2008/01/31 by A. J. Leverentz, C. M. Topaz, A. J. Bernoff · 1 citation
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #q-bio.PE #nlin.AO
paper · pdf · doi:10.1137/090749037
23 pages, 10 figures; revised version updates the analysis in sec. 2.1 and 2.2, and contains enhanced discussion of the admissible class of social interaction forces
arxiv created 2008/08/07 · arxiv updated 2015/05/12
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.