A Nonlocal Continuum Model for Biological Aggregation

  • Chad M. Topaz
  • Andrea L. Bertozzi
  • Mark A. Lewis
Original Article


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.


Aggregation Integrodifferential equation Pattern Swarm 


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Copyright information

© Society for Mathematical Biology 2006

Authors and Affiliations

  • Chad M. Topaz
    • 1
  • Andrea L. Bertozzi
    • 2
  • Mark A. Lewis
    • 3
    • 4
  1. 1.Rossier School of EducationUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUCLALos AngelesUSA
  3. 3.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  4. 4.Department of Biological SciencesUniversity of AlbertaEdmontonCanada

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