Abstract Dynamics of predator-prey systems under the influence of cooperative hunting among predators and the fear thus imposed on the prey population is of great importance from ecological point of view. The role of hunting cooperation and the fear effect in the predator-prey system is gaining considerable attention by the researchers recently. But the study on combined effect of hunting cooperation and fear in the predator-prey system is not yet studied. In the present paper, we investigate the impact of hunting cooperation among predators and predator induced fear in prey population by using the classical predator-prey model. We consider that predator populations cooperate during hunting. We also consider that hunting cooperation induces fear among prey, which has far richer and complex dynamics. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induced fear may destabilize the system and produce periodic solution via Hopf-bifurcation. The stability of the Hopf-bifurcating periodic solution is obtained by computing the Lyapunov coefficient. The limit cycles thus obtained may be supercritical or subcritical. We also observe that the system undergoes the Bogdanov-Takens bifurcation in two-parameter space. Further, we observe that the system exhibits backward bifurcation between predator-free equilibrium and coexisting equilibrium. The system also exhibits two different types of bi-stabilities due to subcritical Hopf-bifurcation (between interior equilibrium and stable limit cycle) and backward bifurcation (between predator-free and interior equilibrium points). Further, we observe strong demographic Allee phenomenon in the system. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MATCONT softwares.

中文翻译：

---

捕食者-猎物模型中狩猎合作和恐惧的影响

摘要 从生态学的角度来看，在捕食者之间合作狩猎的影响下捕食者-猎物系统的动态以及因此对猎物种群施加的恐惧具有重要意义。狩猎合作的作用和捕食者-猎物系统中的恐惧效应最近引起了研究人员的极大关注。但尚未研究捕食者-猎物系统中狩猎合作和恐惧的联合效应。在本文中，我们使用经典的捕食者-猎物模型研究了捕食者之间的狩猎合作以及捕食者对猎物种群的恐惧的影响。我们认为捕食者种群在狩猎期间会合作。我们还认为狩猎合作会引起猎物的恐惧，而猎物的动态要丰富得多。我们观察到，如果没有狩猎合作，唯一的共存平衡点是全局渐近稳定的。然而，增加狩猎合作引起的恐惧可能会破坏系统的稳定性并通过 Hopf 分叉产生周期性解决方案。Hopf 分岔周期解的稳定性是通过计算李雅普诺夫系数获得的。由此获得的极限循环可以是超临界或亚临界的。我们还观察到系统在两参数空间中经历了 Bogdanov-Takens 分岔。此外，我们观察到该系统在无捕食者平衡和共存平衡之间表现出向后分叉。由于亚临界 Hopf 分岔（内部平衡和稳定极限循环之间）和后向分岔（无捕食者和内部平衡点之间），该系统还表现出两种不同类型的双稳定性。此外，我们在系统中观察到强烈的人口统计学 Allee 现象。为了可视化系统的动态行为，使用 MATLAB 和 MATCONT 软件进行了大量的数值实验。