Graphtheoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models
 Maya Mincheva,
 Marc R. Roussel
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A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical massaction kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a massaction model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.
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 Title
 Graphtheoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models
 Journal

Journal of Mathematical Biology
Volume 55, Issue 1 , pp 6186
 Cover Date
 20070701
 DOI
 10.1007/s0028500700991
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Chemical reactions
 Graph
 Multistability
 Oscillations
 34C23
 Authors

 Maya Mincheva ^{(1)} ^{(2)}
 Marc R. Roussel ^{(1)}
 Author Affiliations

 1. Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
 2. Department of Mathematics, University of WisconsinMadison, Madison, WI, 537061388, USA