Integro-diierential Model for Pattern Formation in Bacterial Swarm

This paper introduces a one-dimensional, partial integro-diierential model for bacterial aggregation. Integro-diierential equation is derived from a chemotactic partial diierential system, and stability and numerical analyses are applied to study pattern formation. We demonstrate that an interplay between positive and negative chemotaxis can generate both periodic patterns and aggregation. In the limit of slow diiusion we use a Lagrangian approach to derive a gradient system and analyze periodic peak-like patterns. We discuss the possibility of existence of other types of patterns, and the applicability of the model to the pattern formation in bacterial swarms, in particular the rippling phenomenon in Myxobacteria.