The Deficiency Zero Theorem

  • Martin Feinberg
Part of the Applied Mathematical Sciences book series (AMS, volume 202)


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.


  1. 2.
    Anderson, D.F.: Global asymptotic stability for a class of nonlinear chemical equations. SIAM Journal on Applied Mathematics 68(5), 1464 (2008)MathSciNetCrossRefGoogle Scholar
  2. 3.
    Anderson, D.F.: A proof of the global attractor conjecture in the single linkage class case. SIAM Journal on Applied Mathematics 71(4), 1487–1508 (2011)MathSciNetCrossRefGoogle Scholar
  3. 6.
    Anderson, D.F., Shiu, A.: The dynamics of weakly reversible population processes near facets. SIAM Journal on Applied Mathematics 70(6), 1840–1858 (2010)MathSciNetCrossRefGoogle Scholar
  4. 30.
    Brauer, F., Nohel, J.A.: The Qualitative Theory of Ordinary Differential Equations: An Introduction. Dover Publications (1989)Google Scholar
  5. 34.
    Chicone, C.: Ordinary Differential Equations with Applications, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  6. 45.
    Craciun, G.: Toric differential inclusions and a proof of the global attractor conjecture. arXiv:1501.02860 (2015)Google Scholar
  7. 50.
    Craciun, G., Nazarov, F., Pantea, C.: Persistence and permanence of mass-action and power-law dynamical systems. SIAM Journal on Applied Mathematics 73(1), 305–329 (2013)MathSciNetCrossRefGoogle Scholar
  8. 67.
    Feinberg, M.: Complex balancing in general kinetic systems. Archive for Rational Mechanics and Analysis 49(3), 187–194 (1972)MathSciNetCrossRefGoogle Scholar
  9. 68.
    Feinberg, M.: On chemical kinetics of a certain class. Archive for Rational Mechanics and Analysis 46(1), 1–41 (1972)MathSciNetCrossRefGoogle Scholar
  10. 71.
    Feinberg, M.: Lectures on Chemical Reaction Networks (1979). Written version of lectures given at the Mathematical Research Center, University of Wisconsin, Madison, WI Available at
  11. 72.
    Feinberg, M.: Chemical oscillations, multiple equilibria, and reaction network structure. In: W.E. Stewart, W.H. Ray, C. Conley (eds.) Dynamics and Modeling of Reactive Systems, Mathematics Research Center Symposia Series, pp. 59–130. Academic Press (1980). Available at
  12. 73.
    Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors I. The deficiency zero and deficiency one theorems. Chemical Engineering Science 42(10), 2229–2268 (1987)Google Scholar
  13. 75.
    Feinberg, M.: Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chemical Engineering Science 44(9), 1819–1827 (1989)CrossRefGoogle Scholar
  14. 79.
    Feinberg, M., Horn, F.J.M.: Dynamics of open chemical systems and the algebraic structure of the underlying reaction network. Chemical Engineering Science 29(3), 775–787 (1974)CrossRefGoogle Scholar
  15. 104.
    Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third Edition. Academic Press (2012)zbMATHGoogle Scholar
  16. 107.
    Horn, F.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Archive for Rational Mechanics and Analysis 49(3), 172–186 (1972)MathSciNetCrossRefGoogle Scholar
  17. 108.
    Horn, F.: The dynamics of open reaction systems. In: Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, Proceedings of the SIAM-AMS Symposium on Applied. Mathematics, vol. 8, pp. 125–137 (1974)Google Scholar
  18. 109.
    Horn, F., Jackson, R.: General mass action kinetics. Archive for Rational Mechanics and Analysis 47(2), 81–116 (1972)MathSciNetCrossRefGoogle Scholar
  19. 135.
    Pantea, C.: On the persistence and global stability of mass-action systems. SIAM Journal on Mathematical Analysis 44(3), 1636–1673 (2012)MathSciNetCrossRefGoogle Scholar
  20. 152.
    Shapiro, A.: Statics and dynamics of multicell reaction systems. Ph.D. thesis, University of Rochester (1975)Google Scholar
  21. 153.
    Shapiro, A., Horn, F.J.M.: On the possibility of sustained oscillations, multiple steady states, and asymmetric steady states in multicell reaction systems. Mathematical Biosciences 44(1–2), 19–39 (1979). [See Errata in 46(2):157 (1979).]CrossRefGoogle Scholar
  22. 161.
    Siegel, D., MacLean, D.: Global stability of complex balanced mechanisms. Journal of Mathematical Chemistry 27(1), 89–110 (2000)MathSciNetCrossRefGoogle Scholar
  23. 162.
    Sontag, E.D.: Structure and stability of certain chemical networks and applications to the kinetic proofreading model of t-cell receptor signal transduction. IEEE Transactions on Automatic Control 46(7), 1028–1047 (2001)MathSciNetCrossRefGoogle Scholar
  24. 164.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second Edition. Westview Press (2014)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Martin Feinberg
    • 1
  1. 1.Chemical & Biomolecular Engineering, The Ohio State UniversityColumbusUSA

Personalised recommendations