### 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} gt; c Aa^{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.

Original language | English (US) |
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Pages (from-to) | 353-389 |

Number of pages | 37 |

Journal | Journal of Mathematical Biology |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2003 |

### Keywords

- Animal groups
- Individual distance
- Individual-based model
- Lyapunov function
- Social aggregation

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

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## Cite this

*Journal of Mathematical Biology*,

*47*(4), 353-389. https://doi.org/10.1007/s00285-003-0209-7