In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. The dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay passes through a sequence of critical values. The explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. Thus, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions.

中文翻译：

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具有恐惧效应和时滞的捕食者-食饵模型的动力学行为。

在本文中，我们提出了一种具有Holling-II型功能性反应的时滞捕食者-捕食者模型，该模型将孕育期和恐惧成本纳入了猎物繁殖中。从稳定性，持久性和分叉的角度，都对该系统的动力学行为进行了分析和数值研究。我们发现存在稳定性开关，并且当延迟通过一系列关键值。通过使用范式理论和中心流形定理，给出了确定分支周期解的方向，稳定性和其他性质的显式公式。我们进行了广泛的数值模拟，以探索一些重要参数对系统动力学的影响。数值模拟表明，高水平的恐惧对稳定者有稳定作用，而相对低水平的恐惧对掠食者与猎物之间的相互作用则有破坏作用，从而导致极限循环振荡。我们还发现，具有或不具有依赖于延迟的因素的模型可以具有显着不同的动力学。因此，忽略延迟或不包括依赖于延迟的因素可能会导致建模预测不准确。