The Threshold of Survival: Transcritical Bifurcation and Regime Shifts in Holling Type II Predator-Prey Systems

This research provides a rigorous mathematical investigation into the structural stability and qualitative transitions of a two-dimensional prey-predator system incorporating a Holling Type II functional response. While classical models often yield neutrally stable dynamics, the integration of predator satiation and handling time introduces complex nonlinearities that more accurately reflect empirical biological constraints. The primary focus of this study is the formal characterization of the transcritical bifurcation as a mechanism for species persistence and extinction. By isolating the predator mortality rate (d) as the master control parameter, we identify a critical threshold dc = \(\frac{aek}{1+hk}\) that dictates a fundamental stability exchange between the prey-only boundary equilibrium E1(k, 0) (with eigenvalues λ1 = −r and λ2 = e \(\frac{ak}{1+hk}\) − d) and the interior coexistence state E∗(x∗, y∗). We employ Sotomayor’s Theorem to analytically verify the transversality and non-degeneracy conditions of this bifurcation, providing a robust proof of the system’s topological regime shifts. While Hopf bifurcations and oscillatory dynamics frequently occur in such systems, our study explicitly focuses on the extinction-invasion boundary. Our analytical findings are corroborated by extensive numerical simulations, including time-series panels and phase portraits, which illustrate the catastrophic collapse of predator populations when environmental stressors push mortality beyond the calculated threshold. This work fills a significant gap in the literature by prioritizing the extinction-invasion gateway over oscillatory dynamics, offering actionable quantitative insights for biodiversity conservation and the management of fragile ecosystems.