Mutual interactions, potentials, and individual distance in a social aggregation.
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.
Mogilner
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Dept. of Mathematics and Center for Genetics and Development, Univ. of California, Davis, CA 95616, USA. mogilner@math.ucdavis.edu
Edelstein-Keshet
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Bent
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Spiros
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D000465:Algorithms; D000818:Animals; D001522:Behavior, Animal; D003198:Computer Simulation; D008124:Locomotion; D008960:Models, Psychological; D009043:Motor Activity; D011156:Population Density; D011157:Population Dynamics; D012919:Social Behavior; D013037:Spatial Behavior