Super-lattice patterns in two-layered coupled non-symmetric reaction diffusion systems

Abstract

The coupling mechanism is one of most important approaches to generating multiple-scaled spatial-temporal patterns. In this paper, the mode interaction between two different Turing modes and the pattern forming mechanisms in the non-symmetric reaction diffusion system are numerically investigated by using a two-layered coupled model. This model is comprised of two different reaction diffusion models: the Brusselator model and the Lengyel-Epstein model. It is shown that the system gives rise to superlattice patterns if these two Turing modes satisfy the spatial resonance condition, otherwise the system yields simple patterns or superposition patterns. A suitable wave number ratio and the same symmetry are two necessary conditions for the spatial resonance of Turing modes. The eigenvalues of these two Turing modes can only vary in a certain range in order to make the two sub-system patterns have the same symmetry. Only when the long wave mode becomes the unstable mode, can it modulate the other Turing mode and result in the formation of spatiotemporal patterns with multiple scale. As the wave number ratio increases, the higher-order harmonics of the unstable mode appear, and the sub-system with short wave mode undergoes a transition from the black-eye pattern to the white-eye pattern, and finally to a temporally oscillatory hexagon pattern. It is demonstrated that the resonance between the Turing mode and its higher-order harmonics located in the wave instability region is the dominant mechanism of the formation of this oscillatory hexagon pattern. Moreover, it is found that the coupling strength not only determines the amplitudes of these patterns, but also affects their spatial structures. Two different types of white-eye patterns and a new super-hexagon pattern are obtained as the coupling strength increases. These results can conduce to understanding the complex spatial-temporal behaviors in the coupled reaction diffusion systems.