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We consider the local bifurcation and global dynamics of a predator-preymodel with cooperative hunting and Allee effect. For the model with weakcooperation, we prove the existence of limit cycle, heteroclinic cycle at athreshold of conversion rate $p=p^{\#}$. When $p>p^{\#}$, both species goextinct, and when $p<p^{\#}$, there is a separatrix. The species with initialpopulation above the separatrix finally become extinct; otherwise, they coexistor oscillate sustainably. In the case with strong cooperation, we exhibit thecomplex 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-Hopfbifurcation, leaving the system with spatially inhomogeneous distribution ofthe species, coexistence of two different spatial-temporal oscillations.Finally, we investigate Hopf and double Hopf bifurcations of the diffusivesystem induced by two delays.

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