Uniqueness and stability of positive solutions for a diffusive predator–prey model in heterogeneous environment

Abstract

In this paper, we study a spatially heterogeneous predator–prey model where the interaction is governed by a Crowley-Martin type functional response. It is shown that the predator competition in this model plays an extremely important role. A good understanding of the existence, uniqueness and stability of positive solutions is obtained when the predator competition is strong. In particular, we show that when the predator competition is strong, the model has at most one positive steady-state solution for any $$\mu \in {{\mathbb {R}}}$$ , and it is globally asymptotically stable for any $$\mu >0$$ (if it exists). Additionally, we show that when the prey growth rate is over a critical value, the two populations stabilize at a unique coexistence state if the predator growth rate is strong.