DOI QR코드

DOI QR Code

GLOBAL ASYMPTOTIC STABILITY FOR A DIFFUSION LOTKA-VOLTERRA COMPETITION SYSTEM WITH TIME DELAYS

  • Zhang, Jia-Fang ;
  • Zhang, Ping-An
  • Received : 2011.01.14
  • Published : 2012.11.30

Abstract

A type of delayed Lotka-Volterra competition reaction-diffusion system is considered. By constructing a new Lyapunov function, we prove that the unique positive steady-state solution is globally asymptotically stable when interspecies competition is weaker than intraspecies competition. Moreover, we show that the stability property does not depend on the diffusion coefficients and time delays.

Keywords

global asymptotic stability;delays;Lyapunov function

References

  1. P. N. Brown, Decay to uniform states in ecological interaction, SIAM J. Appl. Math. 38 (1980), no. 1, 22-37. https://doi.org/10.1137/0138002
  2. Y. H. Fan and L. L. Wang, Global asymptotical stability of a logistic model with feedback control, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2686-2697. https://doi.org/10.1016/j.nonrwa.2009.09.016
  3. T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks, J. Differential Equations 244 (2008), no. 5, 1049-1079. https://doi.org/10.1016/j.jde.2007.12.005
  4. S. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal term: a competition model, SIAM J. Math. Anal. 35 (2003), no. 3, 806-822. https://doi.org/10.1137/S003614100139991
  5. H. F. Huo and W. T. Li, Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model, Math. Comput. Modelling 40 (2004), no. 3-4, 261-269. https://doi.org/10.1016/j.mcm.2004.02.026
  6. Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
  7. S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc. 96 (1986), no. 1, 75-78. https://doi.org/10.1090/S0002-9939-1986-0813814-3
  8. W. T. Li, G. Lin, and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity 19 (2006), no. 6, 1253-1273. https://doi.org/10.1088/0951-7715/19/6/003
  9. Z. Ma, Mathematical Modeling and Research on the Population Ecology, AnHui educational press, Hefei, 1996.
  10. R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math. 413 (1991), 1-35.
  11. C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. 48 (2002), no. 3, Ser. A: Theory Methods, 349-362. https://doi.org/10.1016/S0362-546X(00)00189-9
  12. R. Peng and M. Wang, Note on a ratio-dependent predator-prey system with diffusion, Nonlinear Anal. Real World Appl. 7 (2006), no. 1, 1-11. https://doi.org/10.1016/j.nonrwa.2004.11.008
  13. S. Ruan and J.Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart. 2 (1994), no. 4, 485-550.
  14. S. Ruan and X. Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations 156 (1999), no. 1, 71-92. https://doi.org/10.1006/jdeq.1998.3599
  15. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41, American Mathematical Society, Providence, RI, 1995.
  16. Y. Song, M. Han, and Y. Peng, Stability and Hopf bifurcation in a competitive Lotka-Volterra system with two delays, Chaos Solitons Fractals 22 (2004), no. 5, 1139-1148. https://doi.org/10.1016/j.chaos.2004.03.026
  17. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B 237 (1952), 37-72. https://doi.org/10.1098/rstb.1952.0012
  18. L. L. Wang and Y. H. Fan, Note on permanence and global stability in delayed ratio-dependent predator-prey models with monotonic functional response, J. Comput. Appl. Math. 234 (2010), no. 2, 477-487. https://doi.org/10.1016/j.cam.2009.12.039
  19. Y. M. Wang, Asymptotic behavior of solutions for a cooperation-diffusion model with a saturating interaction, Comput. Math. Appl. 52 (2006), no. 3-4, 339-350. https://doi.org/10.1016/j.camwa.2006.03.016
  20. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
  21. J. Zhen and Z. Ma, Stability for a competitive Lotka-Volterra system with delays, Non-linear Anal. 51 (2002), no. 7, 1131-1142. https://doi.org/10.1016/S0362-546X(01)00881-1