Dynamics of a Prey-Predator Model with Group Defense for Prey, Cooperative Hunting for Predator, and Levy Jump

A stochastic predator-prey system with group cooperative behavior, white noise, and Levy noise is considered. In group cooperation, we introduce the Holling IV interaction term to reflect group defense of prey, and cooperative hunting to reflect group attack of predator. Firstly, it is proved that the system has a globally unique positive solution. Secondly, we obtain the conditions of persistence and extinction of the system in the sense of time average. Under the condition that the environment does not change dramatically, the intensity of cooperative hunting and group defense needs to meet certain conditions to make both predators and preys persist. In addition, considering the system without Levy jump, it is proved that the system has a stationary distribution. Finally, the validity of the theoretical results is verified by numerical simulation.