Abstract: The behavior of cooperation is common in animals. A lot of work showed that cooperative predation in predator-prey systems will induce complicated dynamics such as complicated coexistence of equilibria and population oscillations. In this paper, we consider a predator-prey system with cooperative breeding of predators, and focus on the oscillations related to the Hopf bifurcations of the model. Firstly, we discuss the number of internal equilibrium and the qualitative properties of internal equilibrium, and show that the equilibrium ~$E_{1}$~ is of center type under certain parameter conditions. Secondly, we analyze the Hopf bifurcation at $E_{1}$, and give the corresponding bifurcation conditions. Finally, we give an example of a weak focus of order 3, and study the complex oscillation behaviors of the model through this example. Because of the coexistence of multiple limit cycles, even under the same parameter conditions, the periods and amplitudes vary with initial values. It also reflects the high sensitivity of solutions on initial values.

Key words: Hopf bifurcation, Oscillation, Cooperative breeding