Skip to main content
Log in

Stability of Lotka–Volterra system

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The stability of Lotka–Volterra system has been discussed by many authors for two and three species. In this paper, we will discussed the notion of stability for a Lotka–Volterra system with four species. Some criteria and results are given. Our technique depends on the Lyapunov–Razumikhin method.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. M. Cushing, “Forced asymptotically periodic solutions of predator-prey system with or without hereditary effects,” SIAM J. Appl. Math., 29, 665–674 (1976).

    Article  MathSciNet  Google Scholar 

  2. S. Elaydi, “Asymptotic stability for linear systems with infinite delay,” Comput. Math. Appl. (in press).

  3. K. Gopalsamy, “Stability criteria for the linear system x′(t)+A(t)x(tτ ) = 0, and an application to a nonlinear system,” Int. J. Syst. Sci., 21, 1841–1853 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Hernandez and V. Fairen, “Lotka–Volterra representaion of general nonlinear systems,” Math. Biosci., 140, 1–32 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Hernandez and V. Fairen, “Local stability and Lyapunov functionals for n-dimensional quasipolynomial conservative systems,” J. Math. Anal. Appl., 256, 242–256 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Hernandez, “Stability conditions and Lyapunov functions for quasi-polynomial systems,” Appl. Math. Lett. 15, 25–28 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  7. X. Li, C. Tang, and X. Ji, “The criteria for globally stable equilibrium in n-dimensional Lotka–Volerra systems,” J. Math. Anal. Appl., 240, 600–606 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  8. R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press (1973).

  9. R. M. May and W. J. Leonard, “Nonlinear aspects of competition between three species,” SIAM J. Appl. Math., 29, 243–253 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Mangel and L. Donald, “Probability of extinction in a stochastic competition,” J. Appl. Math., 33, 256–266 (1977).

    MATH  Google Scholar 

  11. R. K. Miller, “Asymptotic stability properties of linear Volterra integrodifferential equations,” J. Differ. Equ., 10, 485–506 (1971).

    Article  MATH  Google Scholar 

  12. G. Pimbley, “Periodic solution of third order predator-prey equations simulating an immune response,” Arch. Rat. Mech. Anal., 35, 93–123 (1974).

    Article  MathSciNet  Google Scholar 

  13. M. R. Rama, Theory of Ordinary Differential Equations and Applications, New York (1980).

  14. R. Redheffer, “A new class of Volterra differential equations for which the solutions are globally asymptotically stable,” J. Differ. Equ., 82, 251–268 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Schuster, K. Sygmund, and R. Wolf, “On ω-limits for competition between three species,” SIAM J. Appl. Math., 37, 49–54 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems , World Scientific, Singapore (1996).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. A. Soliman.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Soliman, A.A. Stability of Lotka–Volterra system. J Math Sci 161, 308–319 (2009). https://doi.org/10.1007/s10958-009-9554-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-009-9554-4

Keywords

Navigation