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Abstract and Applied Analysis
Volume 2015, Article ID 964735, 11 pages
http://dx.doi.org/10.1155/2015/964735
Research Article

Stability and Sensitive Analysis of a Model with Delay Quorum Sensing

1School of Science, Xi’an University of Science and Technology, Xi’an, Shaanxi 710054, China
2Department of Mathematics, North China Electric Power University, Beijing 102206, China

Received 26 August 2014; Accepted 17 December 2014

Academic Editor: Yanni Xiao

Copyright © 2015 Zhonghua Zhang 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. J. M. Henke and B. L. Bassler, “Bacterial social engagements,” Trends in Cell Biology, vol. 14, no. 11, pp. 648–656, 2004. View at Publisher · View at Google Scholar · View at Scopus
  2. K. H. Nealson, T. Platt, and J. W. Hastings, “Cellular control of the synthesis and activity of the bacterial luminescent system,” Journal of Bacteriology, vol. 104, no. 1, pp. 313–322, 1970. View at Google Scholar · View at Scopus
  3. A. Eberhard, “Inhibition and activation of bacterial luciferase synthesis,” Journal of Bacteriology, vol. 109, no. 3, pp. 1101–1105, 1972. View at Google Scholar · View at Scopus
  4. J. P. Braselton and P. Waltman, “A competition model with dynamically allocated inhibitor production,” Mathematical Biosciences, vol. 173, no. 2, pp. 55–84, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. D. Dockery and J. P. Keener, “A mathematical model for quorum sensing in Pseudomonas aeruginosa,” Bulletin of Mathematical Biology, vol. 63, no. 1, pp. 95–116, 2001. View at Publisher · View at Google Scholar · View at Scopus
  6. A. J. Koerber, J. R. King, J. P. Ward, P. Williams, J. M. Croft, and R. E. Sockett, “A mathematical model of partial-thickness burn-wound infection by Pseudomonas aeruginosa: quorum sensing and the build-up to invasion,” Bulletin of Mathematical Biology, vol. 64, no. 2, pp. 239–259, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. P. Fergola, E. Beretta, and M. Cerasuolo, “Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1081–1095, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. K. Anguige, J. R. King, and J. P. Ward, “A multi-phase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm,” Mathematical Biosciences, vol. 203, no. 2, pp. 240–276, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. K. Anguige, J. R. King, J. P. Ward, and P. Williams, “Mathematical modelling of therapies targeted at bacterial quorum sensing,” Mathematical Biosciences, vol. 192, no. 1, pp. 39–83, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. P. Fergola, J. Zhang, M. Cerasuolo et al., “On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, collective dynamic: topics on copetition and coopertion in the biosciences: a selection of papers,” in Proceedings of the BIOCOMP International Conference, vol. 1028 of AIP Conference Proceedings, pp. 215–232, 2007.
  11. Z. H. Zhang, J. G. Peng, and J. Zhang, “Analysis of a bacteria-immunity model with delay quorum sensing,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 102–115, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Z. Juan, Z. Zhonghua, and S. Yaohong, “Hopf bifurcation of a bacteria-immunity system with delayed quorum sensing,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3936–3949, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z. H. Zhang, J. H. Wu, Y. H. Suo, and Z. Juan, “Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2422–2435, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z. H. Zhang, J. G. Peng, and J. Zhang, “Melnikov method to a bacteria-immunity model with bacterial quorum sensing mechanism,” Chaos, Solitons & Fractals, vol. 40, no. 1, pp. 414–420, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Z. H. Zhang, J. Zhang, and J. G. Peng, “A bacteria-immunity system with delayed quorum sensing,” Journal of Applied Mathematics and Computing, vol. 40, pp. 414–420, 2009. View at Google Scholar
  16. T. Faria and L. T. Magalhaes, “Normal forms for retarded differential equations and applications to Bogdanov-Takens singularity,” Journal of Differential Equations, vol. 122, no. 2, pp. 201–224, 1995. View at Google Scholar
  17. T. Faria and L. T. Magalhaes, “Normal forms for functional differential equations with parameters and applications to Hopf bifurcation,” Journal of Differential Equations, vol. 122, pp. 180–200, 1995. View at Google Scholar
  18. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  19. N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus