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Asymptotic dynamics of attractive-repulsive swarms
SIAM Journal on Applied Dynamical Systems (2009)
  • Andrew Leverentz, Harvey Mudd College
  • Chad M. Topaz, Macalester College
  • Andrew J. Bernoff, Harvey Mudd College
Abstract

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.

Disciplines
Publication Date
2009
Citation Information
Andrew Leverentz, Chad M. Topaz and Andrew J. Bernoff. "Asymptotic dynamics of attractive-repulsive swarms" SIAM Journal on Applied Dynamical Systems Vol. 8 Iss. 3 (2009)
Available at: http://works.bepress.com/chad_topaz/2/