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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 442-459. doi: 10.3934/mbe.2020024
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Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability

  • 1.
    Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
  • 2.
    Institute for Algebra, Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria
  • Received: 28 June 2019 Accepted: 26 September 2019 Published: 14 October 2019

  • It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply abstract results on matrix stability to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).

    Citation: Balázs Boros, Stefan Müller, Georg Regensburger. Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 442-459. doi: 10.3934/mbe.2020024

    Related Papers:

  • It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized mass-action systems, even if there exists a unique complex-balanced equilibrium (in every stoichiometric class and for all rate constants), it need not be stable. We first discuss several notions of matrix stability (on a linear subspace) such as D-stability and diagonal stability, and then we apply abstract results on matrix stability to complex-balanced equilibria of generalized mass-action systems. In particular, we show that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness. For cyclic networks, we characterize linear stability (in terms of D-stability of the Jacobian matrix); and for weakly reversible networks, we give necessary conditions for linear stability (in terms of D-semistability of the Jacobian matrices of all cycles in the network). Moreover, we show that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable). Finally, we recall and extend characterizations of D-stability and diagonal stability for matrices of dimension up to three, and we illustrate our results by examples of irreversible cycles (of dimension up to three) and of reversible chains and S-systems (of arbitrary dimension).


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