Steady states of a predator–prey model with prey-taxis

We study the steady states of a predator–prey model with prey-taxis incorporating Holling type II functional response under the homogeneous Neumann boundary condition. The stability of equilibrium points and the existence of non-constant steady states are investigated. We obtain that the prey-tactic sensitivity coefficient delays the stability of the unique positive constant solution, but for other equilibrium points’ stability, the prey-tactic sensitivity coefficient does not influence on it. Furthermore, we derive some sufficient conditions relative to the prey-tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large, there also exist non-constant positive steady states under some conditions.