Giuseppe Toscani

A kinetic model for flocking

The description of emerging collective behaviors and self–organization in multi-
agent interactions has gained increasing interest from various research commu-
nities in biology, ecology, robotics and control theory, as well as sociology and
economics. In the biological context, the emergent behavior of bird flocks, fish
schools or bacteria aggregations, among others, is a ma jor research topic in
population and behavioral biology and ecology. In this talk [1], we introduce
and analyze a continuous version of the flocking model of Cucker and Smale
[2], which describes the state of a population of birds. Within the same idea
that a bird adjusts its velocity towards the average of neighbors’ velocities, we
construct a spatially dependent Boltzmann-type equation which describes the
behavior of the flock in terms of a density f = f (x, v, t). The large-time behav-
ior of f is subsequently studied by means of mass transportation techniques.
In particular, a continuous analogue of the theorems of [2] is shown to hold for
the solution. These results generalize the approach by Ha and Tadmor [3], who
investigated the large-time behavior of the solution by different techniques.

[1] J.A. Carrillo, M. Fornasier, G. Toscani: A kinetic model for flocking (preprint) (2008)
[2] F. Cucker, S. Smale: IEEE Trans. Automat. Control 52 (2007) 852–862.
[3] Seung-Yeal Ha, E. Tadmor: Kinetic and Related Models 1(3) (2008) 415-435.