Dynamic Interplay: Cooperative Hunting Strategies in a Predator-Prey Model with Smith Growth Dynamics

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 \(a-K\) $a-K$ 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.