# Mathematics of Reaction Networks

Mathematical modeling of chemical reaction networks consists of a variety of methods for approaching questions about the dynamical behaviour of chemical reactions arising in real world applications.

This wiki is intended to serve the dual purpose of being an accessible primer for students and researchers new to the area of mathematical modeling of chemical reactions, and a summary of the current state of the discipline for those who are active in the field.

## Resources for research

The following resources are intended to assist people who are familiar with, and actively involved in, research in modeling of reaction networks. See the active research topics pages for discussion of current research initiatives.

## Resources for education

This section is intended as a primer for those who are curious about the mathematics underlying the study of reaction networks.

### Chemical reaction network theory

Chemical reaction network theory is a framework for modeling the evolution of chemical concentrations resulting from simultaneously occurring chemical reactions. A key feature of the theory is the relationship between the graphical structure of the reaction network and the resulting dynamics. A strong emphasis, consequently, is placed on results which hold regardless of the parameter values of the network, i.e. results which depend on the network structure alone.

The foundations of chemical reaction network theory were laid down in a series of seminal papers by Fritz Horn, Roy Jackson and Martin Feinberg in the early 1970's [1][2][3]. In these papers, the authors were primarily focused on developing conditions sufficient for uniqueness and stability of equilibrium concentrations, but their foundation has since between adapted to questions of multistability, injectivity, monotonicity, persistence, equivalence of mass-action systems, model reduction, oscillations, and applications. The models have also been adapted to the stochastic framework, reaction-diffusion equations, and kinetic schemes other than mass-action (e.g. Michaelis-Menten, Hill kinetics, etc.).

## References

1. R. Horn, R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal. 47 (1972) 81-116
2. F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Ration. Mech. Anal., 49 (1972) 172-186
3. M. Feinberg, Complex balancing in general kinetic systems, Arch. Ration. Mech. Anal., 49 (1972) 187-194

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