Darboux first integral conditions and integrability of the 3D lotka-volterra system

Journal of Nonlinear Mathematical Physics

Volumn 7, Issue 4, 2000, Pages 511-531

  cairó, Laurent ^a  

^a MAPMO–UMR 6628 Université d’Orléans UFR Sciences ^*  (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034553953     PISSN: 14029251     EISSN: 17760852     Source Type: Journal    
DOI: 10.2991/jnmp.2000.7.4.7     Document Type: Article

References (31)

1. Lotka, A.J., 1920. Analytical Note on Certain Rhythmic Relations in Organic Systems. Proc. Natl. Acad. Sci. U.S., 6:410–415.
2. Volterra, V., 1931. Leçons sur la Théorie Mathématique de la Lutte pour la vie, Paris:Gauthier Villars.
3. May, R.M., and Leonard, W.J., 1975. Nonlinear Aspects of Competition Between Three Species. Siam J. Appl. Math., 29:243–253.
4. Cairó, L., and Llibre, J., 2000. Darboux Integrability for the 3-Dimensional Lotka–Volterra Systems. J. Physics A, Math. and Gen., 33:2395–2406.
5. Lamb, W.E., 1964. Theory of an Optical Maser. Phys Rev. A, 134:1429
6. Laval, G., and Pellat, R., 1975. Plasma Physics, in Proceedings of Summer School of Theoretical Physics, New York:Gordon and Breach.
7. Busse, F.H., 1978. Transitions to Turbulence Via the Statistical Limit Cycle Route, in Synergetics, 39Berlin:Springer-Verlag.
8. Lupini, R., and Spiga, G., 1988. Chaotic Dynamics of Spatially Homogeneous Gas Mixtures. Phys. Fluids., 31:2048–2051.
9. Bogoyavlensky, O.I., 1988. Integrable Discretizations of the KdV Equations. Phys. Lett. A, 134:34
10. Noonburg, V.W., 1989. A Neural Network Modeled by an Adaptive Lotka-Volterra System. SIAM J. Appl. Math., 49:1779
11. Brenig, L., 1988. Complete Factorisation and Analytic Aolutions of Generalized Lotka-Volterra Equations. Phys. Lett., 133:378–382.
12. Brenig, L., and Goriely, A., 1989. Universal Canonical Forms for Time-Continuous Dynamical Systems. Phys. Rev. A, 40:4119–4122.
13. Kus`, M., 1983. Integrals of Motion for the Lorenz System. J. Phys. A:Math. Gen., 16:L689–L691.
14. Cairó, L., and Feix, M.R., 1992a. Families of Invariants of the Motion for the Lotka-Volterra Equations:the Linear Polynomials Family. J. Math. Phys., 33:2440–2455.
15. Cairó, L., Feix, M.R., Hua, D.D., and Bouquet, S., 1993a. Hamiltonian Method and Invariant Search for 2D Quadratic Systems. J. Phys. A:Math. Gen., 26:4371–4386.
16. Hua, D.D., Cairó, L., and Feix, M.R., 1993b. Time-Independent Invariants for the Quadratic System. J. Phys. A:Math. Gen., 26:7097–7114.
17. Bountis, T., Ramani, A., Grammaticos, B., and Dorizzi, B., 1984. ‘On the Complete and Partial Integrability of Non-Hamiltonian Systems. Physica A, 128:268–288.
18. Goriely, A., 1992. Investigation of Painlevé Property Under Time Singularities Transformations. J. Math. Phys., 33:2728–2742.
19. Steeb, W.H., 1982. Continuous Symmetries of the Lorenz Model and the Rikitake Two-Disc Dynamo System. J. Phys. A:Math. Gen., 15:L389–L390.
20. Almeida, M.A., Magalhñaes, M.E., and Moreira, I.C., 1995. Lie Symmetries and Invariants of the Lotka–Volterra System. J. Math. Phys., 36:1854–1867.
21. Strelcyn, J.M., and Wojciechowski, S., 1988. A Method of Finding Integrals for Three-Dimensional Dynamical Systems. Phys. Lett. A, 133:207–212.
22. 3:the Lotka-Volterra System. Physica A, 163:683–722.
23. Cairó, L., and Feix, M.R., 1992b. On the Hamiltonian Structure of the 2D ODE Possessing an Invariant. J. Phys. A:Math. Gen., 25:L1287–L1293.
24. Hua, D.D., Cairó, L., and Feix, M.R., 1993c. ‘A General Time-Fependent Invariant for and Integrability of the Quadratic System. Proc. of the Royal Soc. London A, 443:643–650.
25. Darboux, G., 1878. Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. (Mélanges), Bull. Sci. Math., 60–96:123–144. 151–200
26. Schlomiuk, D., 1993. Algebraic Particular Integrals, Integrability and the Problem of the Center. Trans. Amer. Math. Soc., 338:799–841.
27. Cairó, L., Llibre, J., and Feix, M.R., 1999. Darboux Method and Search of Invariants for the Lotka-Volerra and Complex Quadratic Systems. J. of Math. Phys., 40:2074–2091.
28. Cairó, L., Llibre, J., and Feix, M.R., 2000. Integrability and Algebraic Aolutions for Planar Polynomial Differential Systems with Emphasis on the Quadratic Systems. Resenhas IME–USP, 4:127–161.
29. Cairó, L., and Llibre, J., 1999. “Integrability and Algebraic Solutions for the 2-d Lotka–Volterra System”. In Dynamical systems, Plasmas and Gravitation, Vol. 518, 243–254. Springer–Verlag Monograph in Physics.
30. Moulin-Ollagnier, J., 1999. Rational Integration of the Lotka–Volterra System. Bull. Sci. Math., 123:437–466.
31. Cairó, L., and Llibre, J., 2000. Algebraic Integrability for a Class of 3-Dimensional Lotka–Volterra Systems, preprint Univ. Orleans.