Skip to main content
SpringerLink
Log in

Liouvillian first integrals for the planar Lotka-Volterra system

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Abstract

We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almeida M. A., Magalhães M. E., Moreira I. C.,Lie symmetries and invariants of the Lotka-Volterra system, J. Math. Phys.,36 (1995), 1854–1867.

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrade R. F. S., Rauh A.,The Lorenz model and the method of Carleman embedding, Phys. Lett. A,82 (1981), 276–278.

    Article  MathSciNet  Google Scholar 

  3. Bountis T., Grammaticos B., Ramani A.,On the complete and partial integrability of non-Hamiltonian systems, Phys. Rep.,180 (1989), 159.

    Article  MathSciNet  Google Scholar 

  4. Brenig L., Goriely A.,A. Quasimonomial transformations and integrability. Partially integrable evolution equations in physics, (Les Houches, 1989), 571–572, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht, 1990.

    Google Scholar 

  5. Cairó L., Feix M. R., Llibre J.,Integrability and invariant algebraic curves for planar polynomial differential systems with emphasis on the quadratic systems, Resenhas de Universidade de Sao Paulo,4 (1999), 127–161.

    MATH  Google Scholar 

  6. Cairó L., Llibre J.,Darboux integrability for the 3-dimensional Lotka-Volterra systems, J. Phys. A: Math. and Gen.,33 (2000), 2395–2406.

    Article  MATH  Google Scholar 

  7. Cairó L., Llibre J.,Darboux first integrals and invariants for real quadratic systems having an invariant conic, J. Phys. A: Math. and Gen.,35 (2002), 589–608.

    Article  MATH  Google Scholar 

  8. Cantrijn F., Sarlet W.,Generalizations of Noether’s theorem in classical mechanics, SIAM Rev.,23 (1981), 467–494.

    Article  MATH  MathSciNet  Google Scholar 

  9. Carleman T.,Application de la théorie des équations intégrales linéaires aux systémes d’équations différentielles non linéaires, Acta Mathematica,59 (1932), 63.

    Article  MathSciNet  Google Scholar 

  10. Chavarriga J., Giacomini H., Giné J., Llibre J.,Darboux integrability and the inverse integrating factor, to appear in J. of Differential Equations.

  11. Chavarriga J., Llibre J.,Invariant algebraic curves and rational first integrals for planar polynomial vector fields, J. Differential Equations,169 (2001), 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  12. Chavarriga J., Llibre J., Sotomayor J.,Algebraic solutions for polynomial vector fields with emphasis in the quadratic case, Expositions Math.,15 (1997), 161–173.

    MATH  MathSciNet  Google Scholar 

  13. Christopher C.,Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburgh,124A (1994), 1209–1229.

    MathSciNet  Google Scholar 

  14. Christopher C. J., Llibre J.,Algebraic aspects of integrability for polynomial systems, Qualitative Theory of Dynamical Systems,1 (1999), 71–95.

    MathSciNet  Google Scholar 

  15. Christopher C., Llibre J.,Integrability via invariant algebraic curves for planar polynomial differential systems, Annals of Differential Equations,16 (2000), 5–19.

    MATH  MathSciNet  Google Scholar 

  16. Christopher C., Llibre J., Pereira J. V.,Multiplicity of invariant algebraic curves and Darboux integrability, preprint, 2002.

  17. Darboux G.,Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math., 2ème série,2 (1878), 60–96; 123–144; 151–200.

    Google Scholar 

  18. Forgas P., Giacomini H., Llibre J., Viano M.,On the Liouville integrability of Bogomolnyi equations for Einstein-Yang-Mills-dilation theory, preprint.

  19. Giacomini H. J., Repetto C. E., Zandron O. P.,Integrals of motion of three-dimensional non-Hamiltonian dynamical systems, J. Phys. A,24 (1991), 4567–4574.

    Article  MATH  MathSciNet  Google Scholar 

  20. Goriely A.,Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys.,37 (1996), 1871–1893.

    Article  MATH  MathSciNet  Google Scholar 

  21. Gradshteyn I. S., Ryzhik I. M.,Table of integrals, series, and products, Translated from the Russian. Sixth edition. Academic Press, Inc., San Diego, CA, 2000.

    MATH  Google Scholar 

  22. Grammaticos B., Moulin Ollagnier J., Ramani A., Strelcyn J. M., Wojciechowski S.,Integrals of quadratic ordinary differential equations in ℝ3 : the Lotka-Volterra system, Physica A,163 (1990), 683–722.

    Article  MATH  MathSciNet  Google Scholar 

  23. Gutierrez C., Llibre J.,Darboux integrability for polynomial vector fields on the 2-dimensional sphere, Extracta Mathematicae,17 (2002), 289–301

    MATH  MathSciNet  Google Scholar 

  24. Hietarinta J.,Direct methods for the search of the second invariant, Phys. Rep.,147 (1987), 87–154.

    Article  MathSciNet  Google Scholar 

  25. Jouanolou J. P.,Equations de Pfaff algébriques, in “Lectures Notes in Mathematics,”708, Springer-Verlag, New York/Berlin, 1979.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Lax P. D.,Integrals of Nonlinear Equations of Evolution and Solitary Waves, Commun. Pure Appl. Math.,21 (1968), 467–490.

    Article  MATH  MathSciNet  Google Scholar 

  28. Llibre J., Rodríguez G.,Invariant hyperplanes and Darbox integrability for d-dimensional polynomial differential systems, Bull. Sci. Math.,124 (2000), 1–21.

    Article  MathSciNet  Google Scholar 

  29. Llibre J., Rodríguez G.,Darboux integrability of polynomial vector fields on 2-dimensional surfaces, to appear in J. of Bifurcation and Chaos,12 (2002).

  30. Llibre J., Schlomiuk D.,The geometry of differential quadratic systems with a weak focus of third order, to appear in Canadian J. of Math.

  31. Llibre J., Zhang X.,Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces, Rendiconti del circolo matematico di Palermo, Serie II,LI (2002), 109–126.

    Article  MathSciNet  Google Scholar 

  32. Lotka A. J.,Analytical note on certain rhythmic relations in organic systems, Proc. Nat. Acad. Sci.,6 (1920), 410.

    Article  Google Scholar 

  33. Moulin Ollagnier J.,Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. math.,121 (1997), 463–476.

    MATH  MathSciNet  Google Scholar 

  34. Moulin Ollagnier J.,Rational integration of the Lotka-Volterra system, Bull. Sci. math.,123 (1999), 437–466.

    Article  MATH  MathSciNet  Google Scholar 

  35. Moulin Ollagnier J.,Liouvillian Integration of the Lotka-Volterra system, Qualitative Theory of Dynamical Systems,2 (2001), 307–358.

    MathSciNet  Google Scholar 

  36. Moulin Ollagnier J.,Some remarks about the integration of polynomial planar vector fields, Qualitative Theory of Dynamical Systems,3 (2002), 19–28.

    Article  MATH  MathSciNet  Google Scholar 

  37. Moulin Ollagnier J.,About a conjecture on quadratic vector fields, Journal of Pure and Applied Algebra,165 (2001), 227–234.

    Article  MATH  MathSciNet  Google Scholar 

  38. Olver P. J.,Applications of Lie groups to differential equations, Springer, New York, 1986.

    MATH  Google Scholar 

  39. Poincaré H.,Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo,5 (1891), 161–191;11 (1897), 193–239.

    Google Scholar 

  40. Prelle M. J., Singer M. F.,Elementary first integrals of differential equations, Trans. Amer. Math. Soc.,279 (1983), 613–636.

    Article  MathSciNet  Google Scholar 

  41. Schlomiuk D.,Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc.,338 (1993), 799–841.

    Article  MATH  MathSciNet  Google Scholar 

  42. Schlomiuk D.,Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk (ed.), 1993, 429–467.

  43. Singer M. F.,Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc.,333 (1992), 673–688.

    Article  MATH  MathSciNet  Google Scholar 

  44. Strelcyn J. M., Wojciechowski S.,A method of finding integrals for three-dimensional dynamical systems, Phys. Lett. A,133 (1988), 207–212.

    Article  MathSciNet  Google Scholar 

  45. Volterra V.,Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931.

    Google Scholar 

  46. Yablonskii A. I.,Algebraic integrals of a differential-equation system, Differential Equations,6 (1970), 1326–1333.

    Google Scholar 

  47. Weil J. A.,Constantes et polynômes de Darboux en algèbre différentielle: applications aux systèmes différentiels linéaires, Ph. D., École Polytecnique, 1995.

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université d’Orléans, MAPMO, UMR 6628, BP 6759, 45067, Orléans, Cédex 2, France

    Laurent Cairó

  2. Laboratoire de Mathématique et Physique Théorique CNRS(UPRES-A 6083) Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200, Tours, France

    Hector Giacomini

  3. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Jaume Llibre

Authors
  1. Laurent Cairó
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hector Giacomini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jaume Llibre
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cairó, L., Giacomini, H. & Llibre, J. Liouvillian first integrals for the planar Lotka-Volterra system. Rend. Circ. Mat. Palermo 52, 389–418 (2003). https://doi.org/10.1007/BF02872763

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02872763

Keywords

Search

Navigation