A proof of the Global Attractor Conjecture in the single linkage class case

TL;DR

This paper proves the long-standing Global Attractor Conjecture for chemical reaction systems with a single linkage class, introducing a novel monomial partitioning method to analyze trajectory convergence and Lyapunov functions.

Contribution

It provides the first proof of the Global Attractor Conjecture in the single linkage class case, advancing understanding of chemical reaction network dynamics.

Findings

Confirmed global stability for single linkage class systems.

Developed a new monomial partitioning analytical method.

Showed Lyapunov functions decrease along trajectories approaching the boundary.

Abstract

This paper is concerned with the dynamical properties of deterministically modeled chemical reaction systems. Specifically, this paper provides a proof of the Global Attractor Conjecture in the setting where the underlying reaction diagram consists of a single linkage class, or connected component. The conjecture dates back to the early 1970s and is the most well known and important open problem in the field of chemical reaction network theory. The resolution of the conjecture has important biological and mathematical implications in both the deterministic and stochastic settings. One of our main analytical tools, which is introduced here, will be a method for partitioning the relevant monomials of the dynamical system along sequences of trajectory points into classes with comparable growths. We will use this method to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory. This will allow us to overcome the fact that the usual Lyapunov functions of chemical reaction network theory are bounded on the boundary of the positive orthant, which has been the technical sticking point to a proof of the Global Attractor Conjecture in the past.