A non-local model for a swarm

This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to in uence the velocity of the organisms. The model consists of integro-di erential advection-di usion equations, with convolution terms that describe long range attraction and repulsion. We nd that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm pro le is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent di usion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the di usion term, we nd that true, locally stable traveling band solutions occur. We further explore the e ects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and nd that such models cannot account for cohesive, nite swarms with realistic density pro les.