Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 025, 25 pages      arXiv:1208.0874      https://doi.org/10.3842/SIGMA.2013.025

A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics

Manoj Gopalkrishnan a, Ezra Miller b and Anne Shiu c
a) School of Technology and Computer Science, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India
b) Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA
c) Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA

Received August 07, 2012, in final form March 23, 2013; Published online March 26, 2013

Abstract
Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks – including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks – that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates.

Key words: differential inclusion; mass-action kinetics; reaction network; persistence; global attractor conjecture.

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