Advertisement

Generalizing the May-Leonard System to Any Number of Species

  • Genaro de la VegaEmail author
  • Santiago López de Medrano
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)

Abstract

We construct generalizations of the May-Leonard three species Lotka-Volterra system to any number of species, extending the property of cyclic dominance. First we recall the main properties of the May-Leonard system. Then we construct a Lotka-Volterra system for n species which has an invariant attracting polygon. We construct Lotka-Volterra-Kolmogorov deformations of higher degree of this system which have the same attracting polygon and all the properties of the May-Leonard one. We finish with the construction of Lotka-Volterra deformations of the same system which numerically show a curvilinear attracting polygon and the same properties, but the corresponding analytic results are still not complete.

Keywords

Vector Field Equilibrium Point Invariant Manifold Cyclic Behaviour Cyclic Permutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work owes much to many years of joint work and long discussions of the second author with Marc Chaperon. The first author received a Conacyt grant for writing his Ph. D. Thesis, part of which is included in this article. The second author was partially supported by Conacyt-CNRS and by the UNAM-DGAPA-Papiit grants IN102009 and IN108112.

References

  1. 1.
    Arfken, G.: Mathematical Methods for Physicists, 3rd edn., p. 733. Academic, Orlando (1985)Google Scholar
  2. 2.
    Chaperon, M., López De Medrano, S.: Birth of attracting compact invariant submanifolds diffeomorphic to moment-angle manifolds in generic families of dynamics. C. R. Acad. Sci. París 346(I), 1099–1102 (2008)Google Scholar
  3. 3.
    Chenciner, A.: Comportement asymptotique de systémes differentiels su type “compétition d’especes”. C. R. Acad. Sci. París 284, 313–315 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    De la Vega, G.: Ecuaciones Lotka Volterra Kolmogorov y Sistemas Dinámicos. Ph. D. Dissertation, UNAM (2013)Google Scholar
  5. 5.
    Hirsch, M.W.: On existence and uniqueness of the carrying simplex for competitive dynamical systems. J. Biol. Dyn. 2(2), 169–179 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  7. 7.
    López de Medrano, S.: The Topology of the Intersection of Quadrics in \({\mathbb{R}}^{{}^{n} }\). Lecture Notes in Mathematics, vol. 1370, pp. 280–292. Springer, Berlin (1989)Google Scholar
  8. 8.
    May, R.M., Leonard, W.J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Genaro de la Vega
    • 1
    Email author
  • Santiago López de Medrano
    • 1
  1. 1.Instituto de Matemáticas, UNAMUniversidad Nacional Autónoma de MéxicoMéxico, D.F.México

Personalised recommendations