Optimal Harvesting of a Stochastic Three-Species Predator–Prey Model with Multiplicative Noise

This paper addresses the optimal harvesting of a stochastic three-species predator–prey system with multiplicative noise and harvesting efforts. We first establish sufficient conditions for the persistence and extinction of each species and derive threshold-type criteria in terms of harvesting efforts. Under suitable competition coefficient conditions, we prove that the system is asymptotically stable in distribution. We then characterize the feasible harvesting region and, by applying the ergodic method, obtain the explicit optimal harvesting effort and the corresponding maximum expectation of sustainable yield when the feasibility condition holds; otherwise, we prove that no interior optimal harvesting strategy exists, show that any optimal strategy (if it exists) must lie on the boundary of the feasible region, and provide the corresponding necessary boundary KKT conditions. Numerical simulations validate the theoretical results and illustrate the effect of noise intensity on the existence of the optimal strategy and sustainable yield.