A stratum approach to global stability of complex balanced systems

@article{Siegel2010ASA,
  title={A stratum approach to global stability of complex balanced systems},
  author={David Siegel and Matthew D. Johnston},
  journal={Dynamical Systems},
  year={2010},
  volume={26},
  pages={125 - 146}
}
It has long been known that complex balanced mass-action systems exhibit a restrictive form of behaviour known as locally stable dynamics. This means that within each compatibility class – the forward invariant space where solutions lies – there is exactly one equilibrium concentration and that this concentration is locally asymptotically stable. It has also been conjectured that this stability extends globally to . That is to say, all solutions originating in approach the unique positive… 
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14 Citations
Background Citations
7
Methods Citations
2

14 Citations

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It is conjecture that the result holds when the reaction network is weakly reversible, and it is proved that this conjecture holds in the case when the Reaction network consists of a single linkage class, or connected component.

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References

SHOWING 1-10 OF 14 REFERENCES

Necessary and sufficient conditions for complex balancing in chemical kinetics

SummaryIn a recent publication (Horn & Jackson [1]) it was shown that complex balancing together with mass action type rate laws ensures certain stability properties of a kinetic system, thereby…

The Dynamics of Weakly Reversible Population Processes near Facets

This paper concerns the dynamical behavior of weakly reversible, deterministically modeled population processes near the facets (codimension-one faces) of their invariant manifolds and proves that…

General mass action kinetics

The principal result of this work shows that there exists a simply identifiable class of kinetic expressions, including the familiar detailed balanced kinetics as a proper subclass, which ensure consistency with the extended thermodynamic conditions.

Toric dynamical systems

The existence and uniqueness of steady states for a class of chemical reaction networks

My purpose here is to draw some general relationships between the structure of a chemical reaction network and the nature of the set of equilibrium states for the corresponding system of nonlinear…

Global stability of complex balanced mechanisms

We prove that the ω‐limit set of any solution of a complex balanced chemical reaction mechanism contains either a unique, positive complex balanced equilibrium point, or boundary complex balanced…

A Petri net approach to the study of persistence in chemical reaction networks.

Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction

The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan (1995) for T-cell receptor signal transduction.

quelques extensions d'un théorème de Horn et Jackson

Nous etudions dans cette Note les points d'equilibre de systemes differentiels issus de la cinetique chimique. Horn et Jackson [1] ont mis en evidence des conditions suffisantes d'existence, unicite…

The dynamics of open reaction systems: Mathematical aspects of chemical and biochemical problems and quantum chemistry

  • SIAM-AMS Proceedings, Vol. VIII, American Mathematical Society, Providence, RI
  • 1974