Kinetic and Related Models (KRM)

A kinetic model for the formation of swarms with nonlinear interactions

Pages: 131 - 164, Volume 9, Issue 1, March 2016      doi:10.3934/krm.2016.9.131

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Martin Parisot - INRIA, ANGE Project-Team, Rocquencourt, F-78153 Le Chesnay Cedex, France (email)
Mirosław Lachowicz - Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland (email)

Abstract: The present paper deals with the modeling of formation and destruction of swarms using a nonlinear Boltzmann--like equation. We introduce a new model that contains parameters characterizing the attractiveness or repulsiveness of individuals. The model can represent both gregarious and solitarious behaviors. In the latter case we provide a mathematical analysis in the space homogeneous case. Moreover we identify relevant hydrodynamic limits on a formal way. We introduce some preliminary results in the case of gregarious behavior and we indicate open problems for further research. Finally, we provide numerical simulations to illustrate the ability of the model to represent formation or destruction of swarms.

Keywords:  Kinetic formulation, collective behavior, swarms, self-organisation, orientation interactions, hydrodynamic limits, asymptotic preserving schemes, anti--diffusive schemes.
Mathematics Subject Classification:  47G20, 76N10, 78M35, 82C22, 92C15, 92D25.

Received: May 2015;      Revised: July 2015;      Available Online: October 2015.