Original Article

Bulletin of Mathematical Biology

, 68:1601

First online:

A Nonlocal Continuum Model for Biological Aggregation

  • Chad M. TopazAffiliated withRossier School of Education, University of Southern California Email author 
  • , Andrea L. BertozziAffiliated withDepartment of Mathematics, UCLA
  • , Mark A. LewisAffiliated withDepartment of Mathematical and Statistical Sciences, University of AlbertaDepartment of Biological Sciences, University of Alberta

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Abstract

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.

Keywords

Aggregation Integrodifferential equation Pattern Swarm