ABSTRACT In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.

中文翻译：

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离散时间捕食者-猎物模型的长期动力学：平衡、循环和混沌的稳定性

摘要在 [AS Ackleh、MI Hossain、A. Veprauskas 和 A. Zhang，离散时间捕食者-猎物模型的持久性和稳定性分析：种群和进化动力学研究，J. Differ。设备 应用程序 25 (2019), pp. 1568–1603.]，我们为两个离散时间捕食者-猎物模型（一个没有和有抵抗毒物的进化）建立了内部平衡的持久性和局部渐近稳定性的条件。在当前的论文中，我们对这些模型进行了更深入的分析，包括均衡的全局稳定性、循环和混沌的存在。我们的主要重点是研究进化速度 ν 如何影响种群动态。对于这两种模型，我们使用微扰分析和李雅普诺夫函数的构造建立了内部平衡是全局渐近稳定的条件。对于小 ν，我们表明进化系统的全局动力学只不过是非进化系统的连续扰动。然而，当进化速度增加时，我们进行了数值研究，这表明进化可能会引入丰富的动力学，包括在没有进化时观察到的循环和混沌行为。