  We present a combined numerical and analytical study of pattern formation in
an active system where particles align, possess a density-dependent motility,
and are subject to a logistic reaction. This is a model for suspensions of
reproducing bacteria, but it can also represent, in the ordered phase,
actomyosin gels in vitro or in vivo. In the disordered phase, we find that
motility suppression and growth compete to yield stable or blinking patterns,
which, when dense enough, acquire internal orientational ordering, to yield
asters or spirals. In the ordered phase, the reaction term leads to previously
unobserved never-settling patterns which can provide a simple framework to
understand the formation of motile and spiral patterns in actin.
