Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 958140, 18 pages
http://dx.doi.org/10.1155/2014/958140
Research Article

Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 18 April 2014; Revised 15 June 2014; Accepted 16 June 2014; Published 14 August 2014

Academic Editor: Yongli Song

Copyright © 2014 Xin-You Meng and Hai-Feng Huo. 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. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, NY, USA, 1925.
  2. V. Volterra, “Fluctuations in the abundance of a species considered mathematically,” Nature, vol. 118, no. 2972, pp. 558–560, 1926. View at Publisher · View at Google Scholar · View at Scopus
  3. Z. Jin and Z. E. Ma, “Stability for a competitive Lotka-Volterra system with delays,” Nonlinear Analysis: Theory, Methods and Applications, vol. 51, no. 7, pp. 1131–1142, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. Y. Song, M. Han, and Y. Peng, “Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1139–1148, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. Z. Zhang, Z. Jin, J. R. Yan, and G. Q. Sun, “Stability and Hopf bifurcation in a delayed competition system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 658–670, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. F. Zhang, “Stability and bifurcation periodic solutions in a Lotka-Volterra competition system with multiple delays,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 70, no. 1, pp. 849–860, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. 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 · View at MathSciNet · View at Scopus
  8. Y. L. Song, Y. H. Peng, and J. J. Wei, “Bifurcations for a predator-prey system with two delays,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 466–479, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. 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 MathSciNet · View at Scopus
  10. C. Q. Xu and S. L. Yuan, “Stability and Hopf bifurcation in a delayed predator-prey system with herd behavior,” Abstract and Applied Analysis, vol. 2014, Article ID 568943, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  11. X. Y. Meng, H. F. Huo, X. B. Zhang, and H. Xiang, “Stability and Hopf bifurcation in a three-species system with feedback delays,” Nonlinear Dynamics, vol. 64, no. 4, pp. 349–364, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Y. L. Song and J. 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 · View at MathSciNet · View at Scopus
  13. L. S. Chen, Z. Y. Lu, and W. D. Wang, “The effect of delays on the permanence for Lotka-Volterra systems,” Applied Mathematics Letters, vol. 8, no. 4, pp. 71–73, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977. View at MathSciNet
  15. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993. View at MathSciNet
  16. J. D. Gause, Mathematical Biology, Springer, New York, NY, USA, 1989.
  17. X. Z. He and K. Gopalsamy, “Persistence, attractivity, and delay in facultative mutualism,” Journal of Mathematical Analysis and Applications, vol. 215, no. 1, pp. 154–173, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. X. Z. Meng and J. J. Wei, “Stability and bifurcation of mutual system with time delay,” Chaos, Solitons & Fractals, vol. 21, no. 3, pp. 729–740, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. F. Q. Zhang and Y. J. Zhang, “Hopf bifrucation for Lotka-Volterra mutualistic systems with three delays,” Jouranl of Biomathematics, vol. 26, no. 2, pp. 223–233, 2011. View at Google Scholar
  20. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet
  21. J. H. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799–4838, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S. G. Ruan and J. 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 A: Mathematical Analysis, vol. 10, no. 6, pp. 863–874, 2003. View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82–95, 1977. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. X. Z. He, “Stability and delays in a predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 355–370, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. C. W. Clark, Bioeconomic Modelling and Fisheries Management, John Wiley & Sons, New York, NY, USA, 1985.