Journal of Mathematical Chemistry

, Volume 52, Issue 2, pp 399–406 | Cite as

Solvable model for chemical oscillations

  • Eisuke ChikayamaEmail author
  • Yasuhiro Sunaga
  • Shigeho Noda
  • Hideo Yokota
Brief Communication


The Lotka–Volterra equation, proposed first with two variables by A. J. Lotka, underpins the well-known classic model for chemical oscillations. The general solutions of the Lotka–Volterra equation, with \(n\) variables, however, remain unknown. We describe a solvable nonlinear model and general solution, previously unstudied for chemical oscillations, that is analogous to the Lotka–Volterra equations with \(n\) variables. This model approximates the Lotka–Volterra equations in the neighbourhood of an equilibrium point and is solvable because it can be shown to be linearized to a set of first-order linear differential equations. The purpose of this report is a description of the general solution of the model.


Solvable model Chemical oscillation Lotka–Volterra equation Nonlinear differential equation 

Mathematics Subject Classification

80A30 92E20 



We thank Yohei Nanazawa for discussions concerning our study. The numerous numerical simulations performed using the RIKEN Integrated Cluster of Clusters (RICC) improved the simulations performed for this study.


  1. 1.
    A.J. Lotka, J. Am. Chem. Soc. 42, 1595 (1920)CrossRefGoogle Scholar
  2. 2.
    R.M. May, W.J. Leonard, SIAM J. Appl. Math. 29, 243 (1975)CrossRefGoogle Scholar
  3. 3.
    A. Arneodo et al., J. Math. Biol. 14, 153 (1982)CrossRefGoogle Scholar
  4. 4.
    C.M. Evans, G.L. Findley, J. Math. Chem. 25, 105 (1999)CrossRefGoogle Scholar
  5. 5.
    C.M. Evans, G.L. Findley, J. Math. Chem. 25, 181 (1999)CrossRefGoogle Scholar
  6. 6.
    L. Brenig, Phys. Lett. A 133, 378 (1988)CrossRefGoogle Scholar
  7. 7.
    S.A. Levin, L.A. Segel, Nature 259, 659 (1976)CrossRefGoogle Scholar
  8. 8.
    L.J.S. Allen, Math. Biosci. 65, 1 (1983)CrossRefGoogle Scholar
  9. 9.
    F. Rothe, J. Math. Biol. 3, 319 (1976)CrossRefGoogle Scholar
  10. 10.
    A. Hastings, J. Math. Biol. 6, 163 (1978)CrossRefGoogle Scholar
  11. 11.
    E.E. Holmes et al., Ecology 75, 17 (1994)CrossRefGoogle Scholar
  12. 12.
    P.L. Chow, W.C. Tam, Bull. Math. Biol. 38, 643 (1976)Google Scholar
  13. 13.
    K. Orihashi, Y. Aizawa, Phys. D 240, 1853 (2011)CrossRefGoogle Scholar
  14. 14.
    J.A. Sherratt et al., Proc. Nat. Acad. Sci. U.S.A. 92, 2524 (1995)CrossRefGoogle Scholar
  15. 15.
    M.W. Hirsch et al., Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd edn. (Academic Press, San Diego, 2004)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Eisuke Chikayama
    • 1
    • 2
    Email author
  • Yasuhiro Sunaga
    • 1
  • Shigeho Noda
    • 1
  • Hideo Yokota
    • 1
  1. 1.Cell Scale Team, Integrated Simulation of Living Matter GroupRIKENSaitamaJapan
  2. 2.Department of Information SystemsNiigata University of International and Information StudiesNiigataJapan

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