Critical transitions in a prey–predator model with Allee effect and habitat complexity: Noise-induced tipping and early warning signals

Understanding how interacting ecological processes influence prey–predator dynamics is crucial for predicting population persistence, anticipating multistability, identifying critical transitions, predicting tipping points, and guiding conservation strategies under environmental fluctuations. We examine a prey–predator model that integrates the Allee effect in the predator population and habitat complexity, which may operate simultaneously. To incorporate the environmental stochasticity, we extend the deterministic framework by introducing multiplicative white noise into the growth rate of the prey and the death rate of the predator. The deterministic model exhibits rich dynamics, including bistability, tristability, and both local and global bifurcations, such as saddle–node, Hopf, Bogdanov–Takens, homoclinic, generalised Hopf, and saddle–node bifurcations of limit cycles. Varying the strengths of the Allee effect and habitat complexity partition the parameter space into six subregions, each with distinct stability properties, including the extinction risk of the predator population. Most importantly, we critically examine how environmental stochasticity drives tipping between two alternative stable states in the system. By analysing the relationships among noise intensity, tipping probability, and tipping time, we also quantify predator extinction risk and assign extinction warning levels for different initial population sizes. For three cases of multistability, the probability of tipping to predator extinction from coexistence generally becomes more likely and faster with increasing noise, while the tipping probability from predator extinction to the coexistence state peaks at intermediate noise levels. Prey noise dominates tipping towards coexistence, whereas predator noise delays the transition. Warning levels vary most near the basin boundary with low noise intensities, although strong prey-dominated noise can alter risks even far from the basin boundary. Extensive numerical simulations further elucidate the influence of the Allee effect strength, habitat complexity, and noise intensity on system dynamics. These results highlight the critical roles of the Allee effect and habitat complexity in guiding conservation strategies to maintain biodiversity and prevent undesirable noise-induced critical transitions in stochastic ecological systems.