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

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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

Abstract
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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 and 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, 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. Zeeman, On the classification of dynamical systems, Bull. London Math. Soc., 20 (1988a), 545-557. doi: 10.1112/blms/20.6.545.
[27]	E. C. Zeeman, Stability 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.

show all references

References:

[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. Zeeman, On the classification of dynamical systems, Bull. London Math. Soc., 20 (1988a), 545-557. doi: 10.1112/blms/20.6.545.
[27]	E. C. Zeeman, Stability 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.

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