Collective Behavior: Macroscopic versus Kinetic Descriptions

On the analysis of coherent patterns in swarming models

Stephan Martin

Imperial College London

In my talk I will present some recent advances in the study of interacting particle systems, which are frequently used to model the collective behavior of bird swarms or fish schools. Coherent patterns such as global consensus in orientation (“flocks”) or rotating vortices (“mills”), that are observed in nature, arise from surprisingly simple interaction rules: attraction, repulsion and self-propulsion. Mean-field kinetic and macroscopic equations lead to a compact description of such particular solutions. It is well known that the particle system approaches compactly supported continuous solutions for suitable interaction potentials. I will first discuss the analysis of a particular class of interaction potentials, where, in contrast to the general situation, explicit density profiles and some uniqueness results for the coherent patterns can be established. Secondly, I will talk about the stability of flock solutions in the particle system and show how, in a general setting, nonlinear stability inherits from the first-order discrete aggregation equation to the second-order model considered. Joint work with: José A. Carrillo, Yanghong Huang, Vladislav Panferov