We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodiﬀerential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for ﬂuid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale aﬀects the degree of spiral formation. For the purely potential case, the model resembles a nonlocal (forwards or backwards) porous media equation. The sign of the social interaction term controls whether the population aggregates or disperses, and the interaction length scale controls the balance between transport and smoothing of the density proﬁle. For the aggregative case, the population clumps into regions of high and low density. The characteristic length scale of the density pattern is predicted and conﬁrmed by numerical simulations.
Available at: http://works.bepress.com/chad_topaz/7/