Uniqueness and non-uniqueness of steady states for a diffusive predator-prey-mutualist model with a protection zone

This paper is concerned with the stationary problem for a diffusive Lotka-Volterra predator-prey-mutualist model with a protection zone under homogeneous Neumann boundary conditions. Compared with the case where the mutualist is absent in [12], this paper aims to reveal the effects of mutualism coefficients α and β on the existence, number and stability of steady states. It turns out that when α is large, the model has at most one steady state and it is stable (if it exists); however existence of multiple steady states is examined for large β, moreover asymptotic profiles of steady states are established as β tends to infinity.