Mean-field theory of collective motion due to velocity alignment
Authored by Lutz Schimansky-Geier, Pawel Romanczuk
Date Published: 2012
DOI: 10.1016/j.ecocom.2011.07.008
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Platforms:
XPPAUT
Model Documentation:
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Mathematical description
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Abstract
Establishing a direct link between individual based models and the
corresponding population description is a common challenge in
theoretical ecology. Swarming is a prominent example, where collective
effects arising from interactions of individuals are essential for the
understanding of large-scale spatial population dynamics, and where both
levels of modeling have been often employed without establishing this
connection.
Here, we consider a system of self-propelled agents. with velocity
alignment in 20 and derive a mean-field theory from the microscopic
dynamics via a nonlinear Fokker-Planck equation and a moment expansion
of the probability density. We analyze the stationary solutions
corresponding to macroscopic collective motion (ordered state) and the
disordered solution with no collective motion in the spatially
homogeneous system. In particular, we discuss the impact of different
propulsion functions governing individual dynamics. Our results predict
a strong impact of individual dynamics on the mean field onset of
collective motion (continuous vs discontinuous). In addition to the
macroscopic density and velocity fields, we consider the effective
``temperature{''} field, measuring velocity fluctuations around the mean
velocity. We show that the temperature decreases strongly with
increasing level of collective motion despite constant fluctuations on
individual level, which suggests that extreme caution should be taken in
deducing individual behavior, such as, state-dependent individual
fluctuations from mean-field measurements (Yates et al., 2009). (C) 2011
Elsevier B.V. All rights reserved.
Tags
Dynamics
noise
Model
hydrodynamics
Self-propelled particles
Flocks
Active brownian particles
Fokker-planck
equation
Langevin equation