Regular and Chaotic Dynamics

, Volume 13, Issue 6, pp 543–556 | Cite as

Integrable Lotka-Volterra systems

Jürgen Moser - 80


Infinite- and finite-dimensional lattices of Lotka-Volterra type are derived that possess Lax representations and have large families of first integrals. The obtained systems are Hamiltonian and contain perturbations of Volterra lattice. Examples of Liouville-integrable 4-dimensional Hamiltonian Lotka-Volterra systems are presented. Several 5-dimensional Lotka- Volterra systems are found that have Lax representations and are Liouville-integrable on constant levels of Casimir functions.

Key words

Lax representation Hamiltonian structures Casimir functions Riemannian surfaces Lotka-Volterra systems integrable lattices 

MSC2000 numbers



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  1. 1.
    Volterra, V., Leçons sur la théorie mathématique de la lutte pour la vie, Paris: Gauthier-Villars, 1931; Reprinted in: Les Grands Classiques Gauthier-Villars, Sceaux: J. Gabay, 1990.Google Scholar
  2. 2.
    Moser, J., Three Integrable Hamiltonian Systems Connected with Isospectral Deformations, Adv. Math., 1975, vol. 16, pp. 197–220.MATHCrossRefGoogle Scholar
  3. 3.
    Bogoyavlenskij, O. I., Some Constructions of Integrable Dynamical Systems, Izv. Akad. Nauk SSSR, Ser. Mat., 1987, vol. 51, no. 4, pp. 737–766, 910 [Math. USSR-Izv., 1988, vol. 31, no. 1, pp. 47–75].MATHMathSciNetGoogle Scholar
  4. 4.
    Bogoyavlenskij, O. I., Integrable Dynamical Systems Connected with the KdV Equation, Izv. Akad. Nauk SSSR, Ser. Mat., 1987, vol. 51, no. 6, pp. 1123–1141.MATHGoogle Scholar
  5. 5.
    Schouten, J.A., Ueber Differentialkomitanten zweier kontravarianter Grössen, Proc. Nederl. Akad. Wetensch., Amsterdam, 1940, vol. 43, pp. 449–452.MathSciNetGoogle Scholar
  6. 6.
    Bogoyavlenskij, O. I., On Perturbations of the Periodic Toda Lattice, Comm. Math. Phys., 1976, vol. 51, pp. 201–209.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bogoyavlenskij, O. I., Algebraic Constructions of Integrable Dynamical Systems — Extension of the Volterra System, Russian Math. Surveys, 1991, vol. 46, pp. 1–64.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kac, M. and van Moerbeke, P., On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices, Adv. Math., 1975, vol. 16, pp. 160–169.MATHCrossRefGoogle Scholar
  9. 9.
    Itoh, Y., Integrals of a Lotka-Volterra System of Odd Number of Variables, Progr. Theoret. Phys., 1987, vol. 78, no. 3, pp. 507–510.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Itoh, Y., A Combinatorial Method for the Vanishing of the Poisson Brackets of an Integrable Lotka-Volterra System, J. Phys. A, 2009 (to appear).Google Scholar
  11. 11.
    Manakov, S. V., On a Complete Integrability and Stochastization in Discrete Dynamical Systems, J. Exp. Theor. Phys., 1974, vol. 67, pp. 543–555.MathSciNetGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Department of MathematicsQueen’s UniversityKingstonCanada
  2. 2.V.A. Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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