• Yan, Xiang-Ping ;
  • Zhang, Cun-Hua
  • Received : 2010.12.06
  • Published : 2012.07.01


This paper considers the stability of positive steady-state solutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and subject to the homogeneous Dirichlet boundary conditions on a general bounded open spatial domain with smooth boundary. The existence, uniqueness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit function theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.


Lotka-Volterra competition-cooperation system;discrete delay;diffusion effect;positive steady-state solution;stability


  1. S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations 124 (1996), no. 1, 80-127.
  2. S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Spring-Verlag, New York, 1982.
  3. L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
  4. J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, 1985.
  5. W. Huang, Global Dynamics for a Reaction-Diffusion Equation with Time Delay, J. Differential Equations 143 (1998), no. 2, 293-326.
  6. H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Spring-Verlag, New York, 2004.
  7. Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations 4 (1991), no. 1, 117-128.
  8. W. T. Li, X. P. Yan, and C. H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos Solitons Fractals 38 (2008), no. 1, 227-237.
  9. Z. Lin and M. Pedersen, Coexistence of a general elliptic system in population dynamics, Comput. Math. Appl. 48 (2004), no. 3-4, 617-628.
  10. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  11. W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl. 197 (1996), no. 2, 558-578.
  12. K. Ryu and I. Ahn, Positive coexistence of steady state for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc. 38 (2001), no. 4, 643-655.
  13. C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395-418.
  14. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
  15. X. P. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comp. 192 (2007), no. 2, 552-566.
  16. X. P. Yan and C. H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Appl. Math. Model. 34 (2010), no. 1, 184-199.
  17. Q. Ye and Z. Li, An Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990.
  18. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1980.
  19. L. Zhou, Y. Tang, and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals 14 (2002), no. 8, 1201-1225.