  Collectives of actively-moving particles can spontaneously separate into
dilute and dense phases -- a fascinating phenomenon known as motility-induced
phase separation (MIPS). MIPS is well-studied for randomly-moving particles
with no directional bias. However, many forms of active matter exhibit
collective chemotaxis, directed motion along a chemical gradient that the
constituent particles can generate themselves. Here, using theory and
simulations, we demonstrate that collective chemotaxis strongly competes with
MIPS -- in some cases, arresting or completely suppressing phase separation, or
in other cases, generating fundamentally new dynamic instabilities. We
establish quantitative principles describing this competition, thereby helping
to reveal and clarify the rich physics underlying active matter systems that
perform chemotaxis, ranging from cells to robots.
