In this paper, the amplitude equations of a Gray–Scott model without (or with) the feedback time delay are derived based on weakly nonlinear method, by which the selection of Turing patterns for this model can be theoretically determined. As a result, the effects of the diffusion coefficient ratio and the time delay factor on the Turing pattern can be investigated as the main purpose of this paper. If one of the diffusion coefficients is chosen as the bifurcation control parameter in the procedure of the amplitude equation at first, it is proved that the first-order bifurcation of the Turing patterns is only determined by the diffusion coefficient ratio and independent of the concrete value of each diffusion coefficient once the parameters of the reaction terms are fixed as the appropriate constants in the regions of Turing patterns. Furthermore, the feedback time delay factor has no effect on the first-order bifurcation of the Turing patterns, but affects the morphological characteristics of the Turing patterns, especially in the case of large ratio of the diffusion coefficients. With time increasing, the feedback time delay factor can postpone the formation of the Turing patterns and cause the oscillations of Turing patterns at each spatial position. By implementing the numerical calculations for this model, the various Turing patterns with different values of the diffusion coefficient ratios are presented, which really verify the dependence of the diffusion coefficient ratio and independence of the feedback time delay on the first-order bifurcation of the Turing patterns.

中文翻译：

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用振幅方程研究格雷-斯科特模型的图灵模式

本文基于弱非线性方法推导了一个没有（或有）反馈时延的Gray-Scott模型的幅度方程，通过该方法可以从理论上确定该模型的图灵模式的选择。因此，本文的主要目的可以研究扩散系数比和时间延迟因子对图灵图的影响。如果首先在振幅方程的过程中选择一个扩散系数作为分岔控制参数，证明了图灵图的一阶分岔只由扩散系数比决定，与具体值无关一旦反应项的参数固定为图灵模式区域中的适当常数，每个扩散系数的值。此外，反馈时延因子对图灵图的一阶分岔没有影响，但会影响图灵图的形态特征，特别是在扩散系数比值较大的情况下。随着时间的增加，反馈时延因子可以推迟图灵图的形成，并引起图灵图在每个空间位置的振荡。通过对该模型的数值计算，给出了具有不同扩散系数比值的各种图灵模式，真正验证了扩散系数比值的依赖性和反馈时延对图灵一阶分岔的独立性模式。但影响图灵模式的形态特征，特别是在扩散系数比值较大的情况下。随着时间的增加，反馈时延因子可以推迟图灵图的形成，并引起图灵图在每个空间位置的振荡。通过对该模型的数值计算，给出了具有不同扩散系数比值的各种图灵模式，真正验证了扩散系数比值的依赖性和反馈时延对图灵一阶分岔的独立性模式。但影响图灵模式的形态特征，特别是在扩散系数比值较大的情况下。随着时间的增加，反馈时延因子可以推迟图灵图的形成，并引起图灵图在每个空间位置的振荡。通过对该模型的数值计算，给出了具有不同扩散系数比值的各种图灵模式，真正验证了扩散系数比值的依赖性和反馈时延对图灵一阶分岔的独立性模式。反馈时延因子可以推迟图灵图的形成，引起图灵图在每个空间位置的振荡。通过对该模型的数值计算，给出了具有不同扩散系数比值的各种图灵模式，真正验证了扩散系数比值的依赖性和反馈时延对图灵一阶分岔的独立性模式。反馈时延因子可以推迟图灵图的形成，引起图灵图在每个空间位置的振荡。通过对该模型的数值计算，给出了具有不同扩散系数比值的各种图灵模式，真正验证了扩散系数比值的依赖性和反馈时延对图灵一阶分岔的独立性模式。