditions for the nonexistence/existence of nonconstant positive solutions are given.