Complex Turing patterns in two-layer non-linearly coupling reaction diffusion systems

Abstract

The influence of Turing modes in two subsystems on pattern formation is investigated by the two-layer non-linearly coupled Brusselator model. It is found that the coupling coefficient and wave number ratio between two Turing modes take an important role in the pattern formation and pattern selection. The kind of pattern changes from simple pattern to complex one with the increase of wave number ratio. When nonlinear coupling coefficient is smaller than 0.1, the short wave mode in system 1 under the action of instability mode in system 2 can form not only simple pattern (such as simple hexagon and quadrilateral and stripe pattern), but also complex pattern due to the resonance coupling between the two Turing modes (such as honeycomb hexagon and super hexagon and complex black-eye pattern), and the transformation process of pattern from quadrilateral to superlattice pattern is observed for the first time under the specific parameters. When nonlinear coupling coefficient is more than 0.1, the simple patterns such as simple hexagon and stripe pattern are obtained only in system 1, because there is no resonance coupling between the two Turing modes in system 1.