  We consider the patterns of collective motion emerging when many aligning,
self-propelling units move in two dimensions while interacting through a
repulsive potential and are also subject to delays and random perturbations. In
this approach, delay plays the role analogous to reaction time so that a given
particle is influenced by the information about the velocity and the position
of the other particles in its vicinity with some time delay. To get insight
into the involved complex flows and the transitions between them we use a
simple model allowing, by fine-tuning of its few parameters, the observation
and analysis of behaviours that are less accessible by experiments or analytic
calculations and at the same time make the reproduction of experimental results
possible. We report for the first time about a transition from an ordered,
polarized collective motion to disorder as a function of the increasing time
delay. For a fixed intermediate value of the delay similar transition (from
order to disorder) is obtained as the repulsion radius is increased. Our
simulations show a transition from total polarization to two kinds of states:
fully disordered and a kind of state which is a mixture of patches of fully
disordered motion in the background of orderly moving other particles. The
transition occurs as the delay time is increased and is sharp, indicating that
the nature of this order-disorder transition is either of first-order or is
described by a sharply decreasing linear function. Our model is a simplified
version of a practical situation of quickly growing interest because time
delays are expected to play an increasingly important role when the traffic of
many, densely distributed autonomous drones will move around in a
quasi-two-dimensional air space.
