Analytic model for ring pattern formation by bacterial swarmers.
We analyze a model proposed by Medvedev, Kaper, and Kopell (the MKK model) for ring formation in two-dimensional bacterial colonies of Proteus mirabilis. We correct the model to formally include a feature crucial of the ring generation mechanism: a bacterial density threshold to the nonlinear diffusivity of the MKK model. We numerically integrate the model equations, and observe the logarithmic profiles of the bacterial densities near the front. These lead us to define a consolidation front distinct from the colony radius. We find that this consolidation front propagates outward toward the colony radius with a nearly constant velocity. We then implement the corrected MKK equations in two dimensions and compare our results with biological experiment. Our numerical results indicate that the two-dimensional corrected MKK model yields smooth (rather than branched) rings, and that colliding colonies merge if grown in phase but not if grown out of phase. We also introduce a model, based on coupling the MKK model to a nutrient field, for simulating experimentally observed branched rings.