Bifurcation Analysis and Bistability in a Predator-Prey Model with Cosner-Type Functional Response and Michaelis-Menten Prey Harvesting

Abstract

Long-term population survival in a predator-prey model often depends on how a species responds to predation and harvesting within its ecosystem. To capture these dynamics, this research investigates a two-dimensional predator-prey model incorporating a Cosner-type functional response and Michaelis-Menten-type nonlinear prey harvesting. The model reflects cooperative hunting behaviour among predators and the saturation effect in prey harvesting, which can significantly influence population dynamics. The model possesses three types of equilibria, namely extinction, predator-free and coexistence equilibria. Bistability is observed at a high harvesting saturation rate, indicating that initial conditions play a crucial role in determining the long-term outcomes for both species. The local stability of each equilibrium point is analysed using the Jacobian matrix and Routh-Hurwitz criterion. Bifurcation analysis is performed on the system using Sotomayor's theorem and the Hopf bifurcation theorem. Numerical bifurcation analysis is carried out by plotting bifurcation diagrams and time series using MATLAB and MatCont software packages. It is found that, as the harvesting saturation rate increases, the model undergoes several codimension-one bifurcations, including saddle-node, pitchfork and Hopf bifurcations. The results show that a high harvesting saturation rate enhances the likelihood of coexistence, whereas a low harvesting saturation rate eventually leads to the extinction of both populations. These results highlight the importance of controlling the harvesting saturation rate to ensure long-term coexistence and prevent population extinction.