  We investigate a class of continuum models for the motion of a
two-dimensional biological group under the influence of nonlocal social
interactions. The dynamics may be uniquely decomposed into incompressible
motion and potential motion. When the motion is purely incompressible, the
model possesses solutions which have constant population density and sharp
boundaries for all time. Numerical simulations of these "swarm patches'' reveal
rotating mill-like swarms with circular cores and spiral arms. The sign of the
social interaction term determines the direction of the rotation, and the
interaction length scale affects the degree of spiral formation. When the
motion is purely potential, the social interaction term has the meaning of
repulsion or attraction depending on its sign. For the repulsive case, the
population spreads and the density profile is smoothed. With increasing
interaction length scale, the motion becomes more convective and experiences
slower diffusive smoothing. For the attractive case, the population
self-organizes into regions of high and low density. The characteristic length
scale of the density pattern is predicted and confirmed by numerical
simulations.
