- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 801812, 14 pages
The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response
1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
2School of Science, Changchun University of Science and Technology, Changchun, Jilin 130022, China
3Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China
Received 21 October 2012; Accepted 14 December 2012
Academic Editor: Ivanka Stamova
Copyright © 2012 Zhenwen Liu 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.
- H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980.
- C. S. Holling, “The components of predation as revealed by a study of small-mammal predation of the european pine sawy,” Canadian Entomologist, vol. 91, pp. 293–320, 1959.
- N. G. Hairston, F. E. Smith, and L. B. Slobodkin, “Community structure, population control and competition,” The American Naturalist, vol. 94, pp. 421–425, 1960.
- M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,” Science, vol. 171, pp. 385–387, 1969.
- L. S. Chen and Z. J. Jing, “Existence and uniqueness of limit cycles for the differential equations of predator-prey interaction,” Chinese Science Bulletin, vol. 29, no. 9, pp. 521–523, 1984.
- 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 and Fractals, vol. 16, no. 2, pp. 311–320, 2003.
- S. W. Zhang and L. S. Chen, “A Holling II functional response food chain model with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1269–1278, 2005.
- S. W. Zhang, D. J. Tan, and L. S. Chen, “Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 28, no. 2, pp. 367–376, 2006.
- R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, NJ, USA, 1973.
- 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.
- 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.
- C. Y. Ji, D. Q. Jiang, and N. Z. 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.
- 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.
- 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.
- R. Z. Has'meminskii, Stochastic Stability of Differential Equations, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
- R. Rudnicki, “Long-time behaviour of a stochastic prey-predator model,” Stochastic Processes and their Applications, vol. 108, no. 1, pp. 93–107, 2003.
- R. Rudnicki and K. Pichór, “Influence of stochastic perturbation on prey-predator systems,” Mathematical Biosciences, vol. 206, no. 1, pp. 108–119, 2007.
- X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
- T. C. Gard, Introduction to STochastic Differential Equations, vol. 270, Madison Avenue, New York, NY, USA, 1988.
- G. Strang, Linear Algebra and Its Applications, Thomson Learning, 1988.
- C. Zhu and G. Yin, “Asymptotic properties of hybrid diffusion systems,” SIAM Journal on Control and Optimization, vol. 46, no. 4, pp. 1155–1179, 2007.