Preclusion of switch behavior in networks with mass-action kinetics

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We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

Original languageEnglish
JournalApplied Mathematics and Computation
Issue number4
Pages (from-to)1449-1467
Number of pages19
Publication statusPublished - 1 Nov 2012

    Research areas

  • Degenerate steady state, Injectivity, Jacobian criterion, Multiple steady states, Stoichiometric subspace

ID: 200690607