Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2010 (2010), Article ID 287969, 41 pages
http://dx.doi.org/10.1155/2010/287969
Research Article

A Predator-Prey Model in the Chemostat with Time Delay

1Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
2Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3

Received 1 November 2009; Accepted 11 January 2010

Academic Editor: Yuri V. Rogovchenko

Copyright © 2010 Guihong Fan and Gail S. K. Wolkowicz. 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. S. B. Hsu, S. Hubbell, and P. Waltman, “A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,” SIAM Journal on Applied Mathematics, vol. 32, no. 2, pp. 366–383, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, vol. 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, UK, 1995. View at MathSciNet
  3. G. J. Butler, S. B. Hsu, and P. Waltman, “Coexistence of competing predators in a chemostat,” Journal of Mathematical Biology, vol. 17, no. 2, pp. 133–151, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Y. Kuang, “Limit cycles in a chemostat-related model,” SIAM Journal on Applied Mathematics, vol. 49, no. 6, pp. 1759–1767, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. J. Butler and G. S. K. Wolkowicz, “Predator-mediated competition in the chemostat,” Journal of Mathematical Biology, vol. 24, no. 2, pp. 167–191, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. S. K. Wolkowicz and H. Xia, “Global asymptotic behavior of a chemostat model with discrete delays,” SIAM Journal on Applied Mathematics, vol. 57, no. 4, pp. 1019–1043, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G. S. K. Wolkowicz, “Successful invasion of a food web in a chemostat,” Mathematical Biosciences, vol. 93, no. 2, pp. 249–268, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York, NY, USA, 1982. View at MathSciNet
  9. R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963. View at Zentralblatt MATH · View at MathSciNet
  10. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  11. K. Engelborghs, T. Luzyanina, and D. Roose, “Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL,” Transactions on Mathematical Software, vol. 28, no. 1, pp. 1–21, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. K. Verheyden, “New functionality in dde-biftool v.2.03.,” Addendum to the manual of DDE-BIFTOOL v. 2.00 (and v.2.02), unpublished, 2007.
  13. E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144–1165, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. C. F. De Oliveira, “Hopf bifurcation for functional differential equations,” Nonlinear Analysis, vol. 4, no. 2, pp. 217–229, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. M. Hirsch, H. Hanisch, and J.-P. Gabriel, “Differential equation models of some parasitic infections: methods for the study of asymptotic behavior,” Communications on Pure and Applied Mathematics, vol. 38, no. 6, pp. 733–753, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. K. Golpalsalmy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1985.