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Mutual interactions, potentials, and individual distance in a social aggregation

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Abstract.

We formulate a Lagrangian (individual-based) model to investigate the spacing of individuals in a social aggregate (e.g., swarm, flock, school, or herd). Mutual interactions of swarm members have been expressed as the gradient of a potential function in previous theoretical studies. In this specific case, one can construct a Lyapunov function, whose minima correspond to stable stationary states of the system. The range of repulsion (r) and attraction (a) must satisfy r<a for cohesive groups (i.e., short range repulsion and long range attraction). We show quantitatively how repulsion must dominate attraction (Rr d+1>cAa d+1 where R, A are magnitudes, c is a constant of order 1, and d is the space dimension) to avoid collapse of the group to a tight cluster. We also verify the existence of a well-spaced locally stable state, having a characteristic individual distance. When the number of individuals in a group increases, a dichotomy occurs between swarms in which individual distance is preserved versus those in which the physical size of the group is maintained at the expense of greater crowding.

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Authors and Affiliations

  1. Dept. of Mathematics and Center for Genetics and Development, Univ. of California, Davis, CA, 95616, USA

    A. Mogilner

  2. Dept. of Mathematics, Univ. of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    L. Edelstein-Keshet

  3. Dept. of Computer Science and Engineering, Univ. of California, San Diego, CA, 92093, USA

    L. Bent

  4. In Silico Biosci, Portland, OR, USA

    A. Spiros

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  1. A. Mogilner
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  2. L. Edelstein-Keshet
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  3. L. Bent
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  4. A. Spiros
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Correspondence to A. Mogilner.

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Mogilner, A., Edelstein-Keshet, L., Bent, L. et al. Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47, 353–389 (2003). https://doi.org/10.1007/s00285-003-0209-7

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  • DOI: https://doi.org/10.1007/s00285-003-0209-7

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