  We consider a coarse-grained description of a system of self-propelled
particles given by hydrodynamic equations for the density and polarization
fields. We find that the ordered moving or flocking state of the system is
unstable to spatial fluctuations beyond a threshold set by the self-propulsion
velocity of the individual units. In this region, the system organizes itself
into an inhomogeneous state of well-defined propagating stripes of flocking
particles interspersed with low density disordered regions. Further, we find
that even in the regime where the homogeneous flocking state is stable, the
system exhibits large fluctuations in both density and orientational order. We
study the hydrodynamic equations analytically and numerically to characterize
both regimes.
