In this paper, we analyze two stochastic predator–prey models with distributed delay and stage structure for prey. For the nonautonomous periodic case of the model, by using Khasminskii’s theory of periodic solution, we show that the system has at least one positive T-periodic solution. For the model which is disturbed by both white and telegraph noises, we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The positive recurrence implies that both prey and predator populations will be persistent in the long term.

中文翻译：

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具有分布式延迟和猎物阶段结构的随机捕食者-猎物模型动力学

在本文中，我们分析了两种具有分布式延迟和猎物阶段结构的随机捕食者 - 猎物模型。对于模型的非自治周期情况，利用 Khasminskii 的周期解理论，我们证明了系统至少有一个正吨- 周期性解决方案。对于同时受到白噪声和电报噪声干扰的模型，我们通过构造具有状态切换的合适的随机 Lyapunov 函数来获得模型解的正递归的充分标准。正复发意味着猎物和捕食者种群将长期存在。