Table of Contents
ISRN Mathematical Analysis
Volume 2014 (2014), Article ID 347201, 10 pages
http://dx.doi.org/10.1155/2014/347201
Research Article

Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

1Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China
2Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received 2 October 2013; Accepted 20 November 2013; Published 17 February 2014

Academic Editors: N. Shanmugalingam and T. Tran

Copyright © 2014 Zhenguo Luo 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. R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin, Germany, 1977. View at MathSciNet
  2. Y. M. Chen, “Multiple periodic solutions of delayed predator-prey systems with type IV functional responses,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 45–53, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. M. Xiao and S. G. 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
  4. X. Q. Ding and J. F. Jiang, “Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2819–2827, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. L. Hu, G. R. Liu, and J. R. Yan, “Existence of multiple positive periodic solutions of delayed predator-prey models with functional responses,” Computers & Mathematics with Applications, vol. 52, no. 10-11, pp. 1453–1462, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. H. Xia, J. D. Cao, and S. S. Cheng, “Multiple periodic solutions of a delayed stage-structured predator-prey model with non-monotone functional responses,” Applied Mathematical Modelling, vol. 31, no. 9, pp. 1947–1959, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. Z. G. Luo and J. H. Shen, “New Razumikhin type theorems for impulsive functional differential equations,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 375–386, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. G. Ballinger and X. Z. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. Liu, Z. D. Teng, and W. B. Liu, “Dynamic behaviors of the periodic Lotka-Volterra competing system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 31, no. 2, pp. 356–370, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, Longman Sci. Tech., Harlow, UK, 1993. View at MathSciNet
  11. V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Dfferential Equations, vol. 6, World Scientific Publishing, Teaneck, NJ, USA, 1989. View at MathSciNet
  12. Z. G. Luo, B. X. Dai, and Q. Wang, “Existence of positive periodic solutions for a nonautonomous neutral delay n-species competitive model with impulses,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3955–3967, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. X. D. Li and X. L. Fu, “On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 3, pp. 442–447, 2013. View at Publisher · View at Google Scholar
  14. Z. G. Luo and L. P. Luo, “Existence and stability of positive periodic solutions for a neutral multispecies Logarithmic population model with feedback control and impulse,” Abstract and Applied Analysis, vol. 2013, Article ID 741043, 11 pages, 2013. View at Publisher · View at Google Scholar
  15. Z. G. Luo and L. P. Luo, “Global positive periodic solutions of generalized n-species competition systems with multiple delays and impulses,” Abstract and Applied Analysis, vol. 2013, Article ID 980974, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. Liu and K. Wang, “Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 909–925, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Q. Wang, B. X. Dai, and Y. M. Chen, “Multiple periodic solutions of an impulsive predator-prey model with Holling-type IV response,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1829–1836, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. X. Dai, H. Su, and D. W. Hu, “Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 126–134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. M. Moghadas and B. D. Corbett, “Limit cycles in a generalized Gause-type predator-prey model,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1343–1355, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. Q. Ding and J. F. Jiang, “Positive periodic solutions in delayed Gause-type predator-prey systems,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1220–1230, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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
  22. J. F. Andrews, “A mathematical model for the continuous cul ture of microorgnaisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707–723, 1986. View at Google Scholar
  23. W. Sokol and J. A. Howell, “Kinetics of phenol oxidation by washed cells,” Biotechnology and Bioengineering, vol. 23, pp. 2039–2049, 1980. View at Google Scholar
  24. F. E. Smith, “Population dynamics in daphnia magna and a new model for population growth,” Ecology, vol. 44, pp. 651–663, 1963. View at Publisher · View at Google Scholar
  25. K. Gopalsamy and G. Ladas, “On the oscillation and asymptotic behavior of Ṅt=Nt[a+bNt-τ-cN2t-τ],” Quarterly of Applied Mathematics, vol. 3, pp. 433–440, 1990. View at Google Scholar