# American Institute of Mathematical Sciences

March  2016, 9(1): 131-164. doi: 10.3934/krm.2016.9.131

## A kinetic model for the formation of swarms with nonlinear interactions

 1 INRIA, ANGE Project-Team, Rocquencourt, F-78153 Le Chesnay Cedex, France 2 Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received  May 2015 Revised  July 2015 Published  October 2015

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.
Citation: Martin Parisot, Mirosław Lachowicz. A kinetic model for the formation of swarms with nonlinear interactions. Kinetic & Related Models, 2016, 9 (1) : 131-164. doi: 10.3934/krm.2016.9.131
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