Abstract
Recall that the reaction networks of deficiency zero can contain hundreds of species and hundreds of reactions. In such cases, the corresponding differential equations will be extraordinarily complex. When the kinetics is mass action, these will amount to a large and intricate system of coupled polynomial equations in perhaps 100 species concentrations and in which a large number of perhaps unknown parameters (rate constants) appear. Recall too that not much is known in general about systems of polynomial equations, even fairly small ones. Nevertheless, when these derive from deficiency zero networks, quite a lot can be said.
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Notes
- 1.
- 2.
In [107] Horn showed that weak reversibility is necessary for the existence of a complex balanced positive equilibrium, and he also went beyond the zero deficiency case to work out necessary and sufficient conditions the rate constants must satisfy in order that a weakly reversible positive-deficiency mass action system admit a complex balanced positive equilibrium. For any weakly reversible positive-deficiency network, there invariably exists an assignment of rate constants such that the resulting mass action differential equations admit no complex balanced positive equilibrium.
- 3.
From results in Chapter 10, it will follow that the conjecture cannot be violated by any network in the very large class of concordant networks: Weak reversibility is a necessary condition for complex balancing [107]. For any weakly reversible concordant network, no positive stoichiometric compatibility class can have on its boundary a reaction-transitive composition, in particular an equilibrium. Siegel and MacLean [161] showed that any violation of Assertion A requires such a boundary equilibrium.
- 4.
In more recent research, the technical formulation of Assertion B usually requires that, for weakly reversible mass action systems, the ω-limit set associated with any bounded trajectory originating in \({\mathbb {R}_+^{\mathcal {S}}}\) lies entirely in \({\mathbb {R}_+^{\mathcal {S}}}\). A discussion of various technical formulations of Assertion B can be found in [3].
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Feinberg, M. (2019). The Deficiency Zero Theorem. In: Foundations of Chemical Reaction Network Theory. Applied Mathematical Sciences, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-030-03858-8_7
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