The motivation of this paper is inspired by the research of Zanette et al. (2011) , which reveals that the perception of predation risk can reduce the number of juvenile prey. To investigate how fear affects the behavior of prey population, we formulate a predator-prey model with a stage structure for prey in which adult prey is undergoing the interference of fear. First, the dynamic properties of the model are analyzed, including the existence and stability of equilibria, bifurcations dynamics, such as transcritical, saddle-node, and Hopf bifurcations. Our results reveal that fear can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. Numerical simulations and theoretical analysis indicated that when the level of fear reaches 25.4, the number of juvenile prey affected by fear decreases by 40% compared to those not affected by fear. Terrifyingly, the predators are on the brink of extinction in high levels of fear. Moreover, we examine the impact of time delay on fear in the previous model as prey needs some time to assess the predation risk. Some dynamic characteristics of this delayed model are also analyzed. Results indicate that the fear effect is conducive to generating a stable state, while delay is detrimental to generating a stable state, leading to limit cycles or even chaos.

中文翻译：

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恐惧诱导和延迟下阶段结构捕食者-被捕食者模型的稳定性、分岔和混沌


本文的动机受到 Zanette 等人的研究的启发。 (2011)，这揭示了捕食风险的感知可以减少幼年猎物的数量。为了研究恐惧如何影响猎物群体的行为，我们制定了一个捕食者-猎物模型，该模型具有猎物的阶段结构，其中成年猎物正在经历恐惧的干扰。首先，分析了模型的动力学特性，包括平衡态的存在性和稳定性、分岔动力学，如跨临界、鞍结和Hopf分岔。我们的结果表明，恐惧可以通过将模型从不稳定状态转变为稳定状态来促进模型稳定性的形成。数值模拟和理论分析表明，当恐惧水平达到25.4时，受恐惧影响的幼年猎物数量比不受恐惧影响的幼年猎物数量减少了40%。可怕的是，掠食者在高度恐惧中濒临灭绝。此外，我们在之前的模型中研究了时间延​​迟对恐惧的影响，因为猎物需要一些时间来评估捕食风险。还分析了该延迟模型的一些动态特性。结果表明，恐惧效应有利于稳定状态的产生，而延迟效应则不利于稳定状态的产生，导致极限环甚至混沌。