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.

中文翻译：

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扩散 Gierer-Meinhardt 模型中的模式动力学

本文的目的是研究激活剂-抑制剂系统（称为 Gierer-Meinhardt 模型）的模式形成和二次不稳定性，包括 Eckhaus 不稳定性和锯齿形不稳定性。Hopf分岔和图灵不稳定性的条件是通过唯一正平衡的线性稳定性分析得到的。然后，利用弱非线性分析的方法推导出振幅方程。特别是，通过对图灵不稳定性临界波数添加一个小扰动，建立了条纹图案的时空Newell-Whitehead-Segel方程。研究发现，在一定条件下可能会出现埃克豪斯不稳定性和锯齿形不稳定性。最后，通过数值模拟得到图灵和非图灵图案，包括斑点图案、混合图案、Eckhaus 模式、时空混沌、非恒定稳态解、二维或一维空间中的空间齐次周期解和空间非齐次解。对于这种扩散 Gierer-Meinhardt 模型，理论分析和数值结果非常吻合。