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PreprintCC BY 4.0arXiv

Endogenous Feedback in Size-Structured Transport Equations

  • Jiguang Yu
  • Louis Shuo Wang
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Abstract

We study a nonlinear size-structured transport equation where the endogenous scalar output E(t)=∫l0lmχ(l)x(t,l) dl feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point E=𝒦(E). Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation E=Φ(E). We prove uniqueness below the sharp margin 1−Φ'(E)>0 and identify Φ'(E)=1 as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation ℰ(λ)=1, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity ℰ(0)=Φ'(E*)=B(0), linking closure resonance, spectral crossing, and adjoint loop gain.

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Cite

BibTeXRIS
BibTeX
@misc{yu2026endogenous,
  title = {{Endogenous Feedback in Size-Structured Transport Equations}},
  author = {Jiguang Yu and Louis Shuo Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.02877},
  eprint = {2607.02877},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.02877},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Endogenous Feedback in Size-Structured Transport Equations
AU  - Jiguang Yu
AU  - Louis Shuo Wang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2607.02877
UR  - https://arxiv.org/abs/2607.02877
AB  - We study a nonlinear size-structured transport equation where the endogenous scalar output E(t)=∫_l_0^l_mχ(l)x(t,l) dl feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point E=𝒦(E). Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation E=Φ(E). We prove uniqueness below the sharp margin 1−Φ'(E)>0 and identify Φ'(E)=1 as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation ℰ(λ)=1, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity ℰ(0)=Φ'(E^*)=B(0), linking closure resonance, spectral crossing, and adjoint loop gain.
ER  -