Implications of General Functional Response on Stability and Bifurcation in Predator–Prey Systems

This work investigates the dynamical consequences of a nonlinear functional response in a predator–prey system. The model exhibits three equilibria: total extinction, predator extinction, and coexistence. Stability analysis shows that the total extinction equilibrium is inherently unstable, while the predator-extinction equilibrium achieves global asymptotic stability if the functional response is increasing. The coexistence equilibrium, however, can exhibit distinct regimes—local stability, global stability, or oscillatory dynamics via a Hopf bifurcation. A key result demonstrates that a strictly increasing functional response is necessary but insufficient for global stability of the coexistence equilibrium. Conversely, Hopf bifurcation arises under two critical conditions: (1) prey density at coexistence remains below half the carrying capacity, and (2) the functional response’s growth rate at equilibrium does not exceed the prey’s intrinsic growth rate. These findings highlight how the curvature of the functional response governs bifurcation structures and long-term ecological dynamics.