A Proof of the Global Attractor Conjecture in a Special Case

Abstract: We prove the Global Attractor Conjecture for complex balanced mass-action reaction networks whose reachable siphons satisfy two structural conditions, allowing multiple linkage classes. The proof proceeds in two steps. A structural condition relating the stoichiometric space to the reactions active within a boundary face ensures that a stoichiometric compatibility class contains at most one boundary equilibrium with any prescribed zero set. Finiteness of the possible zero sets and connectedness of the $ω$-limit set then imply that any boundary limit set consists of a single equilibrium. To exclude convergence to such an equilibrium, we consider the embedded reaction network obtained by projecting onto the vanishing species and freezing the concentrations of the surviving species at a positive limit. The structural condition guarantees complex balance of this embedded network, while a further condition on its minimal active linkage classes ensures that an explicit Chetaev function is strictly increasing near the boundary. This excludes the boundary point as an accumulation point whenever the surviving concentrations converge. Consequently, every positive trajectory converges to the unique positive equilibrium in its stoichiometric compatibility class. Examples illustrate the hypotheses and their relation to strong endotacticity.