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Time evolutions of u r ! , t and their spatial variation of u and v. (a)-(d) were obtained numerically from Eq. (1) with β ¼À0:08; (e)-(h) were obtained numerically from Eq. (1) with β ¼ 0:01; (i)-(l) were the distributions of u (red line) and v (blue line) along the solid arrow in (e)-(h), respectively.
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Turing demonstrated a coupled reaction-diffusion equation with two components produced steady-state heterogeneous spatial patterns, under certain conditions. The instability found by Turing is now called a diffusion-driven instability or Turing instability. Systems in two dimensions produce spot and stripe patterns, and these systems have been appl...
Contexts in source publication
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... from the random initial distributions shown in Figure 1(a) and (e), the time evolutions of the patterns in a density plot of u over time are shown in Figure 1 (b-d, f-h). Figure 1(j-l) displays the spatial variations of the concentrations u and v for the distributions along the arrow in Figure 1(e-h). ...
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... from the random initial distributions shown in Figure 1(a) and (e), the time evolutions of the patterns in a density plot of u over time are shown in Figure 1 (b-d, f-h). Figure 1(j-l) displays the spatial variations of the concentrations u and v for the distributions along the arrow in Figure 1(e-h). ...
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... from the random initial distributions shown in Figure 1(a) and (e), the time evolutions of the patterns in a density plot of u over time are shown in Figure 1 (b-d, f-h). Figure 1(j-l) displays the spatial variations of the concentrations u and v for the distributions along the arrow in Figure 1(e-h). As shown in Figure 1(i-k), the exponential growth of a specific unstable critical mode revealed by the linear analysis emerges from random initial states in the early stages of pattern formation as explained in Ref. [1]. ...
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... from the random initial distributions shown in Figure 1(a) and (e), the time evolutions of the patterns in a density plot of u over time are shown in Figure 1 (b-d, f-h). Figure 1(j-l) displays the spatial variations of the concentrations u and v for the distributions along the arrow in Figure 1(e-h). As shown in Figure 1(i-k), the exponential growth of a specific unstable critical mode revealed by the linear analysis emerges from random initial states in the early stages of pattern formation as explained in Ref. [1]. ...
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... 1(j-l) displays the spatial variations of the concentrations u and v for the distributions along the arrow in Figure 1(e-h). As shown in Figure 1(i-k), the exponential growth of a specific unstable critical mode revealed by the linear analysis emerges from random initial states in the early stages of pattern formation as explained in Ref. [1]. However, as shown in Figure 1(b-d) or (f-h), the static spot or stripe patterns were eventually obtained. ...
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... shown in Figure 1(i-k), the exponential growth of a specific unstable critical mode revealed by the linear analysis emerges from random initial states in the early stages of pattern formation as explained in Ref. [1]. However, as shown in Figure 1(b-d) or (f-h), the static spot or stripe patterns were eventually obtained. This is because the pattern converges asymptotically through nonlinear responses in the next stage of pattern formation [2,3,16]. ...
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... is because the pattern converges asymptotically through nonlinear responses in the next stage of pattern formation [2,3,16]. Here, we note that the distributions of v follow the same periodic patterns with the same periodicities but with different amplitudes, as shown in Figure 1(j-i). Various discussions have been conducted on pattern selection [1-3, 15, 16]. ...
