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Discrete Dynamics in Nature and Society
Volume 2007, Article ID 81756, 15 pages
http://dx.doi.org/10.1155/2007/81756
Research Article

Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
2Department of Mathematics, College of Huaihua, Huaihua, Hunan 418008, China
3Departamento de Análise Matemática, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago do Compostela 15782, Spain

Received 3 January 2007; Accepted 12 July 2007

Copyright © 2007 Weibing Wang 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. P. A. Abrams and L. R. Ginzburg, “The nature of predation: prey dependent, ratio dependent or neither?,” Trends in Ecology & Evolution, vol. 15, no. 8, pp. 337–341, 2000. View at Publisher · View at Google Scholar
  2. S. Ahmad and A. C. Lazer, “Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 40, no. 1–8, pp. 37–49, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings of the American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. A. Berryman, “The orgins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. View at Publisher · View at Google Scholar
  5. F. 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 Zentralblatt MATH · View at MathSciNet
  6. 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
  7. T.-W. Hwang, “Global analysis of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 395–401, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. López-Gómez, R. Ortega, and A. Tineo, “The periodic predator-prey Lotka-Volterra model,” Advances in Differential Equations, vol. 1, no. 3, pp. 403–423, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. L. A. Real, “The kinetics of functional response,” The American Naturalist, vol. 111, no. 978, pp. 289–300, 1977. View at Publisher · View at Google Scholar
  10. H. R. Thieme, “Uniform persistence and permanence for nonautonomous semiflows in population biology,” Mathematical Biosciences, vol. 166, no. 2, pp. 173–201, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. Wang and J. H. Sun, “On the predator-prey system with Holling-(n+1) functional response,” Acta Mathematica Sinica, vol. 23, no. 1, pp. 1–6, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X.-Q. Zhao, Dynamical Systems in Population Biology, vol. 16 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, NY, USA, 2003. View at Zentralblatt MATH · View at MathSciNet
  13. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at Zentralblatt MATH · View at MathSciNet
  14. L.-L. Wang and W.-T. Li, “Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 341–357, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. W. Jia, “Persistence and periodic solutions for a nonautonomous predator-prey system with type III functional response,” Journal of Biomathematics, vol. 16, no. 1, pp. 59–62, 2001 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. Liu, G. Chen, and C. Li, “Integrability and linearizability of the Lotka-Volterra systems,” Journal of Differential Equations, vol. 198, no. 2, pp. 301–320, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. J. Schreiber, “Coexistence for species sharing a predator,” Journal of Differential Equations, vol. 196, no. 1, pp. 209–225, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993. View at Zentralblatt MATH · View at MathSciNet
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593–610, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. D. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 288–303, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995. View at Zentralblatt MATH · View at MathSciNet
  24. S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, vol. 394 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Zentralblatt MATH · View at MathSciNet
  25. M. Choisy, J.-F. Guégan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects,” Physica D, vol. 223, no. 1, pp. 26–35, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. D'Onofrio, “On pulse vaccination strategy in the SIR epidemic model with vertical transmission,” Applied Mathematics Letters, vol. 18, no. 7, pp. 729–732, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. E. Funasaki and M. Kot, “Invasion and chaos in a periodically pulsed mass-action chemostat,” Theoretical Population Biology, vol. 44, no. 2, pp. 203–224, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar · View at PubMed
  29. S. Gao, Z. Teng, J. J. Nieto, and A. Torres, “Analysis of an SIR epidemic model with pulse vaccination and distributed time delay,” Journal of Biomedicine and Biotechnology, vol. 2007, Article ID 64870, 2007. View at Google Scholar
  30. S. Zhang, D. Tan, and L. Chen, “The periodic n-species Gilpin-Ayala competition system with impulsive effect,” Chaos, Solitons & Fractals, vol. 26, no. 2, pp. 507–517, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977. View at Zentralblatt MATH · View at MathSciNet
  32. X.-Q. Zhao, “Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications,” The Canadian Applied Mathematics Quarterly, vol. 3, no. 4, pp. 473–495, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet