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.

中文翻译：

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捕食者 - 猎物互动中的随机性和合作狩猎

我们基于 Alves 和 Hilker 提出的确定性系统推导了描述捕食者 - 猎物相互作用与捕食者合作狩猎的随机微分方程模型。首先通过提供临界合作程度来分析确定性模型，低于该临界程度的捕食者在全球范围内灭绝。在临界阈值之上，确定性模型具有两个共存的稳态，捕食者可能会根据初始条件持续存在。其中一个随机模型源自连续时间马尔可夫链，而另一个基于均值回归过程。使用 Euler-Maruyama 近似，我们通过在捕食者密切合作的参数方案中提供捕食者灭绝的估计概率来数值研究随机系统。发现捕食者可能会在随机环境中灭绝，否则它们可以在确定性环境中无限期地生存。如果种群在 ODE 系统中振荡，则估计的灭绝概率总体上会更大。