Bulletin of Mathematical Biology

, 68:1601

A Nonlocal Continuum Model for Biological Aggregation


    • Rossier School of EducationUniversity of Southern California
  • Andrea L. Bertozzi
    • Department of MathematicsUCLA
  • Mark A. Lewis
    • Department of Mathematical and Statistical SciencesUniversity of Alberta
    • Department of Biological SciencesUniversity of Alberta
Original Article

DOI: 10.1007/s11538-006-9088-6

Cite this article as:
Topaz, C.M., Bertozzi, A.L. & Lewis, M.A. Bull. Math. Biol. (2006) 68: 1601. doi:10.1007/s11538-006-9088-6


We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. 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 solutions, 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 simulations in two dimensions.


AggregationIntegrodifferential equationPatternSwarm
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© Society for Mathematical Biology 2006