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

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.

💡 Research Summary

This paper develops a spatially explicit bio‑economic model for a fishery that is divided into a harvestable zone and a no‑take marine protected area (MPA). The within‑season dynamics are continuous‑time ordinary differential equations that describe natural mortality, harvest mortality (proportional to effort E and catchability q), and dispersal between the two patches with rate m. The fraction of the total habitat that is protected is denoted by R, and the length of the fishing season (as a proportion of a year) is T. At the end of each season, the adult biomass is aggregated and fed into a discrete‑time Beverton‑Holt recruitment function
(J_{k-1}= \frac{r,x(k-1)}{1+\beta x(k-1)}),
where r is the intrinsic growth rate and β captures density‑dependent recruitment limitation. By solving the linear ODE system analytically (matrix exponential) and substituting the recruitment map, the authors reduce the whole hybrid system to a one‑dimensional difference equation
(x(k)=\alpha x(k-1)+\beta x(k-1),\qquad \alpha=F,r),
where F is a composite survival factor that depends on c, q, E, m, R, and T. The condition for a positive, locally stable equilibrium is (\alpha>1), i.e. (F r>1). This inequality is shown to be both necessary and sufficient for long‑term persistence of the stock under harvesting. When E=0 (no harvest) the condition collapses to the familiar requirement (r>e^{cT}), which depends only on the non‑growth period T.

The authors conduct a thorough bifurcation analysis in the three‑dimensional parameter space (R, E, T). The surface (F r=1) separates extinction from sustainable dynamics. Key insights are: (i) increasing the protected fraction R lowers the effective within‑season mortality, thereby expanding the admissible range of effort E; (ii) for a given T, there is an explicit upper bound on effort when the reserve is small, (E\le (\ln r,T -c)/q); (iii) beyond a critical reserve size (derived analytically) the effort bound disappears, meaning the MPA alone can sustain the population regardless of local harvest intensity. Shortening the fishing season (smaller T) also relaxes the effort bound, illustrating a clear trade‑off between temporal and spatial restrictions.

The optimal management problem is formulated as the maximization of the discounted sum of harvest revenues over an infinite horizon. Because the state at the beginning of each season fully determines future dynamics, the problem possesses a Markov structure and can be decomposed into a sequence of single‑season optimal control problems linked by a Bellman equation. Applying Pontryagin’s Maximum Principle yields the Hamiltonian
(H = \lambda_1\dot x_1 + \lambda_2\dot x_2 + \pi q E x_1)
with (\pi) the price‑cost ratio. The switching function (\phi(t)=\partial H/\partial E = -q x_1\lambda_1 + \pi q x_1) dictates the optimal effort. The optimal control therefore has three regimes: (a) Bang – full effort (E_{\max}) when (\phi>0); (b) Bang – zero effort when (\phi<0); (c) Singular – a state‑feedback effort when (\phi=0). By enforcing (\phi=0) and (\dot\phi=0) the authors derive an explicit singular‑arc law
(E_s(t)=\frac{\pi-\lambda_1(t)}{q,\lambda_1(t)}),
which depends only on the co‑state (\lambda_1). The Generalized Legendre‑Clebsch condition is verified, confirming that the singular arc is indeed locally optimal. Consequently, the optimal harvesting policy is a Bang‑Singular‑Bang sequence: an initial burst of maximal harvesting, a middle phase where effort is continuously adjusted according to the feedback law, and a final phase of zero effort as the stock approaches the safe threshold.

Numerical simulations using baseline ecological parameters (c=1, r=2, m=1, q=0) illustrate the performance of the optimal policy. Compared with a constant‑maximum‑effort strategy, the Bang‑Singular‑Bang control raises the present value of revenue by roughly 15–30 % while keeping the biomass on a stable “saw‑tooth” trajectory that never falls below the collapse threshold. When the protected fraction lies between 20 % and 30 %, the model predicts that the fishery can sustain considerably higher effort levels in the open zone without jeopardizing long‑term viability. The simulations also show that larger reserves eliminate the upper bound on effort, confirming the analytical bifurcation results.

The paper concludes with several management implications. First, the simple inequality (F r>1) provides a quantitative rule for jointly selecting reserve size R and allowable effort E to guarantee sustainability. Second, the presence of a singular control segment demonstrates that optimal management is not a simple “all‑or‑nothing” policy; instead, temporally varying effort that responds to the stock state yields superior economic outcomes. Third, the model explicitly captures the interaction between dispersal (m), reserve size (R), and harvest pressure (E), offering a framework for evaluating trade‑offs in MPA design (size, connectivity, and allowable fishing effort). The authors suggest extensions to age‑structured populations, stochastic environmental fluctuations, and multispecies interactions, as well as empirical validation with real fisheries data.