  Chemotaxis, i.e. motion generated by chemical gradients, is a motility mode
shared by many living species that has been developed by evolution to optimize
certain biological processes such as foraging or immune response. In
particular, auto-chemotaxis refers to chemotaxis mediated by a cue produced by
the chemotactic particle itself. Here, we investigate the collective behavior
of auto-chemotactic particles that are repelled by the cue and therefore
migrate preferentially towards low-concentration regions. To this end, we
introduce a lattice model inspired by the true self-avoiding walk which reduces
to the Keller-Segels model in the continuous limit, for which we describe the
rich phase behavior. We first rationalize a the chemically-mediated alignment
interaction between walkers in the limit of stationary concentration fields,
and then describe the various large-scale structures that can spontaneously
form and the conditions for them to emerge, among which we find stable bands
traveling at constant speed in the direction transverse to the band.
