Pattern Dynamics in a Diffusive Gierer-Meinhardt Model

被引:18
作者
Chen, Mengxin [1 ]
Wu, Ranchao [1 ]
Chen, Liping [2 ]
机构
[1] Anhui Univ, Sch Math Sci, Hefei 230601, Peoples R China
[2] Hefei Univ Technol, Sch Elect Engn & Automat, Hefei 230009, Peoples R China
来源
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 2020年 / 30卷 / 12期
基金
中国国家自然科学基金;
关键词
Gierer; Meinhardt model; pattern formation; secondary instability; amplitude equation; bifurcation; BIFURCATION-ANALYSIS; STABILITY; EXISTENCE;
D O I
10.1142/S0218127420300359
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator-inhibitor system, known as the Gierer-Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell-Whitehead-Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer-Meinhardt model.
引用
收藏
页数:26
相关论文
共 29 条
  • [1] [Anonymous], 2000, PATTERN DYNAMICS REA
  • [2] Pattern selection in a predator-prey model with Michaelis-Menten type nonlinear predator harvesting
    Chen, Mengxin
    Wu, Ranchao
    Liu, Biao
    Chen, Liping
    [J]. ECOLOGICAL COMPLEXITY, 2018, 36 : 239 - 249
  • [3] Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production
    Chen, Shanshan
    Shi, Junping
    Wei, Junjie
    [J]. APPLICABLE ANALYSIS, 2014, 93 (06) : 1115 - 1134
  • [4] An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction-Diffusion Equations
    Doelman, Arjen
    Veerman, Frits
    [J]. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 2015, 27 (3-4) : 555 - 595
  • [5] DELAYED REACTION KINETICS AND THE STABILITY OF SPIKES IN THE GIERER-MEINHARDT MODEL
    Fadai, Nabil T.
    Ward, Michael J.
    Wei, Juncheng
    [J]. SIAM JOURNAL ON APPLIED MATHEMATICS, 2017, 77 (02) : 664 - 696
  • [6] THEORY OF BIOLOGICAL PATTERN FORMATION
    GIERER, A
    MEINHARDT, H
    [J]. KYBERNETIK, 1972, 12 (01): : 30 - 39
  • [7] Turing Patterns of a Lotka-Volterra Competitive System with Nonlocal Delay
    Han, Bang-Sheng
    Wang, Zhi-Cheng
    [J]. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2018, 28 (07):
  • [8] Pattern formation and spatiotemporal chaos in a reaction-diffusion predator-prey system
    Hu, Guangping
    Li, Xiaoling
    Wang, Yuepeng
    [J]. NONLINEAR DYNAMICS, 2015, 81 (1-2) : 265 - 275
  • [9] General soliton solutions for nonlinear dispersive waves in convective type instabilities
    Khater, A. H.
    Callebaut, D. K.
    Seadawy, A. R.
    [J]. PHYSICA SCRIPTA, 2006, 74 (03) : 384 - 393
  • [10] Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system
    Li, Fang
    Peng, Rui
    Song, Xianfa
    [J]. JOURNAL OF DIFFERENTIAL EQUATIONS, 2017, 262 (01) : 559 - 589