  A one species time-delay reaction-diffusion system defined on a complex
networks is studied. Travelling waves are predicted to occur as follows a
symmetry breaking instability of an homogenous stationary stable solution,
subject to an external non homogenous perturbation. These are generalized
Turing-like waves that materialize in a single species populations dynamics
model, as the unexpected byproduct of the imposed delay in the diffusion part.
Sufficient conditions for the onset of the instability are mathematically
provided by performing a linear stability analysis adapted to time delayed
differential equation. The method here developed exploits the properties of the
Lambert W-function. The prediction of the theory are confirmed by direct
numerical simulation carried out for a modified version of the classical Fisher
model, defined on a Watts-Strogatz networks and with the inclusion of the
delay.
