Size-Selective Threshold Harvesting under Nonlocal Crowding and Exogenous Recruitment

We propose a nonlinear size-structured fishery model for externally recruited stocks under size-selective harvesting. The population density satisfies a McKendrick--von Foerster transport equation in which growth and natural mortality depend on a nonlocal crowding index, while harvesting acts as a bounded size-dependent mortality control. Unlike standard self-recruiting formulations, recruitment is prescribed as a lower-boundary inflow, making the model suitable for enhancement fisheries or analyses conditional on juvenile input. For the no-harvest baseline, we derive the stationary size profile and reduce the nonlinear equilibrium problem to a scalar closure equation, proving existence and uniqueness under a net monotonicity condition. We introduce an intrinsic replacement index and show why, in this externally forced setting, it is a viability diagnostic rather than a persistence threshold. A formal state--adjoint system yields a bang--bang switching rule; under weak coupling and single crossing, the optimal policy has a threshold structure. Numerical experiments validate the approximation and sensitivity trends