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$L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems
Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises
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 |
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 (): 545.
doi: 10.1112/blms/20.6.545. |
[27] |
E. C. Zeeman, Stability of dynamical system,, Nonlinearity, 1 (): 115.
|
[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 (): 545.
doi: 10.1112/blms/20.6.545. |
[27] |
E. C. Zeeman, Stability of dynamical system,, Nonlinearity, 1 (): 115.
|
[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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