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Using the models of stochastic population dynamics, the competitions and interactions of interspecies and between species and the stochastic environment are studied. In this paper, the stochastic ecosystems (in Itô or Statonovich model) of two competing species are investigated through evaluating probability densities and information entropy fluxes and productions of two species. The formulas of entropy flux (i.e. expectation of divergence) and entropy production are educed for numerical calculations, through the corresponding Fokker-Planck equation with its condition and the definition of Shannon entropy. The nonlinear characteristics of entropy fluxes are captured and the relationships are found between the extremal points of entropy productions and the rapid transitions or bifurcations. The numerical results obtained with path integration method show that the probability densities and Shannon entropies of these two stochastic models (in Itô or Statonovich meaning) have the same evolutional tendency but with different points of extrema.
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Keywords:
- probability density /
- information entropy /
- ecosystem /
- Gaussian white noise
[1] Chen L S, Wang D D 1994 Phys. 23 408 (in Chinese) [陈兰荪, 王东达 1994 物理 23 408]
[2] Gui Z J 2005 Models of Biological Dynamics and Computer Simulation (1st Ed.) (Beijing: Science Press) (in Chinese) [桂占吉 2005 生物动力学模型与计算机仿真 (第1版) (北京: 科学出版社)]
[3] Chen L S, Meng X Z, Jiao J J 2009 Biological Dynamics (1st Ed.) (Beijing: Science Press) (in Chinese) [陈兰荪, 孟新柱, 焦建军 2009 生物动力学 (第1版) (北京: 科学出版社)]
[4] Dimentberg M F 2002 Phys. Rev. E 65 036204
[5] Pigolotti S, Flammini A, Maritan A 2004 Phys. Rev. E 70 011916
[6] Cai G Q, Lin Y K 2004 Phys. Rev. E 70 041910
[7] Cai G Q, Lin Y K 2007 Phys. Rev. E 76 041913
[8] Wu Y, Zhu W Q 2008 Phys. Rev. E 77 041911
[9] Cai G Q 2009 Int. J. Non-Linear Mech. 44 769
[10] Cai G Q, Lin Y K 2011 Phys. Rev. E 76 041913
[11] Nicolis G, Daems D 1998 Chaos 8 311
[12] Daems D, Nicolis G 1999 Phys. Rev. E 59 4000
[13] Bag B C, Chaudhuri J R, Ray D S 2000 J. Phys. A: Math. and General 33 8331
[14] Bag B C, Banik S K, Ray D S 2001 Phys. Rev. E 64 026110
[15] Bag B C 2002 Phys. Rev. E 66 026122
[16] Xie W X, Xu W, Cai L, Jin Y F 2005 Chin. Phys. 14 1766
[17] Bag B C 2003 J. Chem. Phys. 119 4988
[18] Goswami G, Mukherjee B, Bag B C 2005 Chem. Phys. 312 47
[19] Xie W X, Xu W, Cai L 2006 Acta Phys. Sin. 55 1639 (in Chinese) [谢文贤, 徐伟, 蔡力 2006 物理学报 55 1639]
[20] Xu W, Xie W X, Cai L 2007 Phys. A-Stat. Mech. and its App. 384 273
[21] Xie W X, Xu W, Cai L 2007 Chin. Phys. 16 42
[22] Guo P Y, Xu W, Liu D 2009 Acta Phys. Sin. 58 5179 (in Chinese) [郭培荣, 徐伟, 刘迪 2009 物理学报 58 5179]
[23] Guo Y F, Xu W, Li D X, Wang L 2010 Acta Phys. Sin. 59 2235 (in Chinese) [郭永峰, 徐伟, 李东喜, 王亮 2010 物理学报 59 2235]
[24] Guo Y F, Xu W, Li D X 2009 Chin. J. Applied Mech. 26 264 [郭永峰, 徐伟, 李东喜 2009 应用力学学报 26 264]
[25] Xie W X 2007 Ph. D. Dissertation (Xi'an: Northwestern Polytechnical University) (in Chinese) [谢文贤 2007 博士学位论文 (西安:西北工业大学)]
[26] Zhang L P, Wang H N, Xu M 2011 Acta Phys. Sin. 60 010506 (in Chinese) [张丽萍, 王惠南, 徐敏 2011 物理学报 60 010506]
[27] Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer-Verlag)
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[1] Chen L S, Wang D D 1994 Phys. 23 408 (in Chinese) [陈兰荪, 王东达 1994 物理 23 408]
[2] Gui Z J 2005 Models of Biological Dynamics and Computer Simulation (1st Ed.) (Beijing: Science Press) (in Chinese) [桂占吉 2005 生物动力学模型与计算机仿真 (第1版) (北京: 科学出版社)]
[3] Chen L S, Meng X Z, Jiao J J 2009 Biological Dynamics (1st Ed.) (Beijing: Science Press) (in Chinese) [陈兰荪, 孟新柱, 焦建军 2009 生物动力学 (第1版) (北京: 科学出版社)]
[4] Dimentberg M F 2002 Phys. Rev. E 65 036204
[5] Pigolotti S, Flammini A, Maritan A 2004 Phys. Rev. E 70 011916
[6] Cai G Q, Lin Y K 2004 Phys. Rev. E 70 041910
[7] Cai G Q, Lin Y K 2007 Phys. Rev. E 76 041913
[8] Wu Y, Zhu W Q 2008 Phys. Rev. E 77 041911
[9] Cai G Q 2009 Int. J. Non-Linear Mech. 44 769
[10] Cai G Q, Lin Y K 2011 Phys. Rev. E 76 041913
[11] Nicolis G, Daems D 1998 Chaos 8 311
[12] Daems D, Nicolis G 1999 Phys. Rev. E 59 4000
[13] Bag B C, Chaudhuri J R, Ray D S 2000 J. Phys. A: Math. and General 33 8331
[14] Bag B C, Banik S K, Ray D S 2001 Phys. Rev. E 64 026110
[15] Bag B C 2002 Phys. Rev. E 66 026122
[16] Xie W X, Xu W, Cai L, Jin Y F 2005 Chin. Phys. 14 1766
[17] Bag B C 2003 J. Chem. Phys. 119 4988
[18] Goswami G, Mukherjee B, Bag B C 2005 Chem. Phys. 312 47
[19] Xie W X, Xu W, Cai L 2006 Acta Phys. Sin. 55 1639 (in Chinese) [谢文贤, 徐伟, 蔡力 2006 物理学报 55 1639]
[20] Xu W, Xie W X, Cai L 2007 Phys. A-Stat. Mech. and its App. 384 273
[21] Xie W X, Xu W, Cai L 2007 Chin. Phys. 16 42
[22] Guo P Y, Xu W, Liu D 2009 Acta Phys. Sin. 58 5179 (in Chinese) [郭培荣, 徐伟, 刘迪 2009 物理学报 58 5179]
[23] Guo Y F, Xu W, Li D X, Wang L 2010 Acta Phys. Sin. 59 2235 (in Chinese) [郭永峰, 徐伟, 李东喜, 王亮 2010 物理学报 59 2235]
[24] Guo Y F, Xu W, Li D X 2009 Chin. J. Applied Mech. 26 264 [郭永峰, 徐伟, 李东喜 2009 应用力学学报 26 264]
[25] Xie W X 2007 Ph. D. Dissertation (Xi'an: Northwestern Polytechnical University) (in Chinese) [谢文贤 2007 博士学位论文 (西安:西北工业大学)]
[26] Zhang L P, Wang H N, Xu M 2011 Acta Phys. Sin. 60 010506 (in Chinese) [张丽萍, 王惠南, 徐敏 2011 物理学报 60 010506]
[27] Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer-Verlag)
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