Thresholds for noise-induced instability in a three-patch predator-prey fishery with prey migration

Abstract

Understanding how environmental fluctuations affect multi-patch predator-prey systems is critical for sustainable fisheries management. While deterministic and stochastic models have been studied extensively in single-patch or two-patch settings, the combined effect of prey migration and additive white noise on population persistence in a three-patch ecosystem remains largely unexplored. Here we develop a stochastic predator-prey model for three interconnected patches (cages) where prey fingerlings migrate among patches. We derive conditions for stochastic stability using a Lyapunov function and show that the system is stable when the predator conversion efficiency ei < 1 and unstable when ei > 1. Numerical simulations using the Euler-Maruyama scheme reveal clear thresholds: prey populations remain viable only when harvesting rates are maintained below νi = 0.02 per patch and noise intensities are kept within the range σ ≈ 0.10–0.70. Above these thresholds, population oscillations become erratic and lead to extinction. Our findings provide quantitative guidelines for setting harvesting quotas and acceptable environmental variability windows in multipatch fisheries, highlighting the importance of managing both exploitation and stochastic disturbances concurrently.