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Abstract and Applied Analysis
Volume 2014, Article ID 720283, 10 pages
http://dx.doi.org/10.1155/2014/720283
Research Article

Persistence and Nonpersistence of a Predator Prey System with Stochastic Perturbation

1College of Science, China University of Petroleum (East China), Qingdao 266580, China
2Department of Basic Courses, Air Force Aviation University, Changchun, Jilin 130022, China
3School of Mathematics, Jilin University, Changchun, Jilin 130024, China
4School of Business, Northeast Normal University, Changchun, Jilin 130024, China

Received 13 December 2013; Accepted 25 February 2014; Published 3 April 2014

Academic Editor: Jifeng Chu

Copyright © 2014 Haihong Li 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. T. C. Gard, “Stability for multispecies population models in random environments,” Nonlinear Analysis: Theory, Methods & Applications, vol. 10, no. 12, pp. 1411–1419, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. S. Goh, “Global stability in many species system,” American Naturalist, vol. 111, pp. 135–143, 1997. View at Google Scholar
  3. J. L. Lv and K. Wang, “Asymptotic properties of a stochastic predator-prey system with Holling II functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4037–4048, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. S. Chen and Z. J. Jing, “The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions,” Chinese Science Bulletin, vol. 9, pp. 521–523, 1984. View at Google Scholar · View at MathSciNet
  5. X. R. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and Their Applications, vol. 97, no. 1, pp. 95–110, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Mao, “Delay population dynamics and environmental noise,” Stochastics and Dynamics, vol. 5, no. 2, article 149, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. Polansky, “Invariant distributions for multipopulation models in random environments,” Theoretical Population Biology, vol. 16, no. 1, pp. 25–34, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  8. T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, NY, USA, 1988.
  9. Y. Kuang and H. L. Smith, “Global stability for infinite delay Lotka-Volterra type systems,” Journal of Differential Equations, vol. 103, no. 2, pp. 221–246, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y. Z. Hu, F. K. Wu, and C. Huang, “Stochastic Lotka-Volterra models with multiple delays,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 42–57, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. N. I. Wantanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, The Netherlands, 1981.
  12. C. Y. Ji and D. Q. Jiang, “Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 441–453, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. M. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, USA, 2001.
  14. C. Y. Ji, D. Q. Jiang, and N. Z. 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 · View at MathSciNet
  15. C. Y. Ji, D. Q. Jiang, and X. Y. Li, “Qualitative analysis of a stochastic ratio-dependent predator-prey 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
  16. X. N. Liu and L. S. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 311–320, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
  18. D.-Q. Jiang, B.-X. Zhang, D.-H. Wang, and N.-Z. Shi, “Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation,” Science in China A: Mathematics, vol. 50, no. 7, pp. 977–986, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. L. S. Chen and J. Chen, Nonlinear Biological Dynamical System, Science Press, Beijing, China, 1993.
  20. P. Y. Xia, X. K. Zheng, and D. Q. Jiang, “Persistence and nonpersistence of a nonautonomous stochastic mutualism system,” Abstract and Applied Analysis, vol. 2013, Article ID 256249, 13 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. D. Q. Jiang, C. Y. Ji, X. Y. Li, and D. O'Regan, “Analysis of autonomous Lotka-Volterra competition systems with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 390, no. 2, pp. 582–595, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. W. Feller, An Introduction to Probability Theory and Its Application, vol. 2, John Wiley & Sons, New York, NY, USA, 1971. View at MathSciNet
  23. A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, no. 3, pp. 876–902, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. 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 · View at MathSciNet