Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2009, Article ID 543187, 16 pages
http://dx.doi.org/10.1155/2009/543187
Research Article

Analysis of an Impulsive Predator-Prey System with Monod-Haldane Functional Response and Seasonal Effects

1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
2Department of Mathematics, Pusan National University, Pusan 609-735, South Korea

Received 5 December 2008; Revised 23 March 2009; Accepted 27 April 2009

Academic Editor: J. Rodellar

Copyright © 2009 Hunki Baek 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. C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulations,” Mem. Ent. Sec. Can, vol. 45, pp. 1–60, 1965. View at Google Scholar
  2. J. F. Andrews, “A mathematical model for the continuous culture of macroorganisms untilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707–723, 1968. View at Google Scholar
  3. W. Sokol and J. A. Howell, “Kineties of phenol oxidation by ashed cell,” Biotechnology and Bioengineering, vol. 23, pp. 2039–2049, 1980. View at Google Scholar
  4. S.-B. Hsu and T.-W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995. View at Google Scholar · View at MathSciNet
  5. 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 Google Scholar · View at MathSciNet
  6. E. Sáez and E. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999. View at Google Scholar · View at MathSciNet
  7. J. Sugie, R. Kohno, and R. Miyazaki, “On a predator-prey system of Holling type,” Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 2041–2050, 1997. View at Google Scholar · View at MathSciNet
  8. J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82–95, 1977. View at Google Scholar · View at MathSciNet
  9. S. Gakkhar and R. K. Naji, “Chaos in seasonally perturbed ratio-dependent prey-predator system,” Chaos, Solitons & Fractals, vol. 15, no. 1, pp. 107–118, 2003. View at Google Scholar · View at MathSciNet
  10. G. C. W. Sabin and D. Summers, “Chaos in a periodically forced predator-prey ecosystem model,” Mathematical Biosciences, vol. 113, no. 1, pp. 91–113, 1993. View at Publisher · View at Google Scholar
  11. G. J. Ackland and I. D. Gallagher, “Stabilization of large generalized Lotka-Volterra foodwebs by evolutionary feedback,” Physical Review Letters, vol. 93, no. 15, Article ID 158701, 4 pages, 2004. View at Publisher · View at Google Scholar
  12. G. Jiang and Q. Lu, “The dynamics of a prey-predator model with impulsive state feedback control,” Discrete and Continuous Dynamical Systems. Series B, vol. 6, no. 6, pp. 1301–1320, 2006. View at Google Scholar · View at MathSciNet
  13. X. Liu and L. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 311–320, 2003. View at Google Scholar · View at MathSciNet
  14. B. Liu, Y. Zhang, and L. Chen, “Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 2, pp. 517–531, 2005. View at Google Scholar · View at MathSciNet
  15. K. Negi and S. Gakkhar, “Dynamics in a Beddington-DeAngelis prey-predator system with impulsive harvesting,” Ecological Modelling, vol. 206, no. 3-4, pp. 421–430, 2007. View at Publisher · View at Google Scholar
  16. X. Song and Y. Li, “Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect,” Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 64–79, 2008. View at Google Scholar · View at MathSciNet
  17. D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Science & Technical, Harlo, UK, 1993.
  18. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. View at MathSciNet
  19. H. Baek, “Dynamic complexites of a three-species Beddington-DeAngelis system with impulsive control strategy,” Acta Applicandae Mathematicae, pp. 1–16, 2008. View at Publisher · View at Google Scholar
  20. W. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1772–1785, 2007. View at Google Scholar · View at MathSciNet
  21. Z. Xiang and X. Song, “The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2282–2293, 2009. View at Publisher · View at Google Scholar
  22. S. Zhang, D. Tan, and L. Chen, “Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 980–990, 2006. View at Google Scholar · View at MathSciNet
  23. S. Zhang and L. Chen, “A study of predator-prey models with the Beddington-DeAnglis functional response and impulsive effect,” Chaos, Solitons & Fractals, vol. 27, no. 1, pp. 237–248, 2006. View at Google Scholar · View at MathSciNet