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1.A. Chemical Reaction Network Theory: Early Review Articles

    Although they are now a little outdated, the following reviews for a general audience give a fairly good summary of some early results in chemical reaction network theory. These early results tie the qualitative behavior of mass action systems to a property of the underlying network called the network’s deficiency. (The deficiency is a non-negative integer index with which reaction networks can be classified.)  A very early deficiency-oriented article can be found here:

[1.A1] M. Feinberg and F. J. M. Horn. Dynamics of open chemical systems and the algebraic structure of the underlying reaction network. Chemical Engineering Science, 29:775–787, 1974.

The following review is, I think, a good place to begin. It was written in connection with a symposium attended by a nice mix of chemists, engineers, and mathematicians.

[1.A2] Martin Feinberg. Chemical oscillations, multiple equilibria, and reaction network structure. In Warren E. Stewart, W. Harmon Ray, and Charles C. Conley, editors, Dynamics and Modeling of Reactive Systems, pages 59–130. Academic Press, New York, 1980.

The following articles constitute a two-part review, written a little later than the 1980 review. They take things somewhat further and provide much more detail.

[1.A3] M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors I. The deficiency zero and deficiency one theo­rems. Chemical Engineering Science, 42:2229–2268, 1987.

[1.A4] M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors II. Multiple steady states for networks of defi­ciency one. Chemical Engineering Science, 43:1–25, 1988.

The next review was written even earlier than any of the previous ones. Among other things, it attempts to explain for a chemical engineering audi­ence a little bit about the proof of the Deficiency Zero Theorem. This review highlights the crucial work of Horn and Jackson, but it also uses a line of argument that’s a little different from theirs.

[1.A5] M. Feinberg. Mathematical aspects of mass action kinetics. In N. Amund­son and L. Lapidus, editors, Chemical Reactor Theory: A Review, pages 1–78. Prentice-Hall, Englewood Cliffs, NJ, 1977.

If, however, you are a mathematician seeking proofs, then I’d strongly recommend Lectures on Chemical Reaction Networks, which are a written ver­sion of about half the lectures I gave in 1979 at the Mathematics Research Center of the University of Wisconsin. (I got distracted by foundations of classical thermodynamics before I wrote the remaining half. You’ll see some of that work, appearing at about the same time, elsewhere in this bibliog­raphy.) Lectures will take you up through a proof of the Deficiency Zero Theorem.

Here is a link to the written lectures:

[1.A6] Feinberg, M., Lectures on Chemical Reaction Networks, written versions of lectures given at the Mathematics Research Center of the University of Wisconsin, 1979.

Although it was written substantially later, I am including with this set of articles the following one because it travels further down the same road. The article describes progress, based on the PhD work of Phillipp Ellison, toward extending previous results about deficiency one networks to networks of higher deficiency. In particular, Phillipp’s work, like deficiency one theory, aims to translate questions about a network's capacity for multiple steady states (and therefore questions about systems of nonlinear equations) into questions about systems of linear inequalities, which are far more tractable. Phillipp’s work in both theory and software development provided the basis for a major improvement in The Chemical Reaction Network Toolbox.

[1.A7] P. Ellison and M. Feinberg. How catalytic mechanisms reveal themselves in multiple steady-state data: I. Basic principles. Journal of Molecular Catal­ysis. A: Chemical, 154(1-2):155–167, 2000.