In this study, we examine the dynamics of a discrete-time predator–prey system with prey refuge. We discuss the stability prerequisite for effective fixed points. The existence criteria for period-doubling (PD) bifurcation and Neimark–Sacker (N–S) bifurcation are derived from the center manifold theorem and bifurcation theory. Examples of numerical simulations that demonstrate the validity of theoretical analysis, as well as complex dynamical behaviors and biological processes, include bifurcation diagrams, maximal Lyapunov exponents, fractal dimensions (FDs), and phase portraits, respectively. From a biological perspective, this suggests that the system can be stabilized into a locally stable coexistence by the tiny integral step size. However, the system might become unstable because of the large integral step size, resulting in richer and more complex dynamics. It has been discovered that the parameter values have a substantial impact on the dynamic behavior of the discrete prey–predator model. Finally, to control the chaotic trajectories that arise in the system, we employ a feedback control technique.

中文翻译：

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具有猎物避难所的离散捕食者-被捕食模型的动力学和控制：Holling I 型功能响应

在这项研究中，我们研究了带有猎物避难所的离散时间捕食者-被捕食者系统的动力学。我们讨论有效不动点的稳定性先决条件。倍周期（PD）分岔和内马克-萨克（N-S）分岔的存在准则源自中心流形定理和分岔理论。证明理论分析以及复杂动力学行为和生物过程有效性的数值模拟示例分别包括分岔图、最大李雅普诺夫指数、分形维数 (FD) 和相图。从生物学角度来看，这表明系统可以通过微小的积分步长稳定到局部稳定共存。然而，由于积分步长较大，系统可能会变得不稳定，从而导致更丰富、更复杂的动力学。人们发现参数值对离散捕食者模型的动态行为有很大影响。最后，为了控制系统中出现的混沌轨迹，我们采用了反馈控制技术。