In this research paper, we delve into the critical examination of interactions between predators and prey within ecological systems. Drawing on principles from theoretical biology and mathematical ecology, our study introduces a predator-prey model that incorporates the Smith growth rate of prey and Holling type II functional response, while uniquely considering the influence of hunting cooperation between predators. We rigorously explore various aspects such as positivity, boundedness, local stability, and different bifurcation types, including transcritical, saddle-node, Hopf, cusp, and Bogdanov-Takens bifurcation. Both theoretical analyses and graphical representations are employed to offer a comprehensive understanding of the system dynamics. To unravel the complexity of the interactions, we present two-parametric bifurcation diagrams, utilizing a and K as bifurcation parameters. The first quadrant of the \(a-K\) plane is systematically partitioned into seven regions, demarcated by distinct bifurcation curves. Bistable dynamics are exhibited in the ecological system. A notable observation stemming from our investigation is the vulnerability of the predator population to extinction at low carrying capacity values. Furthermore, our numerical findings underscore the enhanced stability of the system with an increase in hunting cooperation. This research contributes valuable insights into the intricate dynamics of predator-prey interactions, shedding light on the significance of ecological parameters and cooperative behaviors in shaping the stability and persistence of these systems.

在这篇研究论文中，我们深入研究了生态系统中捕食者和猎物之间相互作用的严格检验。我们的研究借鉴理论生物学和数学生态学的原理，引入了捕食者-被捕食者模型，该模型结合了猎物的史密斯增长率和霍林II型功能反应，同时独特地考虑了捕食者之间狩猎合作的影响。我们严格探索各个方面，例如正性、有界性、局部稳定性和不同的分岔类型，包括跨临界、鞍结、Hopf、尖点和 Bogdanov-Takens 分岔。采用理论分析和图形表示来提供对系统动力学的全面理解。为了阐明相互作用的复杂性，我们提出了双参数分岔图，利用a和K作为分岔参数。\(aK\)平面的第一象限被系统地划分为七个区域，由不同的分叉曲线划分。生态系统中表现出双稳态动力学。我们的调查得出的一个值得注意的观察结果是，在低承载能力值下，捕食者种群很容易灭绝。此外，我们的数值研究结果强调了随着狩猎合作的增加，系统的稳定性增强。这项研究为捕食者与猎物相互作用的复杂动态提供了宝贵的见解，揭示了生态参数和合作行为在塑造这些系统的稳定性和持久性方面的重要性。