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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 245421, 10 pages
http://dx.doi.org/10.1155/2015/245421
Research Article

Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

1College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2Institute of Systems Biology, Shanghai University, Shanghai 200444, China
3School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, China

Received 11 August 2014; Accepted 29 October 2014

Academic Editor: Mustafa Kulenović

Copyright © 2015 Xia Liu 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. J. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975. View at Google Scholar
  2. D. DeAngelis, R. Goldstein, and R. V. O'Neill, “A model for tropic interaction,” Ecology, vol. 56, no. 4, pp. 881–892, 1975. View at Google Scholar
  3. P. Crowley and E. Martin, “Functional responses and interference within and between year classes of a dragonfly population,” Journal of the North American Benthological Society, vol. 8, pp. 211–221, 1989. View at Publisher · View at Google Scholar
  4. 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. 45, pp. 1–60, 1965. View at Google Scholar
  5. M. Sambath and K. Balachandran, “Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge,” The Journal of Applied Analysis and Computation, vol. 3, no. 1, pp. 71–80, 2013. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Liu and M. Han, “Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1047–1061, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. X. Liu, Y. Liu, and J. Wang, “Bogdanov-Takens bifurcation of a delayed ratio-dependent Hollig-TANner predator prey system,” Abstract and Applied Analysis, vol. 2013, Article ID 898015, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Y. Lamontagne, C. Coutu, and C. Rousseau, “Bifurcation analysis of a predator-prey system with generalised Holling type III functional response,” Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 535–571, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S.-R. Zhou, Y.-F. Liu, and G. Wang, “The stability of predator-prey systems subject to the Allee effects,” Theoretical Population Biology, vol. 67, no. 1, pp. 23–31, 2005. View at Publisher · View at Google Scholar · View at Scopus
  10. D. Hadjiavgousti and S. Ichtiaroglou, “Allee effect in a prey-predator system,” Chaos, Solitons and Fractals, vol. 36, no. 2, pp. 334–342, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. X. Zhang, Q.-L. Zhang, and Z. Xiang, “Bifurcation analysis of a singular bioeconomic model with Allee effect and two time delays,” Abstract and Applied Analysis, vol. 2014, Article ID 745296, 12 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. G. A. van Voorn, L. Hemerik, M. P. Boer, and B. W. Kooi, “Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect,” Mathematical Biosciences, vol. 209, no. 2, pp. 451–469, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. M. S. Fowler and G. D. Ruxton, “Population dynamic consequences of Allee effects,” Journal of Theoretical Biology, vol. 215, no. 1, pp. 39–46, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. A. McCarthy, “The Allee effect, finding mates and theoretical models,” Ecological Modelling, vol. 103, no. 1, pp. 99–102, 1997. View at Publisher · View at Google Scholar · View at Scopus
  15. C. E. Brassil, “Mean time to extinction of a metapopulation with an Allee effect,” Ecological Modelling, vol. 143, no. 1-2, pp. 9–16, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. F. Courchamp, L. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Biology, Oxford University Press, New York, NY, USA, 2009.
  17. B. Dennis, “Allee effects: population growth, critical density, and the chance of extinction,” Natural Resource Modeling, vol. 3, no. 4, pp. 481–538, 1989. View at Google Scholar · View at MathSciNet
  18. P. Aguirre, E. González-Olivares, and E. Sáez, “Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1401–1416, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. R. López-Ruiz and D. Fournier-Prunaret, “Indirect Allee effect, bistability and chaotic oscillations in a predator-prey discrete model of logistic type,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 85–101, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. C. Celik and O. Duman, “Allee effect in a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1956–1962, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. Merdan and O. Duman, “On the stability analysis of a general discrete-time population model involving predation and Allee effects,” Chaos, Solitons & Fractals, vol. 40, no. 3, pp. 1169–1175, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. H. T. M. Eskola and P. Kalle, “On the mechanistic underpinning of discrete-time population models with Allee effect,” Theoretical Population Biology, vol. 72, no. 1, pp. 41–51, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. W.-X. Wang, Y.-B. Zhang, and C.-Z. Liu, “Analysis of a discrete-time predator-prey system with Allee effect,” Ecological Complexity, vol. 8, no. 1, pp. 81–85, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. I. Djellit, M. L. Sahari, and A. Hachemi, “Complex dynamics in 2-species predator-prey systems,” The Journal of Applied Analysis and Computation, vol. 3, no. 1, pp. 11–20, 2013. View at Google Scholar · View at MathSciNet
  25. Z. Jing and J. Yang, “Bifurcation and chaos in discrete-time predator-prey system,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 259–277, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. X. Zhang, Q.-l. Zhang, and V. Sreeram, “Bifurcation analysis and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1076–1096, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. H. N. Agiza, E. M. ELabbasy, H. EL-Metwally, and A. A. Elsadany, “Chaotic dynamics of a discrete prey-predator model with Holling type II,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 116–129, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. Z. He and X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-prey system,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 403–417, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. D. Blackmore, J. Chen, J. Perez, and M. Savescu, “Dynamical properties of discrete Lotka-Volterra equations,” Chaos, Solitons & Fractals, vol. 12, no. 13, pp. 2553–2568, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. D. Summers, J. G. Cranford, and B. P. Healey, “Chaos in periodically forced discrete-time ecosystem models,” Chaos, Solitons & Fractals, vol. 11, no. 14, pp. 2331–2342, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. X. Liao, S. Zhou, and Z. Ouyang, “On a stoichiometric two predators on one prey discrete model,” Applied Mathematics Letters, vol. 20, no. 3, pp. 272–278, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. W. Yang and X. Li, “Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1068–1072, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. X. T. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. X. Yang, Y. Liu, and J. Chen, “Uniform persistence for a discrete predator-prey system with delays,” Applied Mathematics and Computation, vol. 218, no. 4, pp. 1174–1179, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. F. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  37. X. Chen, “Periodicity in a nonlinear discrete predator-prey system with state dependent delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 435–446, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. Y. Xia, J. Cao, and M. Lin, “Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1079–1095, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983.
  40. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1999. View at MathSciNet
  41. J. Carr, Application of Center Manifold Theory, Springer, New York, NY, USA, 1981. View at MathSciNet