Rippling patterns of myxobacteria appear in starving colonies before they
aggregate to form fruiting bodies. These periodic traveling cell density waves
arise from the coordination of individual cell reversals, resulting from an
internal clock regulating them, and from contact signaling during bacterial
collisions. Here we revisit a mathematical model of rippling in myxobacteria
due to Igoshin et al.\ [Proc. Natl. Acad. Sci. USA {\bf 98}, 14913 (2001) and
Phys. Rev. E {\bf 70}, 041911 (2004)]. Bacteria in this model are phase
oscillators with an extra internal phase through which they are coupled to a
mean-field of oppositely moving bacteria. Previously, patterns for this model
were obtained only by numerical methods and it was not possible to find their
wavenumber analytically. We derive an evolution equation for the reversal point
density that selects the pattern wavenumber in the weak signaling limit, show
the validity of the selection rule by solving numerically the model equations
and describe other stable patterns in the strong signaling limit. The nonlocal
mean-field coupling tends to decohere and confine patterns. Under appropriate
circumstances, it can annihilate the patterns leaving a constant density state
via a nonequilibrium phase transition reminiscent of destruction of
synchronization in the Kuramoto model.Comment: Revtex 26 pages, 7 figure
