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In this paper, we consider the dynamics of a predator-prey system of Gausetype with cooperative hunting among predators and Holling III functionalresponse. The known work numerically shows that the system exhibits saddle-nodeand Hopf bifurcations except homoclinic bifurcation for some special parametervalues. Our results show that there are a weak focus of multiplicity three anda degenerate equilibrium with double zero eigenvalues (i.e., a cusp ofcodimension two) for general parameter conditions and the system can exhibitvarious bifurcations as perturbing the bifurcation parameters appropriately,such as the transcritical and the pitchfork bifurcations at the degenerateboundary equilibrium, the saddle-node and the Bogdanov-Takens bifurcations atthe degenerate positive equilibrium and the Hopf bifurcation around the weakfocus. The comparative study demonstrates that the dynamics are far richer andmore complex than that of the system without cooperative hunting amongpredators. The analysis results reveal that appropriate intensity ofcooperative hunting among predators is beneficial for the persistence ofpredators and the diversity of ecosystem.

Bifurcations of a predator-prey system with cooperative hunting and Holling III functional response