Numerical simulations of hexagonal grid state patterns

The four hexagonal grid state patterns and a variety of non-grid states are obtained by changing the values of intensity ratio between two Turing modes in the two-layer coupled Lengel-Epstein model system. Results of numerical investigation show that those grid states in reaction diffusion are interleaving structures of three sets of different sublattices, which result from the interaction of both the wave number ratio and intensity ratio between Turing modes in the two subsystems; and the specific expressions of three-wave resonance in physical space are governed by the mode intensity ratio. Furthermore, the value of intensity ratio between the two Turing modes in the grid state patterns is greater than that of non-grid state structures, and the type of pattern selected by the system changes from complex to simple pattern with the increase of mode intensity ratio. Finally, it is found that these four hexagonal grid states correspond to different number pair (a, b) having different stability, and the grid state with the number pair (3, 2) is the most stable structure.