# A primer of swarm equilibria - Nonlinear Sciences > Pattern Formation and Solitons

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Abstract: We study equilibrium configurations of swarming biological organisms subjectto exogenous and pairwise endogenous forces. Beginning with a discretedynamical model, we derive a variational description of the correspondingcontinuum population density. Equilibrium solutions are extrema of an energyfunctional, and satisfy a Fredholm integral equation. We find conditions forthe extrema to be local minimizers, global minimizers, and minimizers withrespect to infinitesimal Lagrangian displacements of mass. In one spatialdimension, for a variety of exogenous forces, endogenous forces, and domainconfigurations, we find exact analytical expressions for the equilibria. Theseagree closely with numerical simulations of the underlying discrete model.Theexact solutions provide a sampling of the wide variety of equilibriumconfigurations possible within our general swarm modeling framework. Theequilibria typically are compactly supported and may contain$\delta$-concentrations or jump discontinuities at the edge of the support. Weapply our methods to a model of locust swarms, which are observed in nature toconsist of a concentrated population on the ground separated from an airbornegroup. Our model can reproduce this configuration; quasi-two-dimensionality ofthe model plays a critical role.

Autor: Andrew J. Bernoff, Chad M. Topaz

Fuente: https://arxiv.org/