Self-similar Turing patterns: An anomalous diffusion consequence

In this work, we show that under specific anomalous diffusion conditions, chemical systems can produce well-ordered self-similar concentration patterns through diffusion-driven instability. We also find spiral patterns and patterns with mixtures of rotational symmetries. The type of anomalous diffusion discussed in this work, either subdiffusion or superdiffusion, is a consequence of the medium heterogeneity, and it is modeled through a space-dependent diffusion coefficient with a power-law functional form.

Figure 1

Real (continuous lines) and imaginary part (dotted lines) of the dispersion relation for different values of $\lambda$. The parameter values used for this figure are $\alpha =0.899, \beta =-0.91, \delta =0.01011$, and $D=0.516$.

Figure 2

Turing pattern rotational symmetries for normal diffusion. The dashed vertical lines correspond to patterns that are not centrosymmetric (not invariant under any rotation) but have a clear symmetry in the sense that each spot is surrounded by the same number of spots. The patterns in these cases would also be invariant to specific rotations in an infinite domain or after a specific spatial translation of the pattern in the finite circular domain. Vertical continuous lines correspond to centrosymmetric patterns, i.e., patterns that are invariant under specific rotations; the bullet at the end of the vertical line in the $y$ axis indicates the symmetry or number of rotations that leave the pattern invariant.

Figure 3

Centrosymmetric Turing patterns under normal diffusion conditions, $\lambda =0$. (a) Pentagonal Turing pattern, $\delta =0.0036$. (b) Hexagonal Turing pattern, $\delta =0.0070$. (c) Twofold-symmetric Turing pattern, for $\delta =0.0086$. (d) Tetrasymmetric Turing pattern, $\delta =0.0044$.

Figure 4

Turing pattern rotational symmetries for subdiffusive regimes. (a) Rotational symmetries for $\lambda =0.1$. (b) Rotational symmetries for $\lambda =0.5$.

Figure 5

Centrosymmetric Turing patterns under subdiffusion conditions: $\lambda =0.5$. (a) Threefold-symmetric Turing pattern, $\delta =0.0053$. (b) Eightfold-symmetric Turing pattern, $\delta =0.0039$.

Figure 6

Turing patterns under superdiffusion conditions. (a) Turing patterns with mixed symmetries, $\lambda =-0.66, \delta =0.0041$. (b) Spiral-like Turing patterns, $\lambda =-1\text{.}5, \delta =0.0007$. (c) Double-arm spiral-like Turing pattern, $\lambda =-1\text{.}95, \delta =0.0057$.

Figure 7

Turing patterns under strong superdiffusion, $\lambda =-1.95$. (a) Spiral-like pattern, $\delta =0.0053$. (b) Self-similar hexagonal Turing pattern, $\delta =0.0072$.

Figure 8

Centrosymmetric self-similar Turing patterns under strong superdiffusion conditions. For each figure, $\lambda =-1\text{.}95$. (a) Twofold self-similar symmetric Turing pattern, $\delta =0.0107$. (b) Threefold self-similar symmetric Turing pattern, $\delta =0.0110$. (c) Tetrasymmetric self-similar Turing pattern, $\delta =0.0069$. (d) Fivefold-symmetric self-similar Turing pattern, $\delta =0.0115$.

Figure 9

Turing patterns under superdiffusion. (a) Almost fivefold centrosymmetric pattern with no repetition of the same motif for $\lambda =-0.5$ and $\delta =0.0014$. (b) Tetrafold centrosymmetric pattern with no repetition of the same motif for $\lambda =-0.1$ and $\delta =0.0023$.