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Reaction-diffusion systems with time-delay defined on complex networks havebeen studied in the framework of the emergence of Turing instabilities. The useof the Lambert W-function allowed us get explicit analytic conditions for theonset of patterns as a function of the main involved parameters, thetime-delay, the network topology and the diffusion coefficients. Depending onthese parameters, the analysis predicts whether the system will evolve towardsa stationary Turing pattern or rather to a wave pattern associated to a Hopfbifurcation. The possible outcomes of the linear analysis overcome therespective limitations of the single-species case with delay, and that of theclassical activator-inhibitor variant without delay. Numerical results gainedfrom the Mimura-Murray model support the theoretical approach.

Pattern formation in a two-component reaction-diffusion system with delayed processes on a network