Flocking and swarming models which seek to explain pattern formation in mathematical biology often assume that organisms interact through a force which is attractive over large distances yet repulsive at short distances. Suppose this force is given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. We detail a phase transition in the mildly repulsive range of exponents, which separates a region where the minimum energy configuration is uniquely attained by a uniform distribution of organisms over a spherical shell, from a region in which it is uniquely attained --- apart from translations and rotations --- by equidistributing the organisms over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three).

We explore the sense in which such configurations are stable fixed points for the dynamics of the aggregation equation, also known as the 2-Wasserstein gradient flow of the associated interaction energy.

Based on work with Cameron Davies (University of Toronto) and Tongseok Lim (of Purdue's Krannert School of Management) [78][79] at http://www.math.toronto.edu/mccann/publications