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Bistability and oscillations in chemical reaction networks. (English) Zbl 1311.92088
Summary: Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in [B. D. Aguda and B. L. Clarke, “Bistability in chemical reaction networks: theory and application to the peroxidase-oxidase reaction”, J. Chem. Phys. 87, No. 6, 3461–3470 (1987); B. L. Clarke and W. Jiang, “Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov-Zhabotinskii reaction”, J. Chem. Phys. 99, No. 6, 4464–4476 (1993); K. Gatermann et al., J. Symb. Comput. 40, No. 6, 1361–1382 (2005; Zbl 1120.13033)]. In these papers, some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92E20 Classical flows, reactions, etc. in chemistry
34C23 Bifurcation theory for ordinary differential equations
37G10 Bifurcations of singular points in dynamical systems
37N25 Dynamical systems in biology
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