HEXAGONbeta

Endotactic permanence and quantitative recovery with vanishing reverse rates

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by Michael Mislan ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

For a finite ordinary endotactic reaction network, measurable reaction rates in a fixed positive compact box admit a compact convex invariant absorbing set in every positive stoichiometric class. Global positive Carathéodory solutions are unique, and entry times are uniform over compact initial sets and all admitted rate histories. The argument combines the finite affine construction of OpenAI with an endotactic witness estimate that retains the reaction projection lengths. We then quantify what permanence permits as reverse rates shrink. For autonomous weakly reversible networks with ke=kˉeεaek_e=\bar k_e\varepsilon^{a_e}, eventual reciprocal concentration bounds depend on the exponent spread Δ=a+−a−\Delta=a_+-a_- through εΔ/(2H)\varepsilon^{\Delta/(2H)}, where H>0H>0 is a network-dependent geometric constant chosen before the individual rates. In an explicit three-species system with small positive rate parameters pp and hh, prepared concentrations have scales ph3ph^3, ph2ph^2, phph, and the worst-case prepared recovery time lies between (400/1313)(p2h7)−1(400/1313)(p^2h^7)^{-1} and (31/10)(p2h7)−1(31/10)(p^2h^7)^{-1}. A weighted inventory gives necessary restoration and depletion times from arbitrary positive seeds. Further results treat common clocks, reverse-exponent families and a conventional stochastic lower comparison. Two registered Lean developments support the permanence theorem, continuous dependence and selected deterministic quantitative results. Their precise coverage is separated from conventional extensions and numerical illustrations. Permanence at fixed positive rates supplies neither a uniform guarantee at zero reverse rate nor a global sharp-corridor recovery deadline.

Provenance statement

The text is primarily AI-generated. AI systems assisted the mathematical development, computations and Lean formalization under human direction; human understanding covers some parts of the work.

Formalizations

References

  1. Anderson, David F.; Kurtz, Thomas G. Stochastic Analysis of Biochemical Systems. Springer, vol. 1.2. 2015
  2. Anderson, David F.; Cappelletti, Daniele; Kim, Jinsu. Stochastically modeled weakly reversible reaction networks with a single linkage class. 2019. arXiv:1904.08967 (https://arxiv.org/abs/1904.08967)link
  3. Boros, Balázs; Hofbauer, Josef. Permanence of weakly reversible mass-action systems with a single linkage class. SIAM Journal on Applied Dynamical Systems, vol. 19, no. 1, pp. 352–365. 2020
  4. Bureau International des Poids et Mesures. The International System of Units (SI). 2019. SI Brochure, https://www.bipm.org/en/publications/si-brochurelink
  5. Coddington, Earl A.; Levinson, Norman. Theory of Ordinary Differential Equations. McGraw-Hill. 1955
  6. Craciun, Gheorghe; Deshpande, Abhishek. Endotactic networks and toric differential inclusions. SIAM Journal on Applied Dynamical Systems, vol. 19, no. 3, pp. 1798–1822. 2020. arXiv:1906.08384 (https://arxiv.org/abs/1906.08384)arXiv
  7. Craciun, Gheorghe; Nazarov, Fedor; Pantea, Casian. Persistence and permanence of mass-action and power-law dynamical systems. SIAM Journal on Applied Mathematics, vol. 73, no. 1, pp. 305–329. 2013. arXiv:1010.3050 (https://arxiv.org/abs/1010.3050)arXiv
  8. Desoeuvres, Aurélien; Iosif, Alexandru; Lüders, Christoph; Radulescu, Ovidiu; Rahkooy, Hamid; Seiß, Matthias; Sturm, Thomas. Reduction of chemical reaction networks with approximate conservation laws. SIAM Journal on Applied Dynamical Systems, vol. 23, no. 1. 2024. doi:10.1137/22M1543963 (https://doi.org/10.1137/22M1543963); arXiv:2212.13474 (https://arxiv.org/abs/2212.13474)arXiv
  9. Fenichel, Neil. Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations, vol. 31, no. 1, pp. 53–98. 1979
  10. Filippov, A. F. Differential Equations with Discontinuous Righthand Sides. Kluwer. 1988
Show all 22 referencesShow fewer references
  1. Gillespie, Daniel T. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, vol. 81, no. 25, pp. 2340–2361. 1977
  2. Gopalkrishnan, Manoj; Miller, Ezra; Shiu, Anne. A geometric approach to the Global Attractor Conjecture. SIAM Journal on Applied Dynamical Systems, vol. 13, no. 2, pp. 758–797. 2014. arXiv:1305.5303 (https://arxiv.org/abs/1305.5303)arXiv
  3. Hairer, Ernst; Wanner, Gerhard. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, vol. 14. 1996
  4. Johnston, Matthew D.; Pantea, Casian; Donnell, Pete. A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems. Journal of Mathematical Biology, vol. 72, no. 1–2, pp. 467–498. 2016. arXiv:1412.4662 (https://arxiv.org/abs/1412.4662)arXiv
  5. Kurtz, Thomas G. The relationship between stochastic and deterministic models for chemical reactions. The Journal of Chemical Physics, vol. 57, no. 7, pp. 2976–2978. 1972
  6. Moore, Ramon E.; Kearfott, R. Baker; Cloud, Michael J. Introduction to Interval Analysis. SIAM. 2009
  7. OpenAI. Boundedness and persistence of weakly reversible mass-action systems. 2026. OpenAI Math Release preprint, September 25, 2026; pinned public paper and sources (https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026), revision \textttadc7f1241b42link
  8. OpenAI. Mass-action formalization and scope documentation. 2026. Pinned public Lean scope note and source links (https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/149.md), revision \textttadc7f1241b42link
  9. OpenAI. Uniform permanence in weakly reversible mass-action systems. 2026. OpenAI Math Release preprint, October 5, 2026; pinned public paper and sources (https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026), revision \textttadc7f1241b42link
  10. Pantea, Casian. On the persistence and global stability of mass-action systems. SIAM Journal on Mathematical Analysis, vol. 44, no. 3, pp. 1636–1673. 2012. arXiv:1103.0603 (https://arxiv.org/abs/1103.0603)arXiv
  11. Schnakenberg, J. Network theory of microscopic and macroscopic behavior of master equation systems. Reviews of Modern Physics, vol. 48, no. 4, pp. 571–585. 1976
  12. Tucker, Warwick. Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press. 2011

Version history

  1. v1Submitted by Michael MislanCorrected before publicationCurrentOct 18, 2026