Asymptotic Dynamics of Attractive-Repulsive Swarms

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 δ-functions.