  We classify and predict the asymptotic dynamics of a class of swarming
models. The model consists of a conservation equation in one dimension
describing the movement of a population density field. The velocity is found by
convolving the density with a kernel describing attractive-repulsive social
interactions. The kernel's first moment and its limiting behavior at the origin
determine whether the population asymptotically spreads, contracts, or reaches
steady-state. For the spreading case, the dynamics approach those of the porous
medium equation. The widening, compactly-supported population has edges that
behave like traveling waves whose speed, density and slope we calculate. For
the contracting case, the dynamics of the cumulative density approach those of
Burgers' equation. We derive an analytical upper bound for the finite blow-up
time after which the solution forms one or more $\delta$-functions.
