Size-Selective Threshold Harvesting under Nonlocal Crowding and Exogenous Recruitment

TL;DR

This paper develops a mathematical model for size-selective fish harvesting that balances economic gains with ecological sustainability, using advanced control theory and a case study on Atlantic cod.

Contribution

It introduces a novel exogenous recruitment model with a nonlocal crowding index and proves that optimal harvesting follows a bang-bang threshold policy.

Findings

Optimal harvest strategy is a bang-bang threshold policy.

The model's minimum harvest size maintains the fish population's biological viability.

Numerical results align economic objectives with ecological sustainability.

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.