The hydra effect in a cannibalistic reaction–diffusion predator–prey model: Influence of space-time white noise on population dynamics

In ecological systems, the Hydra effect –a counterintuitive phenomenon in which higher mortality rates lead to increased population size –has been extensively documented in both theoretical and empirical studies. This manuscript develops a stochastic reaction–diffusion predator–prey model that incorporates space–time white noise to account for stochastic disturbances in ecosystems, in which the predator exhibits cannibalistic behavior. The theoretical analysis derives the conditions under which the Hydra effect occurs in the deterministic model and establishes the well-posedness of mild solutions in the stochastic model. Furthermore, key conditions for population persistence and extinction are identified, demonstrating that high levels of space–time white noise can lead to predator population extinction. Numerical simulations validate the theoretical findings and reveal that increasing environmental noise intensity causes the minimum mortality rate triggering the Hydra effect to first rise and then fall. This finding is consistent with the Intermediate Disturbance Hypothesis, suggesting that moderate environmental disturbances enhance population adaptability and survival strategies, providing significant insights for population dynamics and species management.