We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the inter-layer coupling. For systems of $n$-component layers and non-identical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a $\sqrt{2}$:1 length scale ratio produces an unusual steady square pattern.
Turing and wave instabilities in hyperbolic reaction-diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation
Turing's diffusive threshold in random reaction-diffusion systems
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）
Chemotaxis-induced linear instabilities and pattern formation in a reaction-diffusion model
Wenzhou Institute of the University of Chinese Academy of Sciences（中国科学院大学温州研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Universitat de Barcelona（巴塞罗那大学）；Saha Institute of Nuclear Physics（萨哈核物理研究所）
