空间齐次和非齐次下活化-抑制模型动力学分析

数学物理学报 ›› 2017, Vol. 37 ›› Issue (2): 390-400.

• 论文 • 上一篇

空间齐次和非齐次下活化-抑制模型动力学分析

杨文彬¹, 吴建华²

1 西安邮电大学理学院西安 710121;
2 陕西师范大学数学与信息科学学院西安 710062

收稿日期:2016-04-26 修回日期:2016-10-20 出版日期:2017-04-26 发布日期:2017-04-26
通讯作者: 杨文彬 E-mail:yangwenbin-007@163.com
作者简介:吴建华,jianhuaw@snnu.edu.cn
基金资助:
国家自然科学基金（11501496）和陕西省教育厅专项科研计划项目（16JK1710）

Some Dynamics in Spatial Homogeneous and Inhomogeneous Activator-Inhibitor Model

Yang Wenbin¹, Wu Jianhua²

1 School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121;
2 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062

Received:2016-04-26 Revised:2016-10-20 Online:2017-04-26 Published:2017-04-26
Supported by:
Supported by the NSFC (11501496) and the Special Fund of Education Department of Shaanxi Province (16JK1710)

摘要/Abstract

摘要： 该文讨论齐次Neumann边界条件下Gierer-Meinhardt活化-抑制扩散模型.对于空间均匀（ODE）系统，分析了内部平衡态的渐近行为及其附近极限环的存在性和稳定性；对于空间异质（PDE）系统，给出了内部平衡态的Turing不稳定性条件，说明了Turing模式和时空周期模式的存在性.最后，通过数值算例验证了相应理论结果.

关键词: 反应扩散方程, 极限环, Turing不稳定, Turing模式, 时空周期模式

Abstract: The diffusive Gierer-Meinhardt activator-inhibitor model system with Neumann boundary condition is investigated. For the spatial homogeneous (ODE) system, we perform the asymptotic behavior of the interior equilibrium and the existence and stability of limit cycle surrounding the interior equilibrium. For the spatial inhomogeneous (PDE) system, we consider the Turing instability of the interior equilibrium and show the existence of Turing pattern and inhomogeneous periodic oscillatory pattern. To verify our theoretical results, some numerical simulations are also done as a complement.

Key words: Reaction-diffusion equations, Limit cycle, Turing instability, Turing pattern, Inhomogeneous periodic oscillatory pattern

中图分类号:

O175.26

杨文彬, 吴建华. 空间齐次和非齐次下活化-抑制模型动力学分析[J]. 数学物理学报, 2017, 37(2): 390-400.

Yang Wenbin, Wu Jianhua. Some Dynamics in Spatial Homogeneous and Inhomogeneous Activator-Inhibitor Model[J]. Acta mathematica scientia,Series A, 2017, 37(2): 390-400.

参考文献

[1] Castets V,Dulos E,Boissonade J,De Kepper P.Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern.Physical Review Letters,1990,64(24):2953-2956
[2] De Kepper P,Castets V,Dulos E,Boissonade J.Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction.Physica D:Nonlinear Phenomena,1991,49(1):161-169
[3] Turing A M.The chemical basis of morphogenesis.Philosophical Transactions of the Royal Society of London B:Biological Sciences,1952,237(641):37-72
[4] Ni W M,Tang M.Turing patterns in the Lengyel-Epstein system for the CIMA reaction.Transactions of the American Mathematical Society,2005,357(10):3953-3969
[5] Jin J,Shi J,Wei J,Yi F.Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions.Rocky Mountain Journal of Mathematics,2013,43(5):1637-1674
[6] 万阿英,衣凤岐,郑立飞.一类扩散的Gierer-Meindardt的模型的振动模式和Hopf分支分析.数学物理学报,2015,35A(1):381-394 Wan E,Yi F,Zheng L.Hopf bifurcation analysis and oscillatory patterns of a diffusive Gierer-Meinhardt model.Acta Mathematica Scientia,2015,35A(1):381-394
[7] Yagi A.Activator-Inhibitor Models.Abstract Parabolic Evolution Equations and Their Applications.Berlin:Springer,2010:357-371
[8] Gierer A,Meinhardt H.A theory of biological pattern formation.Kybernetik,1972,12(1):30-39
[9] Page K,Maini P K,Monk N A M.Pattern formation in spatially heterogeneous Turing reaction-diffusion models.Physica D:Nonlinear Phenomena,2003,181(1):80-101
[10] Guckenheimer J,Myers M.Computing Hopf bifurcations Ⅱ:Three examples from neurophysiology.SIAM Journal on Scientific Computing,1996,17(6):1275-1301
[11] Meiss J D.Differential Dynamical Systems.Philadelphia,PA:SIAM,2007
[12] Liu P,Shi J,Wang Y,Feng X.Bifurcation analysis of reaction-diffusion Schnakenberg model.Journal of Mathematical Chemistry,2013,51(8):2001-2019
[13] Iron D,Wei J,Winter M.Stability analysis of Turing patterns generated by the Schnakenberg model.Journal of Mathematical Biology,2004,49(4):358-390
[14] Tsai L L,Hutchison G R,Peacock-López E.Turing patterns in a self-replicating mechanism with a self-complementary template.Journal of Chemical Physics,2000,113(5):2003-2006
[15] Yi F,Wei J,Shi J.Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system.Journal of Differential Equations,2009,246(5):1944-1977
[16] Peng R,Yi F,Zhao X.Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme.Journal of Differential Equations,2013,254(6):2465-2498

空间齐次和非齐次下活化-抑制模型动力学分析

Some Dynamics in Spatial Homogeneous and Inhomogeneous Activator-Inhibitor Model

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