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Age-Structured Harvesting Models: A Structural Comparison of Rate-Control and Effort-Control Optimality Systems
Journal article   Open access   Peer reviewed

Age-Structured Harvesting Models: A Structural Comparison of Rate-Control and Effort-Control Optimality Systems

Jiguang Yu, Louis Shuo Wang and Ye Liang
International journal of mathematics and mathematical sciences, Vol.2026, 6212516
08/20/2026
DOI: 10.1155/ijmm/6212516
Appears in  UI Libraries Support Open Access
url
https://doi.org/10.1155/ijmm/6212516View
Published (Version of record) Open Access

Abstract

We study optimal harvesting in continuous-time, age-structured population models of McKendrick-von Foerster type, and we compare two harvesting mechanisms. In the rate-control formulation, harvesting enters the state equation as an additive removal term; in the effort-control formulation, harvesting acts multiplicatively as an additional mortality intensity, and the mortality coefficient depends on the aggregate stock. For the rate-control problem, we first establish the existence of an optimal control for the infinite-horizon discounted problem, and we then derive, under explicitly stated regularity and constraint-qualification assumptions, a conditional Pontryagin-type necessary optimality system consisting of the adjoint equation, the terminal-age and transversality conditions, the switching relations for the distributed and boundary controls and the complementary-slackness relation for the state constraint. For the effort-control problem, we formally derive the associated adjoint equation and identify the nonlocal coupling term generated by aggregate (density) dependence, with the sign of that term verified by a step-by-step variational computation; a rigorous infinite-horizon maximum principle for this nonlinear, nonlocally coupled problem is beyond the present scope and is stated as such. We complement the analysis with autonomous stationary reductions, with explicit representations of the state and adjoint and with a reproducible numerical study. The central message is structural: The harvesting mechanism is not a cosmetic modelling choice. Rate-control produces an additive/affine/local optimality structure, whereas effort-control produces a multiplicative/nonlinear/nonlocal one, with direct consequences for persistence, stationary profiles and bioeconomic interpretation.
 age-structured population dynamics infinite-horizon optimal control McKendrick–von Foerster equation optimal harvesting Pontryagin maximum principle UIOWA OA Agreement

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