Journal of Mathematical Biology

, Volume 55, Issue 1, pp 87-104

First online:

Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays

  • Maya MinchevaAffiliated withDepartment of Chemistry and Biochemistry, University of LethbridgeDepartment of Mathematics, University of Wisconsin-Madison
  • , Marc R. RousselAffiliated withDepartment of Chemistry and Biochemistry, University of Lethbridge Email author 

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Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.


Chemical reactions Graph Delay-induced instability Oscillations

Mathematics Subject Classification (2000)