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Kinetic description and macroscopic limit of swarming dynamics with continuous leader–follower transitions

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  • Cristiani, Emiliano
  • Loy, Nadia
  • Menci, Marta
  • Tosin, Andrea

Abstract

In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.

Suggested Citation

  • Cristiani, Emiliano & Loy, Nadia & Menci, Marta & Tosin, Andrea, 2025. "Kinetic description and macroscopic limit of swarming dynamics with continuous leader–follower transitions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 228(C), pages 362-385.
    • Handle: RePEc:eee:matcom:v:228:y:2025:i:c:p:362-385
    DOI: 10.1016/j.matcom.2024.09.006
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    1. Andrea Cavagna & Antonio Culla & Xiao Feng & Irene Giardina & Tomas S. Grigera & Willow Kion-Crosby & Stefania Melillo & Giulia Pisegna & Lorena Postiglione & Pablo Villegas, 2022. "Marginal speed confinement resolves the conflict between correlation and control in collective behaviour," Nature Communications, Nature, vol. 13(1), pages 1-11, December.
    2. Iain D. Couzin & Jens Krause & Nigel R. Franks & Simon A. Levin, 2005. "Effective leadership and decision-making in animal groups on the move," Nature, Nature, vol. 433(7025), pages 513-516, February.
    3. Pareschi, Lorenzo & Toscani, Giuseppe, 2013. "Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods," OUP Catalogue, Oxford University Press, number 9780199655465.
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