Spatio-temporal patterns of bacteria caused by collective motion

In incubation experiments on bacterial colonies of Proteus mirabilis, collective motion of bacteria is found to generate macroscopic turbulent patterns on the surface of agar media. We propose a mathematical model to describe the time evolution of the positional and directional distributions of motile bacteria in such systems, and investigate this model both numerically and analytically. It is shown that as the average density of bacteria increases, nonuniform swarming patterns emerge from a uniform stationary state. For a sufficient large density, we find that spiral patterns are caused by interactions between the local bacteria densities and the rotational mode of the collective motion. Unidirectional spiral patterns similar to those observed in experiments appear in the case in which the equilibrium directional distribution is asymmetric.