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Global dynamics in a predator-prey model with cooperative hunting and Allee effect and bifurcation induced by diffusion and delays

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.