Impacts of time delay in a bistable predator-prey system

The effects of time delay on dynamics of a predator-prey system with multiple coexisting equilibria are explored. The system exhibits focus-node and cycle-node bistability in the absence of delay. The delay-induced stability and bifurcations of the coexisting equilibria, and the evolution of the bistability are analyzed. Criteria for different delay-driven stability scenarios including stability and instability switches are derived. Our investigation indicates that the delay controls the bistability through different bistable modes. For focus-node bistability, the system evolves through the bistable modes: focus-node. focus-focus -> focus-cycle as the delay grows. On the other hand, we obtain two different scenarios, viz., cycle-node -> cycle-focus and cycle-node -> cyclefocus -> cycle-cycle for the effect of delay on the cycle-node bistability. We report the existence of a homoclinic bifurcation for transition from bistable to monostable dynamics. Furthermore, it is also revealed by computing mean density that due to bistability, time delay can be beneficial or harmful for biological conservation of the populations.