Nonequilibrium transition and pattern formation in a linear reaction-diffusion system with self-regulated kinetics

We consider a reaction-diffusion system with linear, stochastic activator-inhibitor kinetics where the time evolution of concentration of a species at any spatial location depends on the relative average concentration of its neighbors. This self-regulating nature of kinetics brings in spatial correlation between the activator and the inhibitor. An interplay of this correlation in kinetics and disparity of diffusivities of the two species leads to symmetry breaking non-equilibrium transition resulting in stationary pattern formation. The role of initial noise strength and the linear reaction terms has been analyzed for pattern selection.

Figure 1

Concentration profile of the activator $U$ at 15000 time units for $D=1.0$, $\alpha =400.0$, $\beta =400.0$, $r=1.0$ (units arbitrary).

Figure 2

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =400.0$, $\beta =400.0$, $r=1.0$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and (e) three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 3

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =200.0$, $\beta =400.0$, $r=1.0$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and (e) three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 4

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =600.0$, $\beta =400.0$, $r=1.0$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and (e) three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 5

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =200.0$, $\beta =400.0$, $r=0.5$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and $\left(\mathbf{e}\right)$ three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 6

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =200.0$, $\beta =400.0$, $r=1.5$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and (e) three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 7

Concentration profile of the activator $U$ for $D=0.05$, $\alpha =200.0$, $\beta =400.0$, $r=2.0$ at the time units (a) $t=1000$, (b) $t=5000$, (c) $t=12000$, (d) $t=15000$, and (e) three repeating profiles of (d) in $x$ and $y$ directions (units arbitrary).

Figure 8

Stationary ($t=15000$ t.u.) concentration profile of the activator $U$ for $D=0.05$, $\alpha =400$, $\beta =400$, $r=1.0$ for initial Gaussian noise strength $\sigma$ (a) 0.01, (b) 0.05, (c) 0.10, and (d) 0.20 (units arbitrary).

Figure 9

(a) Three repeating profiles of Fig. 8 in $x$ and $y$ directions showing wavy stripe pattern for $\sigma =0.01$, (b) same as in (a) but for Fig. 8, ($\sigma =0.05$) showing honey-comb-like spots, (c) same as in (a) but for Fig. 8, ($\sigma =0.10$) showing honey-comb-like spots, and (d) same as in (a) but for Fig. 8, ($\sigma =0.20$) showing wavy stripes (units arbitrary).