Sustainable and Optimal Harvesting in a Seasonally Harvested Fishery with a Marine Protected Area: A Two-Patch Model with Bang-Bang and Singular Control

Abstract

We analyze a bioeconomic model for optimal fishery harvesting in a spatially heterogeneous habitat comprising both harvestable and preservation (reserve) zones. The population dynamics are governed by a hybrid system coupling continuous time within-season dynamics -mortality, harvesting, and dispersal -with a discrete-time Beverton-Holt reproduction map. We derive the necessary and sufficient condition $Fr > 1$ for long-term population persistence, where $F$ encapsulates within-season survival including harvesting effects and $r$ is the intrinsic growth rate. Through bifurcation analysis, we demonstrate that marine protected areas (MPAs) significantly expand the sustainable parameter space. Using Pontryagin's Maximum Principle, we characterize the optimal harvesting strategy as a composite Bang-Singular-Bang control. We derive an explicit state-feedback formula for the singular arc and verify its optimality via the Generalized Legendre-Clebsch condition. Numerical simulations reveal that this dynamic strategy significantly outperforms constant maximum-effort policies, yielding higher cumulative revenue while maintaining the population above the critical collapse threshold through a stable "sawtooth" trajectory. Our results highlight that modest preservation (20-30% of habitat) allows for more intensive, profitable harvesting in open zones without risking resource extinction.