We derive models of stochastic differential equations describing predator-prey interactions with cooperative hunting in predators based on a deterministic system proposed by Alves and Hilker. The deterministic model is analyzed first by providing a critical degree of cooperation below which the predators go extinct globally. Above the critical threshold, the deterministic model has two coexisting steady states and predators may persist depending on initial conditions. One of the stochastic models is derived from a continuous-time Markov chain while the other is based on a mean reverting process. Using Euler-Maruyama approximations, we investigate the stochastic systems numerically by providing estimated probabilities of predator extinction in the parameter regimes for which the predators cooperate intensively. It is found that predators may go extinct in the stochastic setting when they can otherwise survive indefinitely in the deterministic setting. The estimated probabilities of extinction are overall larger if populations are oscillating in the ODE system.
- Continuous-time Markov Chains: Itø Differential Equations
- Cooperative Hunting