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Abstract We considertoric dynamical systems, which are also calledcomplex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under the assumption of mass-action kinetics, they can give rise tocomplex-balanced equilibria. Given a reaction network, we study theset of parameter valuesfor which the network gives rise to toric dynamical systems, also calledthe toric locusof the network. The toric locus is an algebraic variety, and we are especially interested in its topological properties. We show that complex-balanced equilibriadepend continuouslyon the parameter values in the toric locus, and, using this result, we prove that the toric locus has a remarkableproduct structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron of the network. In particular, it follows that the toric locus is acontractible manifold. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.
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This content will become publicly available on June 10, 2026
The Dimension of the Disguised Toric Locus of a Reaction Network
ABSTRACT Mathematical models of reaction networks are ubiquitous in applications, especially in chemistry, biochemistry, chemical engineering, ecology, and population dynamics. Under the standard assumption ofmass‐action kinetics, reaction networks give rise to general dynamical systems with polynomial right‐hand side. These depend on many parameters that are difficult to estimate and can give rise to complex dynamics, including multistability, oscillations, and chaos. On the other hand, a special class of reaction systems calledcomplex‐balanced systemsare known to exhibit remarkably stable dynamics; in particular, they have unique positive fixed points and no oscillations or chaotic dynamics. One difficulty, when trying to take advantage of the remarkable properties of complex‐balanced systems, is that the set of parameters where a network satisfies complex balance may have positive codimension and therefore zero measure. To remedy this we are studyingdisguised complex balanced systems(also known asdisguised toric systems), which may fail to be complex balanced with respect to an original reaction network , but are actually complex balanced with respect to some other network , and therefore enjoy all the stability properties of complex‐balanced systems. This notion is especially useful when the set of parameter values for which the network gives rise to disguised toric systems (i.e., thedisguised toric locusof ) has codimension zero. Our primary focus is to compute the exact dimension (and therefore the codimension) of this locus. We illustrate the use of our results by applying them to Thomas‐type and circadian clock models.
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- Award ID(s):
- 2051568
- PAR ID:
- 10600183
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 154
- Issue:
- 6
- ISSN:
- 0022-2526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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