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Kinetic and Related Models (KRM)
 

Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type

Pages: 1 - 16, Volume 4, Issue 1, March 2011      doi:10.3934/krm.2011.4.1

 
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Martial Agueh - Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada (email)
Reinhard Illner - Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada (email)
Ashlin Richardson - Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada (email)

Abstract: The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed, b) a repulsive short range force to avoid collisions, c) an attractive "flocking" force computed from the birds seen by each bird inside its vision cone, and d) a "boundary" force which will entice birds to search for and return to the flock if they find themselves at some distance from the flock. We introduce these forces in detail, discuss the required cutoffs and their implications and show that there are natural bounds in velocity space. Well-posedness of the initial value problem is discussed in spaces of measure-valued functions. We conclude with a series of numerical simulations.

Keywords:  Flocking, swarming, particle model, kinetic equation.
Mathematics Subject Classification:  92D50, 97M10, 82C22, 82C70.

Received: August 2010;      Revised: November 2010;      Available Online: January 2011.

 References