We study equilibrium conﬁgurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding 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 in nature are observed 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.
© 2011 Society for Industrial and Applied Mathematics
A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria," SIAM J. Appl. Dyn. Sys. 1 (2011) 212-250. doi: 10.1137/100804504