  In this paper, we consider the dynamics of a predator-prey system of Gause
type with cooperative hunting among predators and Holling III functional
response. The known work numerically shows that the system exhibits saddle-node
and Hopf bifurcations except homoclinic bifurcation for some special parameter
values. Our results show that there are a weak focus of multiplicity three and
a degenerate equilibrium with double zero eigenvalues (i.e., a cusp of
codimension two) for general parameter conditions and the system can exhibit
various bifurcations as perturbing the bifurcation parameters appropriately,
such as the transcritical and the pitchfork bifurcations at the degenerate
boundary equilibrium, the saddle-node and the Bogdanov-Takens bifurcations at
the degenerate positive equilibrium and the Hopf bifurcation around the weak
focus. The comparative study demonstrates that the dynamics are far richer and
more complex than that of the system without cooperative hunting among
predators. The analysis results reveal that appropriate intensity of
cooperative hunting among predators is beneficial for the persistence of
predators and the diversity of ecosystem.
