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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 547152, 24 pages
http://dx.doi.org/10.1155/2012/547152
Research Article

Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

1School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2School of Science, Changchun University, Changchun 130022, China
3Department of Foundation, Harbin Finance University, Harbin 150030, China

Received 24 July 2012; Accepted 13 September 2012

Academic Editor: Yonghui Xia

Copyright © 2012 Li Zu 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. Y. Kuang and Y. Takeuchi, “Predator-prey dynamics in models of prey dispersal in two-patch environments,” Mathematical Biosciences, vol. 120, no. 1, pp. 77–98, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. Y. Li and Z. Shuai, “Global-stability problem for coupled systems of differential equations on networks,” Journal of Differential Equations, vol. 248, no. 1, pp. 1–20, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. E. Beretta, F. Solimano, and Y. Takeuchi, “Global stability and periodic orbits for two-patch predator-prey diffusion-delay models,” Mathematical Biosciences, vol. 85, no. 2, pp. 153–183, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. H. I. Freedman and Y. Takeuchi, “Global stability and predator dynamics in a model of prey dispersal in a patchy environment,” Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 8, pp. 993–1002, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. V. Padrón and M. C. Trevisan, “Environmentally induced dispersal under heterogeneous logistic growth,” Mathematical Biosciences, vol. 199, no. 2, pp. 160–174, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X. Mao, C. Yuan, and J. Zou, “Stochastic differential delay equations of population dynamics,” Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 296–320, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Z. Teng and L. Chen, “Permanence and extinction of periodic predator-prey systems in a patchy environment with delay,” Nonlinear Analysis. Real World Applications, vol. 4, no. 2, pp. 335–364, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. W. Wendi and M. Zhien, “Asymptotic behavior of a predator-prey system with diffusion and delays,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 191–204, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Xu and L. Chen, “Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment,” Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 577–588, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. Cui, “The effect of dispersal on permanence in a predator-prey population growth model,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1085–1097, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. L. Zhang and Z. Teng, “Boundedness and permanence in a class of periodic time-dependent predator-prey system with prey dispersal and predator density-independence,” Chaos, Solitons and Fractals, vol. 36, no. 3, pp. 729–739, 2008. View at Publisher · View at Google Scholar
  12. Z. Y. Lu and Y. Takeuchi, “Global asymptotic behavior in single-species discrete diffusion systems,” Journal of Mathematical Biology, vol. 32, no. 1, pp. 67–77, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. C. Ji, D. Jiang, and N. Shi, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 482–498, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. C. Ji, D. Jiang, and N. Shi, “A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 435–440, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. G. Cai and Y. Lin, “Stochastic analysis of predator-prey type ecosystems,” Ecological Complexity, vol. 4, no. 4, pp. 242–249, 2007. View at Publisher · View at Google Scholar
  16. X. Li, D. Jiang, and X. Mao, “Population dynamical behavior of Lotka-Volterra system under regime switching,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 427–448, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, NY, USA, 1997.
  18. X. Li and X. Mao, “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 2, pp. 523–545, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. C. Ji, D. Jiang, and H. Liu, “Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation,” Mathematical Problems in Engineering, vol. 10, pp. 1155–1172, 2010.
  20. X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.
  21. D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. H. Guo, M. Y. Li, and Z. Shuai, “A graph-theoretic approach to the method of global Lyapunov functions,” Proceedings of the American Mathematical Society, vol. 136, no. 8, pp. 2793–2802, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.