Morphological transitions and bistability in Turing systems

It is well known that in two dimensions Turing systems produce spots, stripes and labyrinthine patterns, and in three dimensions lamellar and spherical structures, or their combinations, are observed. In this paper we study transitions between these states in both two and three dimensions. First, we derive the regions of stability for different patterns using nonlinear bifurcation analysis. Then, we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects. The method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction-diffusion system.

Figure 1

The set of eigenvalues $\mu \left(C\right)$ determining the stability of different structures, i.e., the stripes and hexagonal patterns in 2D, and lamellae, cylinders, and sc droplets in 3D. The symmetries are stable for $\mu \left(C\right)<0$.

Figure 2

Transition from stripes to spots. The patterns obtained after 50 0000 iterations in a $100\times 100$ system with $k_c=0.45$. Black corresponds to areas dominated by chemical $U$ (zeros) and the lighter color chemical $V$ (ones). $C$ varies from 0.007 to 1.000 from A to I.

Figure 3

The number of $U$ clusters as a function of the nonlinear parameter $C$, which was varied continuously throughout the sweep in a single simulation. The two-dimensional $100\times 100$ pattern was given 250 000 iterations for stabilization at each value of $C$ $(k_c=0.86)$. The arrow heads describe the direction of the sweep implying hysteresis.

Figure 4

The normalized number of $U$ clusters as a function of $C$ in two-dimensional systems. System sizes are $L=75$, $L=100$, $L=175$, $L=250$, and $L=500$. The results were averaged up to 20 simulations and they match with the analytical prediction for the bistable regime given by $0.084<C<0.161$.

Figure 5

Transition from a twisted minimal surface to spherical shapes in a three-dimensional system of size $50\times 50\times 50$. The structures were obtained after 500 000 iterations with $k_c=0.86$. The visualization was carried out by plotting the middle concentration isosurface. Parameter values: (A) $C=0$, (B) $C=0.44$, (C) $C=0.53$, and (D) $C=1.0$

Figure 6

The averaged normalized number of $U$ clusters as a function of $C$ in a three-dimensional systems. System sizes are $L=30$, $L=40$, $L=50$, and $L=75$. The bifurcation analysis predicts unstable lamellar structures and stable sc droplets for $C>0.361$, whereas stable cylindrical structures are predicted for all $C<0.65$.