Hopf Bifurcation as a Driver of Oscillatory Dynamics in Predator-Prey Systems with Holling Type II Response

Abstract

This study addresses a key challenge in ecological modelling: accurately predicting population dynamics in predator-prey systems where predator saturation and nonlinear responses are present. Traditional Lotka-Volterra models fail to account for realistic predator behaviour, particularly the limiting effects of prey handling time, which can distort predictions of ecosystem stability. To resolve this, a modified predator-prey system incorporating a Holling type II functional response is analysed to investigate the conditions under which Hopf bifurcation emerges. The central goal is to determine the precise parameter thresholds that trigger transitions from stable equilibrium to sustained oscillations or chaotic dynamics. These transitions represent critical phenomena in real ecosystems. Using centre manifold and normal form theory, analytical conditions are derived for the occurrence and classification of Hopf bifurcation as supercritical or subcritical. The results demonstrate that predator mortality, saturation levels, and the predator-prey interaction coefficient play decisive roles in shaping long-term population behaviour. Unlike general models, the proposed approach proves particularly effective when system dynamics are governed by nonlinear interactions and time delays, such as in ecosystems with delayed predator responses or environmental fluctuations. Numerical simulations validate the analytical findings and reveal specific regimes where oscillatory or chaotic population cycles emerge. This nuanced understanding offers a practical tool for predicting ecological transitions and supports the development of targeted biodiversity management strategies. Thus, the study contributes a robust analytical-numerical framework for identifying and classifying bifurcations in nonlinear ecological systems subject to stochastic and temporal effects.