\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises

Abstract Related Papers Cited by
  • The existence of a stationary distribution and a stochastic Hopf bifurcation phenomenon for a noisy predator-prey system with Beddington-DeAngelis functional response are studied both theoretically and numerically. Considering the qualitative change of the shape of the stationary distribution, the stochastic Hopf bifurcation appears as a change from a peak-like distribution to a crater-like distribution. Results are obtained through the original niosy system rather than approximations based on stochastic averaging or scaling methods.
    Mathematics Subject Classification: Primary: 60H10, 37H20; Secondary: 60E05, 37N25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley, New York, 1972.

    [2]

    L. Arnold, "Random Dynamical Systems," Springer, New York, 1998.

    [3]

    J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44(1975), 331-340.

    [4]

    R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the beddington-deangelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.doi: 10.1006/jmaa.2000.7343.

    [5]

    C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets, Physica A., 387 (2008), 3837-3846.doi: 10.1016/j.physa.2008.01.078.

    [6]

    D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.

    [7]

    A. Friedman, "Stochastic Differential Equations and Their Applications," Academic Press, New York, 1976.

    [8]

    T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370.doi: 10.1016/S0092-8240(84)80044-0.

    [9]

    T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419.doi: 10.1016/0362-546X(86)90111-2.

    [10]

    T. C. Gard, "Introduction to Stochastic Differential Equation," Madison Avenue 270, New York, 1988.

    [11]

    R. Z. Hasminskii, "Stochastic Stability of Differential Equations, in: Mechanics and Analysis," Sijthoff and Noordhoff, Netherlands, 1980.

    [12]

    D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.doi: 10.1137/S0036144500378302.

    [13]

    D. Huang, H. Wang, J. Feng and Z. Zhu, Hopf bifurcation of the stochastic model on hab nonlinear stochastic dynamics, Chaos, Solitons and Fractals, 27 (2006), 1072-1079.doi: 10.1016/j.chaos.2005.04.086.

    [14]

    Z. Huang, Q. Yang and J. Cao, Stochastic stability and bifurcation for the chronic state in marchuk's model with noise, Appl. Math. Model., 35 (2011), 5842-5855.doi: 10.1016/j.apm.2011.05.027.

    [15]

    T. W. Hwang, Global analysis of the predator-prey system with beddington-deangelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.

    [16]

    T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with beddington-deangelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.doi: 10.1016/j.jmaa.2003.09.073.

    [17]

    N. Ikeda and S. Wantanabe, "Stochastic Differential Equations and Diffusion Processes," Amsterdam, North-Holland, 1981.

    [18]

    C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with beddington-deangelis functional response, J. Math. Anal. Appl., 381 (2011), 441-453.doi: 10.1016/j.jmaa.2011.02.037.

    [19]

    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, J. Math. Anal. Appl., 377 (2011), 435-440.doi: 10.1016/j.jmaa.2010.11.008.

    [20]

    W. Li, W. Xu, J. Zhao and Y. Jin, Stochastic stability and bifurcation in a macroeconomic model, Chaos, Solitons and Fractals, 31 (2007), 702-711.doi: 10.1016/j.chaos.2005.10.024.

    [21]

    X. Mao, "Stochastic Differential Equations and Applications," Horwood Publishing, Chichester, 2007.

    [22]

    X. Mao, Stationary distribution of stochastic population systems, Syst. Control Letters, 60 (2011), 398-405.doi: 10.1016/j.sysconle.2011.02.013.

    [23]

    R. M. May, "Stability and Complexity in Model Ecosystems," Princeton Univ., 1973.

    [24]

    K. R. Schenk-Hoppé, Stochastic hopf bifurcation: an example, Inc. J. Non-Ltnmr Mechanws, 31 (1996), 685-692.doi: 10.1016/0020-7462(96)00030-3.

    [25]

    G. Strang, "Linear Algebra and Its Applications," $2^{nd}$ edition, Harcourt Brace, Watkins, 1980.

    [26]

    E. C. ZeemanOn the classification of dynamical systems, Bull. London Math. Soc., 20 (1988a), 545-557. doi: 10.1112/blms/20.6.545.

    [27]

    E. C. ZeemanStability of dynamical system, Nonlinearity, 1 (1988b), 115-155.

    [28]

    C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.doi: 10.1137/060649343.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(285) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return