Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response

In this paper, we study the global dynamics of a density-dependent predator-prey system with ratio-dependent functional response. The main fea- tures and challenges are that the origin of this model is a degenerate equilibrium of higher order and there are multiple positive equilibria. Firstly, local quali- tative behavior of the system around the origin is explicitly described. Then, based on the dynamics around the origin and other equilibria, global qualitative analysis of the model is carried out. Finally, the existence of Bogdanov-Takens bifurcation (cusp case) of codimension two is analyzed. This shows that the system undergoes various bifurcation phenomena, including saddle-node bifur- cation, Hopf bifurcation, and homoclinic bifurcation along with di_erent topo- logical sectors near the degenerate origin. Numerical simulations are presented to illustrate the theoretical results.