Non-constant Positive Steady State of the Predator-Prey-Mutualist System with Diffusion

The predator-prey model is one of the most famous models in ecology. The research of the predator-prey model has been one of the dominant themes due to its universal existence and importance. And the applications of the predator-prey model play an active role in physics, chemistry, biological populations and medicine. In this paper, a reaction-diffusion predator-prey-mutualist system with the Holling II functional response and homogeneous Neumann boundary condition is considered. First, it is proved that the unique positive constant steady state is stable. The methods to study local stability are based on local linearization techniques. Second, a prior-estimate of positive steady state is given by Harnack inequality and Maximum principle. Finally, the non-existence and the existence of non-constant positive steady state are studied by the degree theorem and bifurcation technique. Through the research, we find out the conditions for the existence of positive steady state, which provide theoretical basis for the improvement of yield predators.