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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 767526, 10 pages
http://dx.doi.org/10.1155/2013/767526
Research Article

Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

Chang Tan1,2 and Jun Cao3

1College of Science, Northeast Forestry University, Harbin 150040, China
2Forestry Engineering Mobile Station, Northeast Forestry University, Harbin 150040, China
3College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China

Received 18 September 2013; Revised 23 November 2013; Accepted 25 November 2013

Academic Editor: Stefan Balint

Copyright © 2013 Chang Tan and Jun Cao. 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. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific Publishing, Teaneck, NJ, USA, 1989. View at MathSciNet
  2. V. Lakshmikantham, X. Liu, and S. Sathananthan, “Impulsive integro-differential equations and extension of Lyapunov's method,” Applicable Analysis, vol. 32, no. 3-4, pp. 203–214, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  3. 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, Harlow, UK, 1993. View at MathSciNet
  4. K. Gopalsamy and B. G. Zhang, “On delay differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 139, no. 1, pp. 110–122, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  5. H. Liang, M. Liu, and M. Song, “Extinction and permanence of the numerical solution of a two-prey one-predator system with impulsive effect,” International Journal of Computer Mathematics, vol. 88, no. 6, pp. 1305–1325, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2005. View at MathSciNet
  7. J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859–872, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Mohamad, “Global exponential stability in discrete-time analogues of delayed cellular neural networks,” Journal of Difference Equations and Applications, vol. 9, no. 6, pp. 559–575, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Mohamad and A. G. Naim, “Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks,” Journal of Computational and Applied Mathematics, vol. 138, no. 1, pp. 1–20, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  10. K. Murakami, “Stability for non-hyperbolic fixed points of scalar difference equations,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 492–505, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, vol. 2, Cambridge University Press, Cambridge, UK, 1996. View at MathSciNet
  12. Q. Zhang, “On a linear delay difference equation with impulses,” Annals of Differential Equations, vol. 18, no. 2, pp. 197–204, 2002. View at MathSciNet
  13. Z. He and X. Zhang, “Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 605–620, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. Z. Abdullin, “Stability of nonlinear difference equations with pulse actions: a comparison method,” Automation and Remote Control 1, vol. 61, no. 11, pp. 1796–1807, 2000.
  15. R. Z. Abdullin, “Stability of difference equations with impulsive actions at the instants of time dependent on the state vector,” Automation and Remote Control 1, vol. 58, no. 7, pp. 1092–1100, 1997.
  16. B. Liu and D. J. Hill, “Uniform stability and ISS of discrete-time impulsive hybrid systems,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 2, pp. 319–333, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Mohamad and K. Gopalsamy, “Exponential stability of continuous-time and discrete-time cellular neural networks with delays,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 17–38, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Mohamad and K. Gopalsamy, “Dynamics of a class of discrete-time neural networks and their continuous-time counterparts,” Mathematics and Computers in Simulation, vol. 53, no. 1-2, pp. 1–39, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  19. B. Liu, Y. Zhang, and L. Chen, “The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management,” Nonlinear Analysis: Real World Applications, vol. 6, no. 2, pp. 227–243, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Z. Zhang and X. Liu, “Robust stability of uncertain discrete impulsive switching systems,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 380–389, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  21. S. Wu, C. Li, X. Liao, and S. Duan, “Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach,” Neurocomputing, vol. 82, pp. 29–36, 2012.
  22. Y. Zhang, “Exponential stability of impulsive discrete systems with time delays,” Applied Mathematics Letters of Rapid Publication, vol. 25, no. 12, pp. 2290–2297, 2012. View at Publisher · View at Google Scholar · View at MathSciNet