Bifurcation analysis of a Leslie-type predator-prey system with prey harvesting and group defense

In this paper, we investigate a Leslie-type predator-prey model that incorporates prey harvesting and group defense, leading to a modified functional response. Our analysis focuses on the existence and stability of the system's equilibria, which are essential for the coexistence of predator and prey populations and the maintenance of ecological balance. We identify the maximum sustainable yield, a critical factor for achieving this balance. Through a thorough examination of positive equilibrium stability, we determine the conditions and initial values that promote the survival of both species. We delve into the system's dynamics by analyzing saddle-node and Hopf bifurcations, which are crucial for understanding the system transitions between various states. To evaluate the stability of the Hopf bifurcation, we calculate the first Lyapunov exponent and offer a quantitative assessment of the system's stability. Furthermore, we explore the Bogdanov-Takens (BT) bifurcation, a co-dimension 2 scenario, by employing a universal unfolding technique near the cusp point. This method simplifies the complex dynamics and reveals the conditions that trigger such bifurcations. To substantiate our theoretical findings, we conduct numerical simulations, which serve as a practical validation of the model predictions. These simulations not only confirm the theoretical results but also showcase the potential of the model for predicting real-world ecological scenarios. This in-depth analysis contributes to a nuanced understanding of the dynamics within predator-prey interactions and advances the field of ecological modeling.