Qualitative behavior of a diffusive predator–prey–mutualist model

Abstract

In this paper, we study a reaction-diffusion system under homogeneous Dirichlet boundary conditions that describes the evolution of population densities of a mutualist–prey species u, a predator species v and a mutualist species w. Firstly, stability properties of the trivial and semi-trivial solutions are determined completely. It is found that when the (local) stability of trivial and semi-trivial solutions change, positive stationary solutions appear and the appropriate expressions of these solutions are derived. Furthermore, we show that for large $\gamma$, there is at most one positive stationary solution for each fixed $b\in \mathbb {R}$, moreover it is asymptotically stable (if it exists). Our results indicate that the dynamics of the predator–prey–mutualist system is much complicated than that of the classic predator–prey system.