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.