American Institute of Mathematical Sciences

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July  2013, 18(5): 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

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

Xiaoling Zou 1, , Dejun Fan 1, and  Ke Wang 2,

1. 

Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China, China
2. 

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

Received  May 2012 Revised  October 2012 Published  March 2013

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.
Keywords: peak-like stationary distribution, numerical simulations., Beddington-DeAngelis functional response, phenomenological bifurcation, crater-like stationary distribution.
Mathematics Subject Classification: Primary: 60H10, 37H20; Secondary: 60E05, 37N2.
Citation: Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley, (1972).   Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer, (1998).   Google Scholar

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

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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.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

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

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A. Friedman, "Stochastic Differential Equations and Their Applications,", Academic Press, (1976).   Google Scholar

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

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

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T. C. Gard, "Introduction to Stochastic Differential Equation,", Madison Avenue 270, (1988).   Google Scholar

[11]

R. Z. Hasminskii, "Stochastic Stability of Differential Equations, in: Mechanics and Analysis,", Sijthoff and Noordhoff, (1980).   Google Scholar

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

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

[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.  doi: 10.1016/j.apm.2011.05.027.  Google Scholar

[15]

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

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

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N. Ikeda and S. Wantanabe, "Stochastic Differential Equations and Diffusion Processes,", Amsterdam, (1981).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.02.037.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.11.008.  Google Scholar

[20]

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

[21]

X. Mao, "Stochastic Differential Equations and Applications,", Horwood Publishing, (2007).   Google Scholar

[22]

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

[23]

R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton Univ., (1973).   Google Scholar

[24]

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

[25]

G. Strang, "Linear Algebra and Its Applications,", $2^{nd}$ edition, (1980).   Google Scholar

[26]

E. C. Zeeman, On the classification of dynamical systems,, Bull. London Math. Soc., 20 (): 545.  doi: 10.1112/blms/20.6.545.  Google Scholar

[27]

E. C. Zeeman, Stability of dynamical system,, Nonlinearity, 1 (): 115.   Google Scholar

[28]

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

show all references

References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley, (1972).   Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer, (1998).   Google Scholar

[3]

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

[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.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

[5]

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

[6]

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

[7]

A. Friedman, "Stochastic Differential Equations and Their Applications,", Academic Press, (1976).   Google Scholar

[8]

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

[9]

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

[10]

T. C. Gard, "Introduction to Stochastic Differential Equation,", Madison Avenue 270, (1988).   Google Scholar

[11]

R. Z. Hasminskii, "Stochastic Stability of Differential Equations, in: Mechanics and Analysis,", Sijthoff and Noordhoff, (1980).   Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1016/j.apm.2011.05.027.  Google Scholar

[15]

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

[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.  doi: 10.1016/j.jmaa.2003.09.073.  Google Scholar

[17]

N. Ikeda and S. Wantanabe, "Stochastic Differential Equations and Diffusion Processes,", Amsterdam, (1981).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.02.037.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.11.008.  Google Scholar

[20]

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

[21]

X. Mao, "Stochastic Differential Equations and Applications,", Horwood Publishing, (2007).   Google Scholar

[22]

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

[23]

R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton Univ., (1973).   Google Scholar

[24]

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

[25]

G. Strang, "Linear Algebra and Its Applications,", $2^{nd}$ edition, (1980).   Google Scholar

[26]

E. C. Zeeman, On the classification of dynamical systems,, Bull. London Math. Soc., 20 (): 545.  doi: 10.1112/blms/20.6.545.  Google Scholar

[27]

E. C. Zeeman, Stability of dynamical system,, Nonlinearity, 1 (): 115.   Google Scholar

[28]

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

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