A Primer of Swarm Equilibria

Start: 03/21/2012 - 1:25pm
End  : 03/21/2012 - 2:15pm

Applied Math Seminar

Andrew Bernoff (HMC)


We study equilibrium configurations of swarming biological organisms subject to external and pairwise interaction forces. Beginning with a discrete dynamical model, we derive a variational description of the continuum population density. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model. The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain δ-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.


This work is a collaborative with Chad Topaz (Macalester College) and Harvey Mudd Undergraduates Sheldon Logan (HMC 2006), Wyatt Toolson (HMC 2007), Andrew Leverentz (HMC 2008) and Louis Ryan (HMC 2012).


Andrew J. Bernoff* (ajb@hmc.edu)

Department of Mathematics,

Harvey Mudd College, Claremont, CA 91711


Louis Ryan* (lryan@hmc.edu)

Department of Mathematics,

Harvey Mudd College, Claremont, CA 91711

RN 103

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