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Keywords:
predator-prey system; fear effect; hunting cooperation; mathematical model; bifurcation; saddle point
predator-prey system; fear effect; hunting cooperation; mathematical model; bifurcation; saddle point
Summary:
This study investigates a predator-prey model that incorporates fear-induced responses in prey along with cooperative hunting behavior among predators. To describe the dynamics of the model, we show that there is a positively invariant region, where the trajectories are bounded, and that the origin is a hyperbolic repeller. We derive necessary and sufficient conditions for the existence and explicit characterization of up to two coexistence equilibria. One equilibrium is always a hyperbolic saddle, whereas the other may act as an attractor, a repeller, or a weak focus. The system can experience bifurcations, including transcritical and Hopf bifurcations, which have an impact on the model's dynamics. Our system crucially depends on the growth rate of prey, i.e. whether the system showcases extinction to coexistence to cyclic dynamics. The system always shows the stable coexistence for lower strength of hunting cooperation and the stable cyclic dynamics for higher strength of hunting cooperation. The bistability appears in the system for certain ranges of prey's growth rate and for hunting cooperation levels as well. Moreover, time delays in the fear response of prey and cooperative hunting are also taken into consideration to examine a more realistic scenario. Our findings demonstrate that both time delays stabilize the system initially by eliminating limit cycle oscillations; for larger time delay magnitudes, stability switching, and chaotic behaviors are seen. The results imply that comprehending the effects of cooperative hunting and prey's fear reaction may be essential for the predator-prey system. Finally, the analytical results are demonstrated to be supported by numerical simulations.
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