Spatiotemporal Dynamics of a Diffusive Predator-Prey System with Allee Effect and Threshold Hunting

In this paper, we study a diffusive predator-prey system with the Allee effect and threshold hunting. First, the number of interior equilibrium points is determined by discussing the relation of parameters. Then, preliminary analysis on the local asymptotic stability and bifurcations of non-spatial system based on ordinary differential equations is presented. It is noted that four stable equilibrium points coexist due to the Allee effect and threshold hunting. The stability of interior equilibrium points and the existence of Turing instability induced by the diffusion, spatially homogeneous and inhomogeneous Hopf bifurcation, Turing-Hopf bifurcation are studied by analyzing the corresponding characteristic equation for spatial system. By constructing generalized Jacobian matrix, we analyze the stability of interior equilibrium point where u-component is equal to the threshold of functional response. These results show that the Allee effect, threshold hunting and diffusion have significant impacts on the dynamics. Last, we present some numerical simulations that supplement the analytic results.