No asymptotic extinction in bounded weakly reversible mass-action kinetics
Let $\mathcal{R}$ be a finite set of reactions $y \to y'$ between complexes $y, y' \in \mathbb{N}_0^d$, each lying on a directed cycle of the complex graph (the network is weakly reversible), and fix positive reaction constants $\kappa_{y \to y'}$. Consider the mass-action system $$\dot x = \sum_{y \to y' \in \mathcal{R}} \kappa_{y \to y'} x^y (y' - y), \qquad x^y = \prod_i x_i^{y_i},$$ with $x(0) \in (0, \infty)^d$. For complex-balanced systems, persistence of every positive trajectory is equivalent to the global attractor conjecture of Horn; Anderson proved it when the network has one linkage class. Pantea proved that in weakly reversible mass-action systems with two-dimensional stoichiometric subspace all bounded trajectories are persistent.
Problem. For every solution defined and bounded on $[0, \infty)$, must $\liminf_{t \to \infty} x_i(t) > 0$ hold for every species $i$? No complex-balance assumption is permitted.
Source: Feinberg (1987), Remark 6.1.E, as the persistence conjecture (Gopalkrishnan, Miller and Shiu (2014), Conjecture 4.4). Stated by AIM — Open Applied Problems for bounded trajectories.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, chemical reaction networks, mass-action kinetics, persistence
References
@article{feinberg1987,
author = {Feinberg, Martin},
title = {Chemical reaction network structure and the stability of complex isothermal reactors--{I}. {T}he deficiency zero and deficiency one theorems},
journal = {Chem. Eng. Sci.},
volume = {42},
number = {10},
year = {1987},
pages = {2229--2268},
doi = {10.1016/0009-2509(87)80099-4},
}
@article{gopalkrishnan2014,
author = {Gopalkrishnan, Manoj and Miller, Ezra and Shiu, Anne},
title = {A geometric approach to the global attractor conjecture},
journal = {SIAM J. Appl. Dyn. Syst.},
volume = {13},
number = {2},
year = {2014},
pages = {758--797},
doi = {10.1137/130928170},
eprint = {1305.5303},
archivePrefix = {arXiv},
}
@article{anderson2011,
author = {Anderson, David F.},
title = {A proof of the global attractor conjecture in the single linkage class case},
journal = {SIAM J. Appl. Math.},
volume = {71},
number = {4},
year = {2011},
pages = {1487--1508},
doi = {10.1137/11082631X},
}
@article{pantea2012,
author = {Pantea, Casian},
title = {On the persistence and global stability of mass-action systems},
journal = {SIAM J. Math. Anal.},
volume = {44},
number = {3},
year = {2012},
pages = {1636--1673},
doi = {10.1137/110840509},
eprint = {1103.0603},
archivePrefix = {arXiv},
}
@misc{aim-reaction-network-persistence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{No asymptotic extinction in bounded weakly reversible mass-action kinetics}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/273-reaction-network-persistence.md},
note = {Entry 273 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00709, "No asymptotic extinction in bounded weakly reversible mass-action kinetics", clopen.solutions.