Recent studies demonstrate that the density of prey population is not only affected by direct killing by the predator, but the fear in prey caused by predator also reduces it by cutting down the reproduction rate of prey community, and prey shows anti-predator behavior in response to this fear. In this study, we propose a prey–predator model with fear in prey due to predator and anti-predator behavior by prey against the predator with fear response delay and gestation delay. It is assumed that the predator consumes prey via simplified Holling Type-IV functional response. We evaluate the equilibrium points and study the local and global stability behavior of the system around them. It is observed that our system undergoes Hopf-bifurcation with respect to the fear parameter. Moreover, the system shows the attribute of bi-stability involving two stable equilibriums. Further, we study the dynamics of the delayed system by incorporating fear response delay and gestation delay. We observe that the delayed system suffers Hopf-bifurcation with respect to both delays. Using the normal form method and center manifold theory, the direction and stability of Hopf-bifurcation are studied. Chaotic behavior for delayed system is observed for large values of fear response delay. All these findings are supported by numerical simulation.

最近的研究表明，捕食者的密度不仅受到捕食者直接杀死的影响，而且捕食者引起的对猎物的恐惧也通过降低猎物群落的繁殖率而降低了，并且猎物表现出反捕食行为对这种恐惧。在这项研究中，我们提出了一个捕食者-捕食者模型，该模型具有捕食者和反捕食者行为的猎物对被捕食者的恐惧，它们具有恐惧反应延迟和孕育延迟。假定捕食者通过简化的Holling IV型功能响应来消耗猎物。我们评估平衡点并研究围绕它们的系统的局部和全局稳定性行为。可以看出，关于恐惧参数，我们的系统经历了Hopf分支。而且，系统显示了涉及两个稳定平衡的双稳定性属性。此外，我们通过结合恐惧反应延迟和妊娠延迟来研究延迟系统的动力学。我们观察到，相对于两个延迟，该延迟系统都遭受Hopf分支。利用范式和中心流形理论，研究了Hopf分支的方向和稳定性。对于较大的恐惧响应延迟，可以观察到延迟系统的混沌行为。所有这些发现均得到数值模拟的支持。研究了Hopf分支的方向和稳定性。对于较大的恐惧响应延迟，可以观察到延迟系统的混沌行为。所有这些发现均得到数值模拟的支持。研究了Hopf分支的方向和稳定性。对于较大的恐惧响应延迟，可以观察到延迟系统的混沌行为。所有这些发现均得到数值模拟的支持。