Acta Applicandae Mathematicae

, Volume 151, Issue 1, pp 53–80 | Cite as

On Global Stability of the Lotka Reactions with Generalized Mass-Action Kinetics

Article

Abstract

Chemical reaction networks with generalized mass-action kinetics lead to power-law dynamical systems. As a simple example, we consider the Lotka reactions with two chemical species and arbitrary power-law kinetics. We study existence, uniqueness, and stability of the positive equilibrium, in particular, we characterize its global asymptotic stability in terms of the kinetic orders.

Keywords

Chemical reaction network Power-law kinetics Andronov-Hopf bifurcation Dulac function 

References

  1. 1.
    Andronov, A.A., Leontovich, E.A.: Some cases of dependence of limit cycles on a parameter. Uch. Zap. Gorkovskogo Univ. 6, 3–24 (1937) Google Scholar
  2. 2.
    Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maĭer, A.G.: Theory of Bifurcations of Dynamic Systems on a Plane. Halsted Press, New York (1973), translated from the Russian, Israel Program for Scientific Translations, Jerusalem-London MATHGoogle Scholar
  3. 3.
    Dancsó, A., Farkas, H., Farkas, M., Szabó, G.: Investigations into a class of generalized two-dimensional Lotka-Volterra schemes. Acta Appl. Math. 23(2), 103–127 (1991) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Farkas, H., Noszticzius, Z.: Generalized Lotka-Volterra schemes and the construction of two-dimensional explodator cores and their Liapunov functions via “critical” Hopf bifurcations. J. Chem. Soc. Faraday Trans. II 81(10), 1487–1505 (1985) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farkas, M.: Periodic Motions. Applied Mathematical Sciences, vol. 104. Springer, New York (1994) MATHGoogle Scholar
  6. 6.
    Frommer, M.: Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmtheitsstellen. Math. Ann. 109(1), 395–424 (1934) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences., vol. 42. Springer, New York (1990), revised and corrected reprint of the 1983 original MATHGoogle Scholar
  8. 8.
    Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 94, 1–22 (1942) MATHGoogle Scholar
  9. 9.
    Llibre, J.: A counterexample to a result on Lotka-Volterra systems. Acta Appl. Math. 142(1), 123–125 (2016) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lotka, A.J.: Contribution to the theory of periodic reactions. J. Phys. Chem. 14(3), 271–274 (1910) CrossRefGoogle Scholar
  11. 11.
    Lotka, A.J.: Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. 6(7), 410–415 (1920) CrossRefGoogle Scholar
  12. 12.
    Lotka, A.J.: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42, 1595–1599 (1920) CrossRefGoogle Scholar
  13. 13.
    Müller, S., Regensburger, G.: Generalized mass action systems: complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math. 72, 1926–1947 (2012) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Müller, S., Regensburger, G.: Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.H. (eds.) Computer Algebra in Scientific Computing. Proceedings of the 16th International Workshop (CASC 2014). Lecture Notes in Comput. Sci., vol. 8660, pp. 302–323. Springer, Berlin (2014) Google Scholar
  15. 15.
    Yin, W., Voit, E.O.: Construction and customization of stable oscillation models in biology. J. Biol. Syst. 16(04), 463–478 (2008) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria

Personalised recommendations