Bulletin of Mathematical Biology (2006) DOI 10.1007/s11538-006-9088-6 ORIGINAL ARTICLE A Nonlocal Continuum Model for Biological Aggregation

Abstract We construct a continuum model for biological aggregations in which in-dividuals experience long-range social attraction and short-range dispersal. For the case of one spatial dimension, we study the steady states analytically and numeri-cally. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady-state clumps are reached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solu-tions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions as well, and is demonstrated via numerical simu-lations in two dimensions