we-solve (beta)

Math-2423 · mathematics · Dynamical systems

Multistationarity and Permanence in MESSI Reaction Networks

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For MESSI mass-action biochemical systems, characterize when a stoichiometric compatibility class can contain more than one positive steady state with the same conserved quantities, and determine the related permanence and invariant-subspace properties.

Formulation references: [S1], [S2].

Short English statement

For MESSI mass-action biochemical systems, characterize when a stoichiometric compatibility class can contain more than one positive steady state with the same conserved quantities, and determine the related permanence and invariant-subspace properties.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2423-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2423

Manuscript 1 (family 149, 2026-09-25): The theorem gives boundedness and persistence (and October gives classwise permanence) for the weakly reversible MESSI subclass, but does not characterize all MESSI multistationarity or invariant-subspace phenomena.

Manuscript 2 (family 149, 2026-10-05): The theorem gives boundedness and persistence (and October gives classwise permanence) for the weakly reversible MESSI subclass, but does not characterize all MESSI multistationarity or invariant-subspace phenomena.

Multistationarity and Permanence in MESSI Reaction Networks

OpenAI's claim

Manuscript 1 (family 149, 2026-09-25): Every positive trajectory of a finite weakly reversible constant-rate mass-action system is claimed to exist forever and keep all concentrations between positive finite bounds, even in unbounded compatibility classes. Manuscript 2 (family 149, 2026-10-05): Each positive stoichiometric class of a weakly reversible constant-rate mass-action system is claimed to have one compact convex forward-invariant set eventually entered by every positive trajectory.

Mathematical statement of the claimed result

Mathematical claim 1.
Take a finite reaction network $\mathcal C\subset\mathbb Z_{\ge0}^d$, $\mathcal R\subset\mathcal C\times\mathcal C$, with positive constant rates $\kappa_{y\to y'}$. Assume weak reversibility: every directed reaction lies on a directed cycle. Its mass-action system is \[ \dot x=\sum_{y\to y'}\kappa_{y\to y'}x^y(y'-y),\qquad x^y=\prod_i x_i^{y_i}. \] OpenAI claims every positive initial point has a global positive solution and a trajectory-dependent $\varepsilon\in(0,1)$ with $\varepsilon\le x_i(t)\le\varepsilon^{-1}$ for all $t\ge0$ and all species $i$. With the extra complex-balance hypothesis (at some positive equilibrium, total inflow equals total outflow at each complex), it records convergence within each positive stoichiometric class to its sole positive equilibrium $x^0+\operatorname{span}\{y'-y\}$ intersected with the positive orthant. The convergence conclusion is not asserted from weak reversibility alone.
Mathematical claim 2.
For a finite weakly reversible mass-action network with fixed positive rates, define its stoichiometric subspace $S=\operatorname{span}\{y'-y\}$ and positive class $P=(x^0+S)\cap\mathbb R_{>0}^d$. OpenAI claims a nonempty compact convex forward-invariant $K_P\subset P$ eventually containing every trajectory from $P$. Each such solution is global and positive, and \[ \exists\varepsilon_P\in(0,1)\ \forall x^0\in P\ \exists T(x^0)<\infty:\quad \varepsilon_P\le x_i(t)\le\varepsilon_P^{-1}\quad(t\ge T(x^0)). \] Weak reversibility means every directed reaction lies on a directed cycle, and the dynamics are $\dot x=\sum\kappa_{y\to y'}x^y(y'-y)$. The bounds depend on the network, rates and class, rather than the initial point; only the entry time may depend on that point.

Outline of the argument

Manuscript 1 (family 149, 2026-09-25): An affine approximation lemma controls exponent pairings separately rather than relying on a single large concentration product. It builds a compact convex polytope inside the positive orthant around each initial condition, with all boundary reaction directions pointing inward; invariance supplies boundedness, persistence, and global existence.

Manuscript 2 (family 149, 2026-10-05): The affine trapping construction is strengthened by strict increase outside a selected trapping set. Connecting such sets across scales removes their dependence on the initial point, giving a common absorbing region and hence concentration bounds uniform over the whole fixed class after trajectory-dependent entry times.

Formal verification

For family 149, the scope page documents the following selections. The selected concentration bounds hold for all forward times with one positive lower bound depending on the initial state, network, and fixed rates. Classwise eventual bounds uniform over initial states are outside this selection. The documented statement covers the main claim of manuscript 1. It does not establish formal coverage of the complete problem clause. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Comparator files: MassAction.lean. Declarations: OAI.Problem326.global_bounded_persistent_solution.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

Go to contribution →