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On the polynomial first integrals of the (a, b, c) Lotka-Volterra system

  • Dynamical Systems
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Dynamical Systems, Plasmas and Gravitation

Part of the book series: Lecture Notes in Physics ((LNP,volume 518))

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Abstract

Using elementary differential algebraic techniques, we prove that the 3-D Lotka-Volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones.

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References

  • Cairó, L., Feix, M.R., Goedert, J. (1989): Invariants for models of interacting populations. Phys. Lett. A 140, 421–427

    Article  ADS  MathSciNet  Google Scholar 

  • Cairó, L., Feix, M.R. (1992): Families of invariants of motion for the Lotka-Volterra equations: the linear polynomials family. J. Math. Phys. 33, 2440–2455

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Darboux, G. (1878): Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. Bull. Sc. Math. 2ème série 2, 60–96, 123–144, 151–200

    Google Scholar 

  • Grammaticos, B., Moulin-Ollagnier, J., Ramani, A., Strelcyn, J.M., Wojciechowski, S. (1990); Integrals of quadratic ordinary differential equations in R 3: the Lotka-Volterra system. Physica A 163, 683–722

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Kummer, M., Churchill, R.C., Rod, D.L. (1990): On a result of Bruns. Can. Math. Bull. 33 No. 2 175–180

    Article  MATH  MathSciNet  Google Scholar 

  • Labrunie, S. (1996): On the polynomial first integrals of the (a, b, c) Lotka-Volterra system. J. Math. Phys. 37, 5539–5550

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Lotka, A.J. (1925): Elements of Physical Biology (Williams and Wilkins, Baltimore)

    MATH  Google Scholar 

  • Man, Y.K. (1994): First integrals of autonomous systems of differential equations and the Prelle-Singer procedure. J. Phys. A 27, L327–L332

    Article  MathSciNet  Google Scholar 

  • Moulin-Ollagnier, J. (1997): Polynomial first integrals of the Lotka-Volterra vector field, to appear in Bull. Sc. Math. 121

    Google Scholar 

  • Volterra, V. (1931): Leçons sur la théorie mathématique de la lutte pour la vie (Gauthier-Villars, Paris)

    Google Scholar 

  • Ziglin, S.L. (1982): Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. Funct. Appl. Anal. 16, 181–189

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. L.P.T.M.C., Université Paris VII, 2 place Jussieu, 75251, Paris Cedex 05, France

    Simon Labrunie

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  1. Simon Labrunie
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Editor information

P. G. L. Leach S. E. Bouquet J.-L. Rouet E. Fijalkow

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© 1999 Springer-Verlag

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Labrunie, S. (1999). On the polynomial first integrals of the (a, b, c) Lotka-Volterra system. In: Leach, P.G.L., Bouquet, S.E., Rouet, JL., Fijalkow, E. (eds) Dynamical Systems, Plasmas and Gravitation. Lecture Notes in Physics, vol 518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105931

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  • DOI: https://doi.org/10.1007/BFb0105931

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65467-4

  • Online ISBN: 978-3-540-49251-1

  • eBook Packages: Springer Book Archive

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