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

Abstract

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 \(μ(E,l)\). 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, $Φ'(E)=\mathsf A(E)-\mathsf C(E)$, 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, $S=S_{\rm red}-\frac{A}{1-B}ψ$, and its zero-discount feedback gain satisfies the identity $B(0)=Φ'(E^*)$. 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.