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Michela Ottobre: Non mean-field Vicsek type models for collective behaviour

Time: Mon 2022-05-09 15.15 - 16.15

Location: Zoom

Video link: Meeting ID: 621 4469 8204

Lecturer: Michela Ottobre (Heriot-Watt)

Abstract

We consider Interacting Particle dynamics with Vicsek type interactions, and their macropscopic PDE limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). It was observed by Motsch and Tadmore that in this non-mean-field regime the influence between agents (i.e. the interaction term) can be scaled either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting at time t (local scaling). We compare the behaviour of the globally scaled and the locally scaled system in many respects; in particular we observe that, while both models exhibit multiple stationary states, such equilibria are unstable (for certain parameter regimes) for the globally scaled model, with the instability leading to travelling wave solutions, while they are always stable for the locally scaled one. This observation is based on a careful numerical study of the model, supported by formal analysis. 
Based on soon to be preprint in collaboration with P. Butta', B. Goddard, T. Hodgson, K.Painter.