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Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability
Balázs Boros1, Stefan Müller1, Georg Regensburger2
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.
Mathematical Biosciences and Engineering : MBE
|November 17, 2019
Summary
Complex-balanced equilibria in generalized mass-action systems are not always stable. This study explores matrix stability concepts to characterize stability, finding linear stability implies uniqueness for these systems.
Area of Science:
- Chemical kinetics
- Systems biology
- Mathematical modeling
Background:
- Complex-balanced equilibria are known to be asymptotically stable in mass-action systems.
- However, in generalized mass-action systems, unique complex-balanced equilibria may not be stable.
- Understanding stability is crucial for predicting system behavior.
Purpose of the Study:
- To investigate the stability of complex-balanced equilibria in generalized mass-action systems.
- To apply matrix stability concepts to analyze these equilibria.
- To establish conditions for linear stability and its relationship with uniqueness.
Main Methods:
- Discussion of matrix stability notions like D-stability and diagonal stability on linear subspaces.
- Application of abstract matrix stability results to generalized mass-action systems.
- Analysis of cyclic and weakly reversible networks, including Jacobian matrices and stoichiometric subspaces.
Main Results:
- Linear stability of complex-balanced equilibria implies uniqueness for generalized mass-action systems.
- Characterization of linear stability for cyclic networks via D-stability of the Jacobian.
- Necessary conditions for linear stability in weakly reversible networks using D-semistability.
- Complex-balanced equilibria in classical mass-action systems are diagonally stable.
Conclusions:
- The study provides a deeper understanding of stability in generalized mass-action systems.
- Matrix stability provides powerful tools for analyzing complex chemical reaction networks.
- Results extend to classical systems and offer insights for network design and analysis.
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