With both hunting cooperation and Allee effects in predators, a predator–prey system was modeled as a planar cubic differential system with three parameters. The known work numerically plots the horizontal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. Transitions among those cases with the rise of limit cycle and homoclinic loop were exhibited by numerical simulations. Although it is hard to obtain the explicit expression of coordinates, in this paper, we give the distribution of equilibria qualitatively, discuss all cases of coexisting equilibria, and obtain the Bogdanov–Takens bifurcation diagram to show analytical parameter conditions for those transitions. Our results give analytical conditions for not only the observed saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation but also the transcritical and pitchfork bifurcations at the predator-extinction equilibrium, which were not considered in the known work. Our analytic conditions provide a quantitative instruction to reduce the risk of predator extinction and promote the ecosystem diversity.

中文翻译：

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捕食者中具有狩猎合作和等位效应的捕食者-猎物模型的动力学

结合捕食者的狩猎合作和 Allee 效应，将捕食者 - 猎物系统建模为具有三个参数的平面立方微分系统。The known work numerically plots the horizo​​ntal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. 数值模拟显示了随着极限环和同宿环上升的这些情况之间的转变。虽然很难得到坐标的明确表达，但在本文中，我们定性地给出了平衡分布，讨论了所有共存平衡的情况，并获得了 Bogdanov-Takens 分岔图来显示这些跃迁的解析参数条件。我们的结果不仅为观察到的鞍节点分叉提供了分析条件，Hopf 分叉和同宿分叉，以及捕食者灭绝平衡处的跨临界和干草叉分叉，这些在已知工作中没有考虑。我们的分析条件为降低捕食者灭绝的风险和促进生态系统多样性提供了定量指导。