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Journal of Applied Mathematics
Volume 2014, Article ID 536019, 10 pages
http://dx.doi.org/10.1155/2014/536019
Research Article

Dynamics of a Predator-Prey System with Mixed Functional Responses

1Department of Mathematics Education, Catholic University of Daegu, Gyeongsan, Gyeongbuk 712-702, Republic of Korea
2Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea

Received 20 May 2014; Revised 5 September 2014; Accepted 5 September 2014; Published 14 September 2014

Academic Editor: Wan-Tong Li

Copyright © 2014 Hunki Baek and Dongseok Kim. 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. M. Baurmann, T. Gross, and U. Feudel, “Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations,” Journal of Theoretical Biology, vol. 245, no. 2, pp. 220–229, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. 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 MathSciNet · View at Scopus
  3. C. Consner, 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
  4. 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, 2000/01. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. 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
  6. 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 MathSciNet · View at Scopus
  7. S. Gakkhar and R. K. Naji, “Seasonally perturbed prey-predator system with predator-dependent functional response,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1075–1083, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. 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 Publisher · View at Google Scholar · View at Scopus
  9. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet
  10. S. Gakkhar and R. K. Naji, “Order and chaos in predator to prey ratio-dependent food chain,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 229–239, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Hsu, T. 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 · View at MathSciNet · View at Scopus
  13. 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 · View at MathSciNet
  14. S. Lv and M. Zhao, “The dynamic complexity of a three species food chain model,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1469–1480, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. R. K. Naji and A. T. Balasim, “Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1853–1866, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. 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 MathSciNet · View at Scopus
  17. 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 MathSciNet · View at Scopus
  18. M. Zhao and S. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2305–2316, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. 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 · View at MathSciNet · View at Scopus
  20. F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, vol. 40 of Texts in applied mathematics, Springer, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  21. 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 · View at Scopus