In this paper, we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting. First, we discuss the existence and stability of all possible equilibria. Next, we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter. Furthermore, we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem. In addition, we give the optimal harvesting strategy of the delayed system based on Pontryagin’s maximum principle with delay. Finally, some numerical simulations are carried out to verify our theoretical results.

中文翻译：

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具有恐惧效应和线性收获的延迟猎物-捕食者-清道夫系统的动力学行为

在本文中，我们提出了一种具有恐惧效应和线性收获的延迟猎物 - 捕食者 - 清道夫系统。首先，我们讨论所有可能的平衡的存在和稳定性。接下来，我们通过将清除剂的孕育期视为分岔参数来研究延迟系统的 Hopf 分岔的存在。此外，我们利用范式理论和中心流形定理获得了Hopf分岔的方向和分岔周期解的稳定性。此外，我们基于Pontryagin的最大延迟原理给出了延迟系统的最优收获策略。最后，进行了一些数值模拟来验证我们的理论结果。