Instabilities and patterns in coupled reaction-diffusion layers

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of $n$-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a $\sqrt{2}$:1 length-scale ratio produces an unusual steady square pattern.

Figure 1

Diagram of resonant triads in Fourier space. The Fourier modes satisfy Eq. (34). Solid circles and vectors indicate neutral stability and dotted ones indicate weak damping. In (a), $|{\mathbf{Q}}_{1,2}|<|{\mathbf{Q}}_3|$ and the resonant angle satisfies $0\le {\theta }_{\mathrm{res}}<2\pi /3$. In (b), $|{\mathbf{Q}}_{1,2}|>|{\mathbf{Q}}_3|$ and $2\pi /3<{\theta }_{\mathrm{res}}<\pi$.

Figure 2

(a) Ratio $r_q$ in Eq. (33) of two (nearly) critical wave numbers near a Turing-Turing bifurcation in the Brusselator. The governing equations are Eqs. (1) and  (39) with $A=3$, $B=9$, $D=2.25$. The coupling parameter $\alpha$ is a free parameter and $\beta =\alpha D$. (b) Angle of triad resonance corresponding to (a). For the lower (dashed) branch, the resonant triad corresponds to Fig. 1a, in which the damped mode has larger wave number than the critical ones. For the upper (dashed) branch, the resonant triad corresponds to Fig. 1b, in which the damped mode has smaller wave number. See Sec. 3b for details.

Figure 3

Numerical simulation of coupled Brusselators given by Eq. (1) with Eq. (39). Parameter values are $A=3$, $B=9$, $D=2.244$, $\alpha =0.723$, and $\beta =1.633$ (to three decimal places). (a) The eigenvalue with maximum real part is plotted as a function of wave number. (b) Striped pattern resulting from this choice of parameters. Dark and light regions indicated variations in concentration of chemical $u$ in the top layer. The bottom layer looks the same but with light and dark regions reversed. (c) The radial power spectrum of the striped pattern with units chosen so that the dominant peak is normalized to unity. (d) The Fourier spectrum of the striped pattern.

Figure 4

Parameter values are $A=3$, $B=9$, $D=2.244$, $\alpha =0.490$, and $\beta =1.113$ (to three decimal places). (a) The eigenvalue with maximum real part is plotted as a function of wave number. (b) Square pattern resulting from this choice of parameters. Dark and light regions indicated variations in concentration of chemical $u$ in the top layer. The bottom layer looks the same but with light and dark regions reversed. (c) The radial power spectrum of the square pattern with units chosen so that the dominant peak is normalized to unity. (d) The Fourier spectrum of the square pattern.

Figure 5

Results analogous to, and with the same parameters as, those described in the legend of Fig 4. However, whereas the square computational domain in Fig. 4 had each side of length eight times the length of the weakly growing mode as determined from linear stability analysis, here we choose $5\sqrt{3}\approx 8.7$ wavelengths per side in order to verify that the box size was not responsible for stabilizing the square pattern. Indeed, here we still obtain a square pattern, albeit one with a different orientation. (a) The eigenvalue with maximum real part is plotted as a function of wave number (identical to Fig. 4a, reproduced here for convenience). (b) Square pattern. Dark and light regions indicated variations in concentration of chemical $u$ in the top layer. (c) The radial power spectrum of the square pattern with units chosen so that the dominant peak is normalized to unity. (d) The Fourier spectrum of the square pattern.