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

TL;DR

The paper develops a two-patch, seasonally driven fishery model incorporating a no-take reserve and Beverton–Holt recruitment to study sustainability and economics. Using persistence analysis, bifurcation diagrams, and Pontryagin's Maximum Principle, it shows that a persistence condition $Fr>1$ governs long-term viability and that MPAs expand the sustainable region. The optimal harvesting strategy is a composite Bang–Singular–Bang control with an explicit state-feedback form for the singular arc, verified by the Generalized Legendre–Clebsch condition. Numerical simulations demonstrate that dynamic strategies outperform constant-effort harvesting and that modest reserves (about 20–30%) can enhance both ecological resilience and economic returns, yielding a robust sawtooth persistence pattern across years.

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.

Figures (3)

• Figure 1: Bifurcation diagrams illustrating the stability boundaries for the harvested and non-harvested models. The shaded regions represent the stable parametric space where $Fr > 1$, ensuring long-term population persistence. (Top-Left): Stability in $(R, T)$ space for the non-harvested case ($E=0$). (Top-Right): Transcritical bifurcation curves in $(E, R)$ space for varying season lengths $T$. (Bottom-Left): Stability regions in $(R, T)$ space for fixed efforts $E=2$ and $E=20$. (Bottom-Right): Impact of reserve ratio $R$ on the $(E, T)$ stability boundary.
• Figure 2: Comparison of harvesting effort strategies and their corresponding state trajectories. Phase (a) illustrates the effort profiles for constant control ($E=3.4$), pure bang-bang control, and the optimal combination of bang-bang and singular control ($E_{sing}$). Phase (b) presents the evolution of the population in the harvested zone ($x_1$) and reserve ($x_2$) under each strategy. The optimal control trajectory (blue) maximizes biomass extraction while ensuring the population remains within sustainable limits at the end of the season.
• Figure 3: Multi-year bioeconomic simulation of the two-patch fishery system over a three-year horizon. The blue and green solid lines represent the population biomass in the harvested zone ($x_1$) and reserve zone ($x_2$), respectively. Shaded red regions indicate the active harvesting seasons ($T=0.5$), during which the population undergoes depletion governed by the optimal control effort $E^*$ (dashed red line). The non-shaded regions represent the recovery phase where $E=0$. Discrete vertical jumps at the start of each year ($t=1, 2, 3$) denote the inter-seasonal recruitment phase defined by the Beverton-Holt reproduction map $J$. For the chosen parameters ($r=5$), the system satisfies the sustainability condition $Fr > 1$, resulting in a stable, periodic 'sawtooth' trajectory that ensures long-term ecological persistence.