Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction–diffusion systems

Abstract

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.