Monotonicity in chemical reaction systems

Abstract

This article discusses the question of when the dynamical systems arising from chemical reaction networks are monotone, preserving an order induced by some proper cone. The reaction systems studied are defined by the reaction network structure while the kinetics is only constrained very weakly. Necessary and sufficient conditions on cones preserved by these systems are presented. Linear coordinate changes which make a given reaction system cooperative are characterized. Also discussed is when a reaction system restricted to an invariant subspace is cone preserving, even when the system fails to be cone preserving on the whole of phase space. Many of the proofs allow explicit construction of preserved cones. Numerous examples of chemical reaction systems are presented to illustrate the results.

Keywords:

• chemical reactions
• monotone dynamical systems
• cones
• order preserving systems
• stability

AMS Subject Classifications::

• 80A30
• 34C12
• 15A48

Acknowledgements

I would like to thank Steve Baigent and Pete Donnell for many hours of helpful discussion on themes in this article.

Notes

Notes

1. Note that as vectors and scalars can be regarded as special cases of matrices, the definitions carry over naturally to these objects.

2. The reader should bear in mind that if we write that a system ‘preserves no cones’, it is still possible that the system is K-cooperative for some non-polyhedral cone K. There is no such ambiguity if we write that a system ‘preserves no polyhedral cones’ or ‘preserves no simplicial cones’.