Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

Abstract

The global attractor conjecture says that toric dynamical systems have a globally attracting point (up to linear conservation relations), or equivalently, complex balanced systems have a globally attracting point within each 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. Toric dynamical systems can be embedded into toric differential inclusions. We show that each bounded positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the boundary of the positive orthant. 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 systems have the global attractor property.

Version 3 is a major revision that completes and clarifies the extension from the 3D and 4D cases to the general n-dimensional case

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