Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 568943, 8 pages
http://dx.doi.org/10.1155/2014/568943
Research Article

Stability and Hopf Bifurcation in a Delayed Predator-Prey System with Herd Behavior

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 24 February 2014; Accepted 13 March 2014; Published 8 April 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Chaoqun Xu and Sanling Yuan. 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. L. Chen, X. Song, and Z. Lu, Mathematical Ecological Models and Research Methods, Sichuan Science and Technology Press, Chengdu, China, 2003.
  2. C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 45, supplement S45, pp. 5–60, 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 Publisher · View at Google Scholar
  4. J. Beddington, “Mutual interference between parasites or predators and its effect on searching effciency,” Journal of Animal Ecology, vol. 44, no. 1, pp. 331–340, 1975. View at Publisher · View at Google Scholar
  5. D. L. DeAngelis, R. A. Goldsten, and R. V. O'Neill, “A model for trophic interaction,” Ecology, vol. 56, no. 4, pp. 881–892, 1975. View at Publisher · View at Google Scholar
  6. R. E. Kooij and A. Zegeling, “A predator-prey model with Ivlev's functional response,” Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 473–489, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. 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
  8. Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 36, no. 4, pp. 389–406, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. D. Xiao and H. Zhu, “Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 802–819, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, “Bifurcation analysis of a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 63, no. 2, pp. 636–682, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. X.-A. Zhang, L. Chen, and A. U. Neumann, “The stage-structured predator-prey model and optimal harvesting policy,” Mathematical Biosciences, vol. 168, no. 2, pp. 201–210, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. M. A. Aziz-Alaoui and M. D. Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1069–1075, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1104–1118, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. L. Han, Z. Ma, and H. W. Hethcote, “Four predator prey models with infectious diseases,” Mathematical and Computer Modelling, vol. 34, no. 7-8, pp. 849–858, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. R. Xu and Z. Ma, “Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1444–1460, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. M. Liu and K. Wang, “Global stability of stage-structured predator-prey models with Beddington-DeAngelis functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3792–3797, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. C. Ji, D. Jiang, and X. Li, “Qualitative analysis of a stochastic ratio-dependent predatorprey system,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1326–1341, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. V. Ajraldi, M. Pittavino, and E. Venturino, “Modeling herd behavior in population systems,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2319–2338, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. P. A. Braza, “Predator-prey dynamics with square root functional responses,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1837–1843, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. E. Venturino and S. Petrovskii, “Spatiotemporal behavior of a prey-predator system with a group defense for prey,” Ecological Complexity, vol. 14, pp. 37–47, 2013. View at Publisher · View at Google Scholar
  22. S. Belvisi and E. Venturino, “An ecoepidemic model with diseased predators and prey group defense,” Simulation Modelling Practice and Theory, vol. 34, pp. 144–155, 2013. View at Publisher · View at Google Scholar
  23. S. Yuan, C. Xu, and T. Zhang, “Spatial dynamics in a predator-prey model with herd behavior,” Chaos, vol. 23, no. 3, Article ID 033102, 2013. View at Publisher · View at Google Scholar
  24. S. J. Brentnall, K. J. Richards, J. Brindley, and E. Murphy, “Plankton patchiness and its effect on larger-scale productivity,” Journal of Plankton Research, vol. 25, no. 2, pp. 121–140, 2003. View at Publisher · View at Google Scholar · View at Scopus
  25. E. A. Fulton, A. D. M. Smith, and C. R. Johnson, “Mortality and predation in ecosystem models: is it important how these are expressed?” Ecological Modelling, vol. 169, no. 1, pp. 157–178, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. 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
  27. N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, UK, 1989. View at MathSciNet
  28. J. K. Hale and S. M. V. Lunel, Introduction to Functional-Differential Equations, Springer, New York, NY, USA, 1993. View at MathSciNet
  29. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  30. J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  31. R. Xu, Q. Gan, and Z. Ma, “Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 187–203, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. Y. Song and S. Yuan, “Bifurcation analysis in a predator-prey system with time delay,” Nonlinear Analysis: Real World Applications, vol. 7, no. 2, pp. 265–284, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Y. Song and J. Wei, “Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 1–21, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. D. Xiao and S. Ruan, “Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,” Journal of Differential Equations, vol. 176, no. 2, pp. 494–510, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159–173, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 183–206, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  37. Z. Liu and R. Yuan, “Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 521–537, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. M. Fan and K. Wang, “Periodicity in a delayed ratio-dependent predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 262, no. 1, pp. 179–190, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. E. Beretta and Y. Kuang, “Global analyses in some delayed ratio-dependent predator-prey systems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 32, no. 3, pp. 381–408, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. T. Zhao, Y. Kuang, and H. L. Smith, “Global existence of periodic solutions in a class of delayed gause-type predator-prey systems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 28, no. 8, pp. 1373–1394, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  41. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at Zentralblatt MATH · View at MathSciNet