  We study competition of two non-motile bacterial strains in a
three-dimensional channel numerically, and analyze how their configuration
evolves in space and time. We construct a lattice model that takes into account
self-replication, mutation, and killing of bacteria. When mutation is not
significant, the two strains segregate and form stripe patterns along the
channel. The formed lanes are gradually rearranged, with increasing length
scales in the two-dimensional cross-sectional plane. We characterize it in
terms of coarsening and phase ordering in statistical physics. In particular,
for the simple model without mutation and killing, we find logarithmically slow
coarsening, which is characteristic of the two-dimensional voter model. With
mutation and killing, we find a phase transition from a monopolistic phase, in
which lanes are formed and coarsened until the system is eventually dominated
by one of the two strains, to an equally mixed and disordered phase without
lane structure. Critical behavior at the transition point is also studied and
compared with the generalized voter class and the Ising class. These results
are accounted for by continuum equations, obtained by applying a mean field
approximation along the channel axis. Our findings indicate relevance of
critical coarsening of two-dimensional systems in the problem of bacterial
competition within anisotropic three-dimensional geometry.
