  Retardation between sensation and action is an inherent biological trait.
Here we study its effect in the Vicsek model, which is a paradigmatic swarm
model. We find that: (i) a discrete time delay in the orientational
interactions diminishes the ability of strongly aligned swarms to follow a
leader and, in return, increases their stability against random orientation
fluctuations; (ii) both longer delays and higher speeds favor ballistic over
diffusive spreading of information (orientation) through the swarm; (iii) for
short delays, the mean change in the total orientation (the order parameter)
scales linearly in a small orientational bias of the leaders and inversely in
the delay time, while its variance first increases and then saturates with
increasing delays; (iv) the linear response breaks down when orientation
conservation is broken.
