Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016, Article ID 9268257, 20 pages
http://dx.doi.org/10.1155/2016/9268257
Research Article

Impulsive Control on Seasonally Perturbed General Holling Type Two-Prey One-Predator Model

Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India

Received 13 April 2016; Revised 10 June 2016; Accepted 16 June 2016

Academic Editor: Zhengqiu Zhang

Copyright © 2016 Chandrima Banerjee and Pritha Das. 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 components of predation as revealed by a study of small-mammal predation of the european pine sawfly,” The Canadian Entomologist, vol. 91, no. 5, pp. 293–320, 1959. View at Publisher · View at Google Scholar
  2. C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, no. 45, pp. 1–60, 1965. View at Publisher · View at Google Scholar
  3. J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707–723, 1968. View at Google Scholar
  4. P. Y. Pang and M. Wang, “Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion,” Proceedings of the London Mathematical Society, vol. 88, no. 1, pp. 135–157, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. Sokol and J. Howell, “Kinetics of phenol oxidation by washed cells,” Biotechnology and Bioengineering, vol. 23, no. 9, pp. 2039–2049, 1981. View at Publisher · View at Google Scholar
  6. S. G. Ruan and D. G. 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 · View at MathSciNet · View at Scopus
  7. J. S. Tener, Muskoxen, Queens Printer, Ottawa, Canada, 1965.
  8. R. E. Kooij and A. Zegeling, “Qualitative properties of two-dimensional predator-prey systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 29, no. 6, pp. 693–715, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  9. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. N. D. Kazarinoff and P. van den Driessche, “A model predator-prey system with functional response,” Mathematical Biosciences, vol. 39, no. 1-2, pp. 125–134, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. A. Gragnani and S. Rinaldi, “A universal bifurcation diagram for seasonally perturbed predator-prey models,” Bulletin of Mathematical Biology, vol. 57, no. 5, pp. 701–712, 1995. View at Publisher · View at Google Scholar · View at Scopus
  12. A. R. Ives, K. Gross, A. Vincent, and A. Jansen, “Periodic mortality events in predator-prey systems,” Ecology, vol. 81, no. 12, pp. 3330–3340, 2000. View at Google Scholar · View at Scopus
  13. S. Rinaldi, S. Muratori, and Y. Kuznetsov, “Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities,” Bulletin of Mathematical Biology, vol. 55, no. 1, pp. 15–35, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. R. A. Taylor, A. White, and J. A. Sherratt, “How do variations in seasonality affect population cycles?” Proceedings of the Royal Society B: Biological Sciences, vol. 280, no. 1754, 2013. View at Publisher · View at Google Scholar · View at Scopus
  15. K. Saleh, “Multiple attractors involving chaos in predator-prey model with non monotonic response function response,” International Journal of Basic and Applied Sciences, vol. 11, no. 4, pp. 38–48, 2011. View at Google Scholar
  16. B. Sahoo, “Predator-prey system with seasonally varying additional food to predators,” International Journal of Basic and Applied Sciences, vol. 1, no. 4, pp. 363–373, 2012. View at Publisher · View at Google Scholar
  17. 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 · View at Zentralblatt MATH · View at Scopus
  18. S. Gakkhar and R. K. Naji, “Chaos in seasonally perturbed ratio-dependent prey-predator system,” Chaos, Solitons and Fractals, vol. 15, no. 1, pp. 107–118, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. H. Yu and M. Zhao, “Seasonally perturbed prey-predator ecological system with the Beddington-DeAngelis functional response,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 150359, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. Y. Pei, G. Zeng, and L. Chen, “Species extinction and permance in a prey-predator model with two-type functional responses and impulsive biological control,” Nonlinear Dynamics, vol. 52, no. 1, pp. 71–81, 2008. View at Publisher · View at Google Scholar
  21. X. Wang, W. Wang, and X. Lin, “Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2392–2404, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. Y. Zhang, Z. Xiu, and L. Chen, “Dynamic complexity of a two-prey one-predator system with impulsive effect,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 131–139, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. X. Song and Y. Li, “Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect,” Chaos, Solitons & Fractals, vol. 33, no. 2, pp. 463–478, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. H. Liu and F. Meng, “Existence of positive periodic solutions for a predator-prey system of Holling type IV function response with mutual interference and impulsive effects,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 138984, 12 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. Li, L. Zhang, Z. Teng, and Y. Jiang, “Dynamic behaviors of Holling type II predator-prey system with mutual interference and impulses,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 793761, 13 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Z. Wang, Y. Shao, X. Fang, and X. Ma, “An impulsive three-species model with square root functional response and mutual interference of predator,” Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3897234, 14 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  27. H. Baek, “Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects,” BioSystems, vol. 98, no. 1, pp. 7–18, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. H. Baek, “An impulsive two-prey one-predator system with seasonal effects,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 793732, 19 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. H. Yu, S. Zhong, R. P. Agarwal, and S. K. Sen, “Effect of seasonality on the dynamical behavior of an ecological system with impulsive control strategy,” Journal of the Franklin Institute, vol. 348, no. 4, pp. 652–670, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. X. Ma, Y. Shao, Z. Wang, X. Fang, and Z. Luo, “Analysis of an impulsive one-predator and two-prey system with stage-structure and generalized functional response,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 864367, 10 pages, 2015. View at Publisher · View at Google Scholar
  31. C. Dai, M. Zhao, and L. Chen, “Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances,” Mathematics and Computers in Simulation, vol. 84, pp. 83–97, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. F. D. Chen, “On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 33–49, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  34. D. Bainov and P. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, vol. 28 of Series on Advances in Mathematics for Applied Sciences, World Scientist, River Edge, NJ, USA, 1995.
  35. G. Birkhorff and G. C. Rota, Ordinary Differential Equations, Ginn Press, Needham Heights, Mass, USA, 1982.