Cooperation and harvesting-induced delays in a predator–prey model with prey fear response: A crossing curves approach

In contrast to solitary hunting, predators can more efficiently capture a larger and challenging prey by dividing the effort, sharing risks, and enhancing reproductive facilitation, which ultimately enhances their hunting success rates. Cooperative hunting though involves a time lag as predators coordinate their positions and adjust their behaviours before initiating the hunt. In predator–prey models, maintaining selective harvesting or a specific time delay is essential for managing the harvest and allowing species to reach a certain age or size threshold. In this work, we look into effects of harvesting-induced delay and cooperation delay within a predator–prey model, considering the presence of predator-induced fear in prey species. We observe that based on the fear level and degree of cooperation, the cooperation delay can result in stability invariance, stability and instability switching, and also the occurrence of degenerate Hopf bifurcation. With fixed harvesting effort, the interior equilibrium point experiences stability change and stability switching, while the predator-free equilibrium point undergoes stability change as harvesting-induced delay increases. In addition, we also notice a bistability between predator-free and interior equilibrium points, a bistability between a stable limit cycle and the predator-free equilibrium point, and the occurrence of a homoclinic orbit, with the variation in harvesting-induced delay. We further examine the simultaneous impact of both harvesting and cooperation delays, and obtain stable and unstable regions in two-delay parametric plane by using the crossing curves approach. Our findings indicate that maintaining controlled hunting cooperation among predators, timely coordination during hunting, regulated selective harvesting of predators, and a significant fear level in prey are essential for the stable coexistence of both species. The rich and complex non-linear dynamics underlines the importance of various factors considered in a predator–prey model.