Complex Dynamics of a Predator–Prey System With Fear Effect in Prey

This study investigates the dynamical consequences of predator-induced fear in a predator–prey system by integrating nonlinear fear effects, intraspecific competition, and Holling type II predation. We analyze both continuous- and discrete-time formulations to capture a wide spectrum of ecological dynamics. The model exhibits rich behaviors, including stable coexistence, bistability, oscillations, and chaotic dynamics, revealing how fear can fundamentally reshape population interactions. Analytical results establish well-posedness and persistence, while numerical bifurcation analysis demonstrates that variations in key ecological parameters can trigger qualitative transitions such as oscillatory outbreaks and complex irregular fluctuations. Global sensitivity analysis identifies the dominant parameters controlling long-term population densities, thereby quantifying system robustness. To account for environmental variability, a stochastic extension with multiplicative noise is examined, showing that moderate fluctuations preserve coexistence, whereas strong disturbances may destabilize populations. Discrete-time analysis further uncovers period-doubling routes to chaos and intricate bifurcation structures. Overall, the results highlight predator-induced fear as a critical regulatory mechanism that can either stabilize or destabilize ecosystems, offering important ecological insights into population management and the prediction of complex biological dynamics.