Stochastic fluctuations drive populations with a strong Allee effect and discontinuous threshold harvesting to inevitable extinction

We study a stochastic population model that incorporates a strong Allee effect, a discontinuous threshold harvesting policy, and multiplicative environmental noise. The drift of the stochastic differential equation has a jump at the harvest threshold, which precludes standard existence–uniqueness arguments. We prove that the equation nevertheless admits a unique global strong solution that stays positive almost surely. Our main result establishes that environmental stochasticity alone forces the population to extinction with probability one for any choice of the harvest threshold, harvest intensity, and initial population size. A comparison with a harvest-free auxiliary process and a scale-function analysis reveals that the Allee threshold acts as a tipping point beyond which an irreversible, harvest-accelerated decline unfolds. Numerical simulations confirm the theoretical results and illustrate the inevitable departure from the deterministic safe region. The findings indicate that threshold-based harvesting is fundamentally fragile under environmental variability for populations subject to a strong Allee effect.