Asymptotic flocking for the three-zone model

Fei Cao, Sebastien Motsch, Alexander Reamy, Ryan Theisen

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)7692-7707
Number of pages16
JournalMathematical Biosciences and Engineering
Issue number6
StatePublished - Nov 5 2020


  • Agent-based models
  • Collective-behavior
  • Energy estimates
  • Flocking
  • Kinetic equations

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Agricultural and Biological Sciences
  • Computational Mathematics
  • Applied Mathematics


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