Global Bifurcation Analysis of a Predator-Prey System With Harvesting and Stocking

Predator-prey interactions play a fundamental role in ecological dynamics and are often influenced by external factors such as harvesting and stocking. Understanding how these interventions impact population stability and oscillatory behavior is crucial for effective resource management and conservation efforts. In this paper, we investigate a two-species Leslie predator-prey model incorporating constant-yield harvesting and stocking terms for both species. Through a detailed bifurcation analysis, we unveil the mechanisms behind the suppression or persistence of multiple oscillations in the system. We demonstrate that the model, under harvesting or stocking, exhibits cusp bifurcations of codimension 2 and 3, leading to a rich sequence of bifurcations, including saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension 2 and 3, and Hopf bifurcation of codimension at least 3. In particular, we focus on the role of the conversion rate and analyze its influence on the system's transition between oscillatory and non-oscillatory coexistence states. Our findings suggest that indiscriminate stocking or harvesting of both species is not necessarily beneficial for population persistence; rather, appropriate management strategies should be designed according to specific parameter conditions. Finally, numerical simulations, including bifurcation diagrams and phase portraits, are presented, which provide valuable insights and potential applications in ecological management and conservation biology.