Wavelength selection of rippling patterns in myxobacteria

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 98, 14913 (2001)PNASA60027-842410.1073/pnas.221579598 and Phys. Rev. E 70, 041911 (2004)PLEEE81539-375510.1103/PhysRevE.70.041911]. 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 wave number analytically. We derive an evolution equation for the reversal point density that selects the pattern wave number in the weak signaling limit, shows the validity of the selection rule by solving numerically the model equations, and describes 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.