  We present a generic mechanism by which reproducing microorganisms, with a
diffusivity that depends on the local population density, can form stable
patterns. It is known that a decrease of swimming speed with density can
promote separation into bulk phases of two coexisting densities; this is
opposed by the logistic law for birth and death which allows only a single
uniform density to be stable. The result of this contest is an arrested
nonequilibrium phase separation in which dense droplets or rings become
separated by less dense regions, with a characteristic steady-state length
scale. Cell division mainly occurs in the dilute regions and cell death in the
dense ones, with a continuous flux between these sustained by the diffusivity
gradient. We formulate a mathematical model of this in a case involving
run-and-tumble bacteria, and make connections with a wider class of mechanisms
for density-dependent motility. No chemotaxis is assumed in the model, yet it
predicts the formation of patterns strikingly similar to those believed to
result from chemotactic behavior.
