  Reaction-diffusion systems with time-delay defined on complex networks have
been studied in the framework of the emergence of Turing instabilities. The use
of the Lambert W-function allowed us get explicit analytic conditions for the
onset of patterns as a function of the main involved parameters, the
time-delay, the network topology and the diffusion coefficients. Depending on
these parameters, the analysis predicts whether the system will evolve towards
a stationary Turing pattern or rather to a wave pattern associated to a Hopf
bifurcation. The possible outcomes of the linear analysis overcome the
respective limitations of the single-species case with delay, and that of the
classical activator-inhibitor variant without delay. Numerical results gained
from the Mimura-Murray model support the theoretical approach.
