### 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) |
---|---|

Pages (from-to) | 353-389 |

Number of pages | 37 |

Journal | Journal of Mathematical Biology |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2003 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Mathematical Biology*,

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

**Mutual interactions, potentials, and individual distance in a social aggregation.** / Mogilner, A.; Edelstein-Keshet, L.; Bent, L.; Spiros, A.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 47, no. 4, pp. 353-389. https://doi.org/10.1007/s00285-003-0209-7

}

TY - JOUR

T1 - Mutual interactions, potentials, and individual distance in a social aggregation

AU - Mogilner, A.

AU - Edelstein-Keshet, L.

AU - Bent, L.

AU - Spiros, A.

PY - 2003/10

Y1 - 2003/10

N2 - 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 (Rrd+1 gt; c Aad+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.

AB - 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 (Rrd+1 gt; c Aad+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.

KW - Animal groups

KW - Individual distance

KW - Individual-based model

KW - Lyapunov function

KW - Social aggregation

UR - http://www.scopus.com/inward/record.url?scp=1342268227&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1342268227&partnerID=8YFLogxK

U2 - 10.1007/s00285-003-0209-7

DO - 10.1007/s00285-003-0209-7

M3 - Article

C2 - 14523578

AN - SCOPUS:1342268227

VL - 47

SP - 353

EP - 389

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 4

ER -