Turing pattern formation in coupled reaction-diffusion system with distributed delays

Turing pattern formation in coupled two-layer system with distributed delayed is investigated. Numerical simulations prove that, when the coupling is weak, it can apparently accelerate the formation process and enhance the spatial amplitude of the pattern. When it is strong, it will prolong the formation process or even inhibit the pattern and turn the whole system into bulk oscillatory state by its influence on the transient oscillatory state. If the coupling covers only part of the system, Turing pattern can be prominently oriented according to the shape of the coupling area at tiny coupling strength. However, if the coupling is too strong, the Turing pattern may also be destroyed. This means that in coupled systems, the delay effect in the cross-layer signal transfer may significantly influence the spatial character and/or the evolution dynamics in Turing pattern formation, even to destroy the pattern. This work is of practical significance in the study of Turing pattern in biosystems, where bilayer membranes or multilayer tissues are often found.