Environmental Stochasticity Reshapes Persistence and Extinction Dynamics in a Fear-Mediated Two-Species Competitive System

Environmental variability can strongly alter coexistence among competing species and their extinction risk, particularly when population dynamics are shaped by behavioral interactions, such as fear. In this work, we develop a novel stochastic differential equation competition model that incorporates both non-consumptive fear effects and environmental variability to investigate how behavioral interactions influence species coexistence under random fluctuations. Our result reveals that environmental stochasticity can drive species to extinction even when the corresponding deterministic system admits coexistence. In particular, under an explicit stability condition on the fear and competition parameters and sufficiently strong averaged noise intensities, we prove that both competing species become extinct exponentially almost surely. Conversely, we derive a stochastic persistence criterion in terms of fear, competition, and noise-induced suppression parameters for the fearful species. We further demonstrate that environmental noise may reverse classical competition-exclusion outcomes, leading to qualitatively different long-term dynamics from those predicted deterministically. These results provide rigorous thresholds separating stochastic extinction from persistence and highlight the critical role of environmental variability in fear-mediated competitive ecosystems. From an applied perspective, these results provide insight into how behavioral interactions and environmental variability influence species survival, with potential applications in ecological management and conservation.