
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 MassAction 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 277080320, 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 massaction 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 massaction
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 lowerdimensional member of the family approaches the
boundary.
With this technology, we make progress on the global attractor conjecture, a central open problem concerning
massaction kinetics systems.
Additionally, we phrase massaction kinetics as a functor on reaction networks with variable rates.
Key words:
differential inclusion; massaction kinetics; reaction network; persistence; global attractor conjecture.
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