Optimal Harvesting for Nonlinear Size-Structured Populations with Nonlocal Environmental Feedback

This paper investigates the optimal harvesting of a nonlinear, size-structured population governed by a first-order transport equation with nonlocal environmental crowding feedback and exogenous inflow. First, we establish finite-horizon well-posedness for the controlled state system in an $L^1$ framework, proving the existence, uniqueness, positivity, and continuous dependence of weak solutions. Second, we show that the infinite-dimensional stationary problem reduces exactly to a scalar nonlinear closure equation, yielding existence and conditional uniqueness results for stationary states. Within this equilibrium framework, we distinguish the persistence of the forced system from intrinsic demographic self-replacement and introduce size-continuous per-recruit and spawning-potential diagnostics. Finally, we formulate a partial differential equation (PDE)-constrained optimal harvesting problem. Under a compactness assumption on the control-to-state map, we establish the existence of optimal controls. We then formally derive a Pontryagin-type first-order optimality system for the harvesting problem. The variation of the nonlocal environmental feedback produces a coupled integral source term in the adjoint equation. The associated pointwise maximization condition yields a bang–bang harvesting structure, while a monotone size-threshold policy is shown to require an additional single-crossing assumption on the switching function. These hypotheses are illustrated using a fisheries model with density-dependent von Bertalanffy growth.