Qualitative Analysis of a Predator-prey Model with Weak Allee Effect

Abstract

In the past decades, the research of reaction-diffusion equations has been one of the dominant themes due to its universal existence and importance. The applications of reaction-diffusion equations play an active role in physics, chemistry, biological populations and medicine. In various ecological systems, the predator- prey model is an important branch. In this paper, a predator-prey reaction-diffusion system with weak Allee effect and homogeneous Neumann boundary condition is considered. First, it is proved that the unique positive constant steady state is stable, and 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. Also, we found that system has no non-constant positive solution when the diffusion coefficients meeting certain conditions. Finally, we establish the existence of positive solutions to the system mainly using the fixed point index theory in cone. 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.