Qualitative analysis for a diffusive predator–prey model

In this paper we study the qualitative properties of a diffusive predator–prey model subject to the homogeneous Neumann boundary condition. By a comparison argument and iteration technique, under some hypotheses we prove that the positive constant steady state is globally asymptotically stable. We also establish the existence and non-existence of non-constant positive steady states (stationary patterns) by use of the degree theory and the a priori estimates.