Size-Selective Threshold Harvesting under Nonlocal Crowding and Exogenous Recruitment

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.

💡 Research Summary

This paper develops and analyzes a novel infinite‑horizon bio‑economic optimal control framework for a size‑structured fish population in which recruitment is imposed exogenously at the lower size boundary and density dependence operates through a non‑local crowding index. The state variable x(t,l) denotes the density of individuals of size l at time t, while the control u(t,l) represents size‑specific fishing mortality. The crowding index E(t)=∫χ(l)x(t,l)dl aggregates the whole population with a weighting kernel χ(l) that can emphasize larger individuals (e.g., χ(l)∝l^β). Growth g(E,l) and natural mortality µ(E,l) are assumed to be continuous, strictly positive (for growth), non‑negative (for mortality), and monotone non‑decreasing in E, reflecting the ecological intuition that higher crowding cannot reduce death risk.

The governing dynamics are a McKendrick–von Foerster transport PDE with non‑local coefficients:
∂_t x + ∂_l