Heteroclinic Bifurcation in the Michaelis-Menten-Type Ratio-Dependent Predator-Prey System

The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratiodependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations. It is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the Hopf bifurcation.