Modeling swarms: from micro to macro
Abstract
A general class of mathematical structures (integro--differential equations)that can model self--organization at the so--called mesoscopic level is proposed.
The equations are of kinetic type and the interactions have nonlinear nature and
may be referred to as the mesoscopic scale of description.
The structures lead to interesting mathematical problems of blow--up of solutions
{[2], [3]) that are directly related to swarming behavior. Both
microscopic and macroscopic levels are also studied ([4]).
References
begin{thebibliography}{0}
bibitem{} J. Banasiak, M. Lachowicz, {Methods of small parameter in
mathematical biology}, Birkh{"a}user, Boston 2014.
bibitem{} M. Lachowicz, H. Leszczy{'n}ski, M. Parisot,
textit{A simple kinetic equation of swarm formation: blow--up and global existence}, Appl. Math. Letters,
textbf{57}, 2016, 104--107.
bibitem{} M. Lachowicz, H. Leszczy{'n}ski, M. Parisot, textit{Blow--up and global existence
for a kinetic equation of swarm formation}, Math. Models Methods Appl. Sci., to appear.
bibitem{} M. Parisot, M. Lachowicz, textit{A kinetic model for the formation of swarms
with nonlinear interactions}, Kinetic Related Models, textbf{9} 1, 131--164, 2016.
end{thebibliography}
