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A Nonlocal Continuum Model for Biological Aggregation

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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.

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Authors and Affiliations

  1. Rossier School of Education, University of Southern California, Los Angeles, CA, 90089, USA

    Chad M. Topaz

  2. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Andrea L. Bertozzi

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

    Mark A. Lewis

  4. Department of Biological Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

    Mark A. Lewis

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  1. Chad M. Topaz
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  2. Andrea L. Bertozzi
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  3. Mark A. Lewis
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Correspondence to Chad M. Topaz.

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Topaz, C.M., Bertozzi, A.L. & Lewis, M.A. A Nonlocal Continuum Model for Biological Aggregation. Bull. Math. Biol. 68, 1601–1623 (2006). https://doi.org/10.1007/s11538-006-9088-6

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  • DOI: https://doi.org/10.1007/s11538-006-9088-6

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