Vital-Rate Feedback and Threshold Harvesting in Size-Structured Populations

Size-selective harvesting is often justified by the intuition that larger individuals have higher value and should therefore be harvested above a critical size. We study a controlled size-structured transport model in which a scalar environmental variable, generated by the population itself, modifies the vital rates $g(E,l)$ and $\mu (E,l)$. The first result is a corrected stationary closure theory: crowding-suppressed growth increases residence density at the inflow, so the closure derivative is not determined by pointwise profile monotonicity. Instead, ${\mathrm{\Phi }}^{\mathrm{\prime }}(E)=\mathsf{A}(E)-\mathsf{C}(E)$, an integrated balance between residence-time amplification and cumulative survival loss. The second result is an exact stationary adjoint reduction. The nonlocal switching correction has rank one, $S=S_{\mathrm{r}\mathrm{e}\mathrm{d}}-\frac{A}{1-B}\psi$, and its zero-discount feedback gain satisfies the identity $B(0)={\mathrm{\Phi }}^{\mathrm{\prime }}(E^{\ast })$. Thus the same scalar governs stationary closure sensitivity and threshold fragility. We also establish finite-horizon well-posedness and compactified optimal-control existence in a spatial-$BV$ policy class. Numerical certification in a density-dependent von Bertalanffy model shows when minimum-size harvesting persists and when vital-rate feedback creates multiple-switch harvest windows.