Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

TL;DR

This paper proves the global attractor conjecture for toric dynamical systems, showing that complex balanced mass-action systems have globally attracting points, implying stable and simple dynamics for a broad class of nonlinear systems.

Contribution

Introduction of toric differential inclusions and a proof of the global attractor conjecture for complex balanced systems.

Findings

All detailed balanced mass action systems are globally attracting.

Deficiency zero weakly reversible networks have the global attractor property.

Solutions are contained in invariant regions preventing approach to the origin.

Abstract

The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. A proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics. The conjecture originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and was formulated in its current form by Horn in 1974. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. We use this result to prove the global attractor conjecture. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible networks have the global attractor property.