Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 934569, 16 pages
http://dx.doi.org/10.1155/2011/934569
Research Article

A Three-Species Food Chain System with Two Types of Functional Responses

1Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Mathematics Education, Catholic University of Daegu, Gyeongsan-si, Gyeongbuk 712-702, Republic of Korea

Received 20 July 2010; Revised 11 December 2010; Accepted 19 January 2011

Academic Editor: H. B. Thompson

Copyright © 2011 Younghae Do 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. P. A. Braza, “The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing,” SIAM Journal on Applied Mathematics, vol. 63, no. 3, pp. 889–904, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. C. Cosner, D. L. Deangelis, J. S. Ault, and D. B. Olson, “Effects of spatial grouping on the functional response of predators,” Theoretical Population Biology, vol. 56, no. 1, pp. 65–75, 1999. View at Publisher · View at Google Scholar · View at Scopus
  3. S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001. View at Publisher · View at Google Scholar
  4. M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15–39, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Gakkhar and R. K. Naji, “Seasonally perturbed prey-predator system with predator-dependent functional response,” Chaos, Solitons & Fractals, vol. 18, no. 5, pp. 1075–1083, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989. View at Google Scholar · View at Scopus
  7. H. I. Freedman and R. M. Mathsen, “Persistence in predator-prey systems with ratio-dependent predator influence,” Bulletin of Mathematical Biology, vol. 55, no. 4, pp. 817–827, 1993. View at Publisher · View at Google Scholar · View at Scopus
  8. L. A. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, Cambridge, UK, 1984.
  9. S. Gakkhar and R. K. Naji, “Order and chaos in predator to prey ratio-dependent food chain,” Chaos, Solitons & Fractals, vol. 18, no. 2, pp. 229–239, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991. View at Google Scholar · View at Scopus
  11. S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “A ratio-dependent food chain model and its applications to biological control,” Mathematical Biosciences, vol. 181, no. 1, pp. 55–83, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427–451, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. S. Lv and M. Zhao, “The dynamic complexity of a three species food chain model,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1469–1480, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. K. Naji and A. T. Balasim, “Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1853–1866, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. C. Shen, “Permanence and global attractivity of the food-chain system with Holling IV type functional response,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 179–185, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, and J. C. Sprott, “Chaos in low-dimensional Lotka-Volterra models of competition,” Nonlinearity, vol. 19, no. 10, pp. 2391–2404, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Zhao and S. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2305–2316, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. B. Liu, Z. Teng, and L. Chen, “Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 347–362, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. F. Cao and L. Chen, “Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model,” Systems Science and Mathematical Sciences, vol. 11, no. 2, pp. 107–111, 1998. View at Google Scholar · View at Zentralblatt MATH
  20. H. I. Freedman and P. Waltman, “Persistence in models of three interacting predator-prey populations,” Mathematical Biosciences, vol. 68, no. 2, pp. 213–231, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. R. M. May, Stablitiy and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, USA, 1973.
  22. M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D, vol. 65, no. 1-2, pp. 117–134, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, New York, NY, USA, 2003.