Study on Turing Patterns of Gray-Scott Model via Amplitude Equation

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.