  We consider the local bifurcation and global dynamics of a predator-prey
model with cooperative hunting and Allee effect. For the model with weak
cooperation, we prove the existence of limit cycle, heteroclinic cycle at a
threshold of conversion rate $p=p^{\#}$. When $p>p^{\#}$, both species go
extinct, and when $p<p^{\#}$, there is a separatrix. The species with initial
population above the separatrix finally become extinct; otherwise, they coexist
or oscillate sustainably. In the case with strong cooperation, we exhibit the
complex dynamics of system in three different cases, including limit cycle,
loop of heteroclinic orbits among three equilibria, and homoclinic cycle.
Moreover, we find diffusion may induce Turing instability and Turing-Hopf
bifurcation, leaving the system with spatially inhomogeneous distribution of
the species, coexistence of two different spatial-temporal oscillations.
Finally, we investigate Hopf and double Hopf bifurcations of the diffusive
system induced by two delays.
