Turing instabilities in two-layer networks

• A methodology is proposed to study Turing instability in reaction-diffusion systems on complex networks with topological disparities between the networks of each concentration. • Classical Turing instability conditions are recovered as a special case when the networks of the concentrations are identical. • The methodology is validated for various types of networks, including higher-order networks. • Turing pattern formation induced by the emergence of nodes is investigated. We studied Turing instabilities in two-layer activator-inhibitor networks under the general assumption that the diffusion operator can be different in each layer, and the chemical kinetics can be different between nodes. A general approach that reduces to the well-known Turing conditions in a particular case was proposed. A numerical implementation of the method is proposed to predict Turing instabilities in complex network configurations. Application to the well-known Schnakenberg reaction-diffusion system showed that under certain conditions, such as changing the connections in one layer or adding new nodes to one of the layers, patterns are formed, even when the standard Turing conditions suggest stability. Our findings highlight the complexity of pattern formation in multilayer networks and suggest that the current analytical tools may be insufficient for predicting instabilities in more complex network structures.