Size-Selective Threshold Harvesting under Nonlocal Crowding and Exogenous Recruitment

Abstract

In this paper, we formulate and analyze an original infinite-horizon bioeconomic optimal control problem for a nonlinear, size-structured fish population. Departing from standard endogenous reproduction frameworks, we model population dynamics using a McKendrick--von Foerster partial differential equation characterized by strictly exogenous lower-boundary recruitment and a nonlocal crowding index. This nonlocal environment variable governs density-dependent individual growth and natural mortality, accurately reflecting the ecological pressures of enhancement fisheries or heavily subsidized stocks. We first establish the existence and uniqueness of the no-harvest stationary profile and introduce a novel intrinsic replacement index tailored to exogenously forced systems, which serves as a vital biological diagnostic rather than a classical persistence threshold. To maximize discounted economic revenue, we derive formal first-order necessary conditions via a Pontryagin-type maximum principle. By introducing a weak-coupling approximation to the adjoint system and applying a single-crossing assumption, we mathematically prove that the optimal size-selective harvesting strategy is a rigorous bang-bang threshold policy. A numerical case study calibrated to an Atlantic cod (\textit{Gadus morhua}) fishery bridges our theoretical framework with applied management. The simulations confirm that the economically optimal minimum harvest size threshold ($66.45$ cm) successfully maintains the intrinsic replacement index above unity, demonstrating that precisely targeted, size-structured harvesting can seamlessly align economic maximization with long-run biological viability.

Figures (6)

• Figure 1: Mechanism diagram for the proposed size-structured fishery model. Exogenous recruitment and harvesting act on the size-structured state $x(t,l)$. The state determines the crowding index $E(t)$, which feeds back into growth $g(E,l)$ and mortality $\mu(E,l)$. The resulting PDE supports steady-state analysis through the closure equation $F(E)=0$, biological viability assessment through the intrinsic replacement index $R(E)$, and optimal-control analysis leading to a threshold harvesting policy.
• Figure 2: No-harvest stationary size profile $x^*(l)$ for the parameter values in Table \ref{['tab:params']}. The figure also reports the corresponding baseline quantities $E^*=103108.17$, $N^*=237005$, and $R(E^*)=1.475$.
• Figure 3: Intrinsic replacement index $R(E)$ as a function of the crowding level. The dashed horizontal line marks $R=1$, and the dashed vertical line marks the no-harvest equilibrium $E^*=103108.17$.
• Figure 4: Time evolution of the crowding index $E(t)$ and total population $N(t)$ under representative threshold harvesting policies.
• Figure 5: Truncated discounted revenue $J_T$ as a function of threshold size $l^*$. The vertical dashed line marks the maximizing threshold $l^*_{\mathrm{opt}}=66.45$ cm. The shaded region corresponds to thresholds for which the resulting steady state would violate the viability target $R(E)\ge 1$. For the present computation, the optimum remains in the viable region, and therefore no shaded region appears.

Theorems & Definitions (7)

• Proposition 4.1: Existence and uniqueness
• proof
• Proposition 7.1: Formal first-order necessary conditions
• proof
• Proposition 7.2: Formal weak-coupling approximation of the adjoint equation
• Proposition 7.4: Threshold structure
• proof