About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 323186, 23 pages
http://dx.doi.org/10.1155/2012/323186
Research Article

Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System

1College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
2School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China

Received 7 June 2012; Revised 4 September 2012; Accepted 22 September 2012

Academic Editor: Malisa R. Zizovic

Copyright © 2012 Yanuo Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G.-P. Hu and W.-T. Li, “Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects,” Nonlinear Analysis. Real World Applications, vol. 11, no. 2, pp. 819–826, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. G. W. Harrison, “Multiple stable equilibria in a predator-prey system,” Bulletin of Mathematical Biology, vol. 48, no. 2, pp. 137–148, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. M. E. Solomon, “The natural control of animal populations,” Journal of Animal Ecology, vol. 18, pp. 1–35, 1949.
  4. T. Faria, “Stability and bifurcation for a delayed predator-prey model and the effect of diffusion,” Journal of Mathematical Analysis and Applications, vol. 254, no. 2, pp. 433–463, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. L. A. F. de Oliveira, “Instability of homogeneous periodic solutions of parabolic-delay equations,” Journal of Differential Equations, vol. 109, no. 1, pp. 42–76, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. Freitas, “Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 59–82, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. T. Faria, “Normal forms and Hopf bifurcation for partial differential equations with delays,” Transactions of the American Mathematical Society, vol. 352, no. 5, pp. 2217–2238, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. R. H. Martin, Jr. and H. L. Smith, “Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,” Journal für die Reine und Angewandte Mathematik, vol. 413, pp. 1–35, 1991. View at Zentralblatt MATH
  9. S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 6, pp. 863–874, 2003. View at Zentralblatt MATH
  10. Y. Song and J. Wei, “Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 1–21, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Y. Song and S. Yuan, “Bifurcation analysis in a predator-prey system with time delay,” Nonlinear Analysis. Real World Applications, vol. 7, no. 2, pp. 265–284, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. X.-P. Yan and W.-T. Li, “Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 427–445, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. K. P. Hadeler and S. Ruan, “Interaction of diffusion and delay,” Discrete and Continuous Dynamical Systems. Series B, vol. 8, no. 1, pp. 95–105, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. Xu and Z. Ma, “Global stability of a reaction-diffusion predator-prey model with a nonlocal delay,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 194–206, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. W. Zuo and J. Wei, “Stability and Hopf bifurcation in a diffusive predatory-prey system with delay effect,” Nonlinear Analysis. Real World Applications, vol. 12, no. 4, pp. 1998–2011, 2011. View at Publisher · View at Google Scholar
  16. J.-F. Zhang, W.-T. Li, and X.-P. Yan, “Multiple bifurcations in a delayed predator-prey diffusion system with a functional response,” Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp. 2708–2725, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. T. K. Kar and A. Ghorai, “Dynamic behaviour of a delayed predator-prey model with harvesting,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9085–9104, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S. Chen, J. Shi, and J. Wei, “A note on Hopf bifurcations in a delayed diffusive Lotka-Volterra predator-prey system,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2240–2245, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1981.
  20. J. Wu, Theory and Applications of Partial Functional-Differential Equations, vol. 119 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar
  21. X. D. Lin, J. W.-H. So, and J. H. Wu, “Centre manifolds for partial differential equations with delays,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 122, no. 3-4, pp. 237–254, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH