Dynamics of a ratio-dependent predator-prey system with a strong Allee effect

A ratio-dependent predator-prey model with a strong Allee effectin prey is studied. We show that the model has a Bogdanov-Takensbifurcation that is associated with a catastrophic crash of thepredator population. Our analysis indicates that an unstable limitcycle bifurcates from a Hopf bifurcation, and it disappears due toa homoclinic bifurcation which can lead to different patterns ofglobal population dynamics in the model. We study the heteroclinicorbits and determine all possible phase portraits when theBogdanov-Takens bifurcation occurs. We also provide the conditionsfor nonexistence of limit cycle under which the global dynamics ofthe model can be determined.