Global Attractor Conjecture for weakly reversible deficiency zero networks

Establish that for every weakly reversible deficiency-zero reaction network under mass-action kinetics, the unique positive equilibrium in each stoichiometric compatibility class is globally asymptotically stable within that class (i.e., the positive equilibrium is a global attractor).

Background

Weakly reversible deficiency-zero networks (abbreviated in the paper as WR networks) are known to possess a unique positive equilibrium in each compatibility class, which is locally asymptotically stable by the Deficiency Zero Theorem. The Global Attractor Conjecture (GAC) seeks to strengthen this by asserting global stability of this equilibrium.

The paper emphasizes the importance of GAC beyond chemical reaction theory, noting connections to algebraic statistics (the Birch point) and to discrete analogues of the Boltzmann equation. Partial results are known (e.g., for stoichiometric subspaces of dimension less than three), and proof strategies via toric differential inclusions have been proposed, but a complete proof in full generality remains open.

References

In fact, it has long been conjectured that equilibria of $WR$ systems are globally stable with respect to their compatibility classes. This is the so-called Global Attractor Conjecture , the focus of much work in recent years.

— Weakly reversible deficiency zero realizations of reaction networks  (2502.17461 - Buxton et al., 10 Feb 2025) in Subsection 2.3, Weakly reversible deficiency zero networks

The case with boundary equilibria gives rise to the so-called {\it Global Attractor Conjecture}, which is still remained open despite many partial answers, see e.g.\ .

— Global classical solutions by transport noise for reaction-diffusion systems with entropy dissipation  (2608.13332 - Agresti et al., 13 Aug 2026) in Section 2, Subsection “Reaction–diffusion systems modelling reaction networks,” final paragraph before “Description of the noise”

The Global Attractor Conjecture (GAC) asserts that every positive trajectory in that class converges to $x*$.

— A Proof of the Global Attractor Conjecture in a Special Case  (2609.24553 - Wiuf, 21 Sep 2026) in Section 1, Introduction