Global Dynamics of 3D Type-K Competitive Lotka-Volterra System with the Identical Intrinsic Growth Rate

• Fengli Liang ^[1] ; Jifa Jiang ^[2] ; Xiang Zhang ^[3]
  1. [1] Anhui Normal University

     Anhui Normal University

     China

     
  2. [2] Henan Normal University

     Henan Normal University

     China

     
  3. [3] Shanghai Jiao Tong University

     Shanghai Jiao Tong University

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
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• Resumen

  • For the three-dimensional type-K competitive Lotka-Volterra system with the identical intrinsic growth rate, we completely characterize its global dynamics in the Poincaré compactification of the system in the positive octant of R3. Precisely, with the help of the replicator equations it is proved that this kind of system can have exactly 44 topologically different phase portraits. As a consequence, we obtain the necessary and sufficient conditions for the system to be bounded in the positive octant and verify that the α and ω limit sets of any orbit of the compactified vector field associated to the system are both equilibria.

• Referencias bibliográficas
  • 1. Alavez-Ramírez, J., Blé, G., Castellanos, V., Llibre, J.: On the global flow of a 3-dimensional LotkaVolterra system. Nonlinear Anal. 75,...
  • 2. Bautin, N.N.: On periodic solutions of a system of differential equations (Russian). Prikl. Mat. Mech. 18, 128 (1954)
  • 3. Blé, G., Castellanos, V., Llibre, J., Quilantán, I.: Integrability and global dynamics of the May-Leonard model. Nonlinear Anal. Real World...
  • 4. Bogoyavlensky, O.I.: Five constructions of integrable dynamical systems connected with the Kortewegde Vries equation. Acta Appl. Math....
  • 5. Bogoyavlensky, O.I.: Integrable discretizations of the KdV equation. Phys. Lett. A 134, 34–38 (1988)
  • 6. Brenig, L.: Complete factorisation and analytic solutions of generalized Lotka-Volterra equations. Phys. Lett. A 133, 378–382 (1988)
  • 7. Cao, F., Jiang, J.: The classiffication on the global phase portraits of two-dimensional Lotka-Volterra system. J. Dyn. Diff. Equat. 20,...
  • 8. Cima, A., Llibre, J.: Bounded polynomial vector fields. Trans. Amer. Math. Soc. 318, 557–579 (1990)
  • 9. Du, Y.: Positive periodic solutions of a competitor-competitor-mutualist parabolic model. Differential Integral Equations 9, 1043–1066...
  • 10. Gyllenberg, M., Wang, Y.: Dynamics of the periodic type-K competitive Kolmogorov systems. J. Differential Equations 205, 50–76 (2004)
  • 11. Gyllenberg, M., Yan, P., Wang, Y.: Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems. Physica D 221, 135–145 (2006)
  • 12. Harrison, G.W.: Comparing predator-prey models to Luckinbill’s experiment with Didinium and Paramecium. Ecology 76, 357–374 (1995)
  • 13. Hirsch, M.W.: Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity 1, 51–71 (1988)
  • 14. Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998)
  • 15. Hou, Z.: On global dynamics of type-K competitive Kolmogorov differential systems. Nonlinearity 36, 3796–3834 (2023)
  • 16. Jiang, J., Liang, F.: Global dynamics of 3D competitive Lotka-Volterra equations with the identical intrinsic growth rate. J. Differential...
  • 17. Liang, F., Jiang, J., Zhang, X.: Global dynamics of 3D cooperative Lotka-Volterra system with the identical intrinsic growth rate. Bull....
  • 18. Liang, F., Jiang, J., Zhang, X.: Global dynamics of 3D type-K monotone Lotka-Volterra system with the identical intrinsic growth rate....
  • 19. Liang, X., Jiang, J.: The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K...
  • 20. Llibre, J., Martínez, Y.P.: Dynamics of a competitive Lotka-Volterra systems in R3. Acta Appl. Math. 170, 569–577 (2020)
  • 21. Llibre, J., Martínez, Y.P.: Dynamics of a family of Lotka-Volterra systems in R3. Nonlinear Anal. 199, 111915 (2020)
  • 22. Llibre, J., Martínez, Y.P., Valls, C.: Global dynamics of a Lotka-Volterra system in R3. J. Nonlinear Math. Phys 27, 509–519 (2020)
  • 23. Llibre, J., Zhang, X.: Dynamics of some three-dimensional Lotka-Volterra systems. Mediterr. J. Math. 14, 126 (2017)
  • 24. Lotka, A.J.: Analytical note on certain rhythmic relations in organic systems. Pro. Natl. Acad. Sci. USA 6, 410–415 (1920)
  • 25. Murray, J.D.: Mathematical Biology. Springer-Verlag, Berlin (1989)
  • 26. Rai, B., Freedman, H.I., Addicott, J.F.: Analysis of three-species models of mutualism in predator-prey and competitive systems. Math....
  • 27. Sigmund, K.: Population dynamics of conflict and cooperation. Documenta Mathematica. Extra Volume ICM 3, 487–506 (1998)
  • 28. Smith, H. L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. (Math. Surveys Monogr....
  • 29. Smith, H.L., Waltman, P.E.: The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, Cambridge (1995)
  • 30. Tineo, A.: Asymptotic behavior of solutions of a periodic reaction-diffusion system of a competitorcompetitor-mutualist model. J. Differential...
  • 31. Volterra, V.: Lecons sur la Théorie Mathématique de la Lutte pour La Vie. Gauthier-Villars, Paris (1931)
  • 32. Zeeman, M.: Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dyn. Stab. Syst. 8, 189–217 (1993)
  • 33. Zhao, X.-Q.: Global asymptotic behavior in a periodic competitor-competitor-mutualist parabolic system. Nonlinear Anal. 29, 551–568 (1997)
  • 34. Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)