  Meso-scale turbulence was originally observed experimentally in various
suspensions of swimming bacteria, as well as in the collective motion of active
colloids. The corresponding large-scale dynamical patterns were reproduced in a
simple model of a polar fluid, assuming a constant density of active particles.
Recent, more detailed experimental studies revealed additional interesting
aspects, such as anomalous velocity statistics and clustering phenomena. Those
phenomena cannot be explained by currently available models for active polar
fluids. Herein, we extend the continuum model suggested by Dunkel et al. to
include density variations and a feedback between the local density and
self-propulsion speed of the active particles. If the velocity decreases strong
enough with the density, a linear stability analysis of the resulting model
shows that, in addition to the short-wavelength instability of the original
model, a long-wavelength instability occurs. This is typically observed for
high densities of polar active particles and is analogous to the well-known
phenomenon of motility-induced phase separation (MIPS) in scalar active matter.
We determine a simple phase diagram indicating the linear instabilities and
perform systematic numerical simulations for the various regions in the
corresponding parameter space. The interplay between the well understood
short-range instability and the long-range instability leads to interesting
dynamics and novel phenomena concerning nucleation and coarsening processes.
Our simulation results display a rich variety of novel patterns, including
phase separation into domains with dynamically changing irregularly shaped
boundaries. Anomalous velocity statistics are observed in all phases where the
system segregates into regions of high and low densities. This offers a simple
explanation for their occurrence in recent experiments with bacterial
suspensions.
