Asymptotic flocking for the three-zone model

Fei Cao; Sebastien Motsch; Alexander Reamy; Ryan Theisen; Fei Cao; Sebastien Motsch; Alexander Reamy; Ryan Theisen

doi:10.3934/mbe.2020391

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7692-7707. doi: 10.3934/mbe.2020391

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Asymptotic flocking for the three-zone model

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA
2.
Department of Statistics, University of California, Berkeley, 367 Evans Hall, Berkeley, CA 94720-3860, USA

Received: 06 July 2020 Accepted: 15 October 2020 Published: 05 November 2020

We prove the asymptotic flocking behavior of a general model of swarming dynamics. The model describing interacting particles encompasses three types of behavior: repulsion, alignment and attraction. We refer to this dynamics as the three-zone model. Our result expands the analysis of the so-called Cucker-Smale model where only alignment rule is taken into account. Whereas in the Cucker-Smale model, the alignment should be strong enough at long distance to ensure flocking behavior, here we only require that the attraction is described by a confinement potential. The key for the proof is to use that the dynamics is dissipative thanks to the alignment term which plays the role of a friction term. Several numerical examples illustrate the result and we also extend the proof for the kinetic equation associated with the three-zone dynamics.
- agent-based models,
- collective-behavior,
- flocking,
- kinetic equations,
- energy estimates
Citation: Fei Cao, Sebastien Motsch, Alexander Reamy, Ryan Theisen. Asymptotic flocking for the three-zone model[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7692-7707. doi: 10.3934/mbe.2020391

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Abstract

We prove the asymptotic flocking behavior of a general model of swarming dynamics. The model describing interacting particles encompasses three types of behavior: repulsion, alignment and attraction. We refer to this dynamics as the three-zone model. Our result expands the analysis of the so-called Cucker-Smale model where only alignment rule is taken into account. Whereas in the Cucker-Smale model, the alignment should be strong enough at long distance to ensure flocking behavior, here we only require that the attraction is described by a confinement potential. The key for the proof is to use that the dynamics is dissipative thanks to the alignment term which plays the role of a friction term. Several numerical examples illustrate the result and we also extend the proof for the kinetic equation associated with the three-zone dynamics.

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