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Differential Equations and Dynamical Systems

, Volume 20, Issue 1, pp 53–66 | Cite as

Global Stability of Interior and Boundary Fixed Points for Lotka–Volterra Systems

  • Stephen BaigentEmail author
  • Zhanyuan Hou
Original Research

Abstract

For permanent and partially permanent, uniformly bounded Lotka–Volterra systems, we apply the Split Lyapunov function technique developed for competitive Lotka–Volterra systems to find new conditions that an interior or boundary fixed point of a Lotka–Volterra system with general species–species interactions is globally asymptotically stable. Unlike previous applications of the Split Lyapunov technique to competitive Lotka–Volterra systems, our method does not require the existence of a carrying simplex.

Keywords

Lotka–Volterra systems Global attractors Global repellors Global asymptotic stability 

Mathematics Subject Classification (2000)

34D05 34D20 34C11 92D25 

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References

  1. 1.
    Ahmad S., Lazer A.C.: Average growth and total permanence in a competitive Lotka–Volterra system. Ann. Mat. 185, S47–S67 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hirsch M.W.: Systems of differential equations that are competitive or cooperative III: competing species. Nonlinearity 1, 51–71 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Hofbauer J., Sigmund K.: The theory of evolution and dynamical systems. Cambridge University Press, New York (1998)Google Scholar
  4. 4.
    Horn R.A., Johnson C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  5. 5.
    Hou Z.: Global attractor in autonomous competitive Lotka–Volterra systems. Proc. Am. Math. Soc. 127, 3633–3642 (1999)zbMATHCrossRefGoogle Scholar
  6. 6.
    Hou Z.: Global attractor in competitive Lotka–Volterra systems. Math. Nachr. 282(7), 995–1008 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hou Z.: Vanishing components in autonomous competitive Lotka–Volterra systems. J. Math. Anal. Appl. 359, 302–310 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hou Z.: Oscillations and limit cycles in competitive Lotka–Volterra systems with delays. Nonlinear Anal. Theory Method Appl. 75, 358–370 (2012)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hou Z., Baigent S.: Fixed point global attractors and repellors in competitive Lotka–Volterra systems. Dyn. Syst. 26(4), 367–390 (2011)zbMATHGoogle Scholar
  10. 10.
    Hou Z.: On permanence of all subsystems of competitive Lotka–Volterra systems with delays. Nonlinear Anal. RWA 11, 4285–4301 (2010)zbMATHCrossRefGoogle Scholar
  11. 11.
    Hou Z.: Permanence and extinction in competitive Lotka–Volterra systems with delays. Nonlinear Anal. RWA 12, 2130–2141 (2011)zbMATHCrossRefGoogle Scholar
  12. 12.
    Jansen W.: A permanence theorem for replicator and Lotka–Volterra systems. J. Math. Biol. 25, 411–422 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Liang X., Jiang J.: The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka–Volterra systems. Nonlinearity 16, 785801 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mierczynski J., Schreiber S.J.: Kolmogorov vector fields with robustly permanent subsystems. J. Math. Anal. Appl. 267, 329–337 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Takeuchi Y.: Global dynamical properties of Lotka–Volterra systems. World Scientific, Singapore (1996)zbMATHCrossRefGoogle Scholar
  16. 16.
    Tineo A.: May Leonard systems. Nonlinear Anal. Real World Appl. 9, 1612–1618 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zeeman E.C., Zeeman M.L.: From local to global behavior in competitive Lotka–Volterra systems. Trans. Am. Math. Soc. 355, 713–734 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2012

Authors and Affiliations

  1. 1.Department of MathematicsUCLLondonUK
  2. 2.Faculty of ComputingLondon Metropolitan UniversityLondonUK

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