Theory of periodic swarming of bacteria: application to Proteus   mirabilis
  The periodic swarming of bacteria is one of the simplest examples for pattern formation produced by the self-organized collective behavior of a large number of organisms. In the spectacular colonies of Proteus mirabilis (the most common species exhibiting this type of growth) a series of concentric rings are developed as the bacteria multiply and swarm following a scenario periodically repeating itself. We have developed a theoretical description for this process in order to get a deeper insight into some of the typical processes governing the phenomena in systems of many interacting living units. All of our theoretical results are in excellent quantitative agreement with the complete set of available observations. 
Andras Czirok, Mitsugu Matsushita and Tamas Vicsek