Abstract This paper investigates a stochastic Markovian switching predator–prey model with infinite memory and general Levy jumps. Firstly, we transfer a classic infinite memory predator–prey model with weak kernel case into an equivalent model through integral transform. Then, for the corresponding stochastic Markovian switching model, we establish the sufficient conditions for permanence in time average and the threshold between stability in time average and extinction. Finally, sufficient criteria for a unique ergodic stationary distribution of the model are derived. Our results show that, firstly, both white noise and infinite memory are unfavorable to the existence of the stationary distribution; secondly, the general Levy jumps could make the stationary distribution vanish as well as happen; finally, the Markovian switching could make the stationary distribution appear.

中文翻译：

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具有无限记忆和一般 Lévy 跳跃的随机马尔可夫切换捕食者-猎物模型的动力学

摘要 本文研究了具有无限记忆和一般 Levy 跳跃的随机马尔可夫切换捕食者-猎物模型。首先，我们通过积分变换将具有弱核情况的经典无限记忆捕食者-猎物模型转换为等效模型。然后，对于相应的随机马尔可夫切换模型，我们建立了时间平均持久性的充分条件和时间平均稳定性与灭绝之间的阈值。最后，推导出模型唯一遍历平稳分布的充分标准。我们的结果表明，首先，白噪声和无限记忆都不利于平稳分布的存在；其次，一般的 Levy 跳跃可以使平稳分布消失和发生；最后，