猎物中的群体防御和捕食者中的狩猎合作是两个重要的生态现象,可以同时发生。在这篇文章中,我们在数学框架下考虑通才捕食者的合作狩猎和猎物的群体防御,以理解模型可以捕获的巨大多样性。要做到这一点,我们考虑了改进的Holling-Tanner模型,在该模型中,我们实施了HollingIV型功能响应,以表征捕食者的放牧模式,其中猎物物种表现出群体防御。此外,我们允许修改捕食者的攻击率,以量化它们之间的狩猎合作。该模型允许三个边界均衡和最多三个共存均衡点。不平凡的猎物和捕食者的几何形状以及共存平衡的数量主要取决于捕食者可替代食物的特定阈值。我们使用线性稳定性分析来确定双曲平衡点的类型,并通过正常形式和中心流形理论来表征非双曲平衡点。模型参数的变化导致非双曲平衡点发生一系列局部分叉,即,超临界,鞍形节点,Hopf,尖点和Bogdanov-Takens分叉;也存在全局分叉,例如极限环的同斜分叉和鞍节分叉。我们观察到由于狩猎合作强度的变化和捕食者可替代食物的可获得性,全球分叉引起的两种有趣的封闭“气泡”形式。三维分岔图,关于原始系统参数,捕获模型公式化中的交替如何诱导分叉场景的逐渐变化。我们的模型强调了群体或群居行为在猎物和捕食者中的稳定作用,因此支持捕食者-食草动物调节假说。此外,我们的模型强调了生态系统中“盐分平衡”的发生,并捕获了观察到的狮子-草食动物相互作用的动力学。
Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed \'bubble\' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of \"saltatory equilibria\" in ecological systems and capture the dynamics observed for lion-herbivore interactions.