Bifurcation analysis in a predator-prey model with strong Allee effect on prey and density-dependent mortality of predator

The influences of Allee effect and density-dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause-type predator-prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density-dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov-Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle-node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density-dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density-dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically.