Traveling waves in a continuum model of schooling swimmers

• 成果类型: Article
• 署名作者: Oza, Anand U.; Kanso, Eva; Shelley, Michael J.
• 署名单位: New Jersey Institute of Technology; University of Southern California; University of Southern California; Simons Foundation; Flatiron Institute
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.2521850123
• 发表日期: 2026-06-02
• 页码: e2521850123
• 关键词: Schooling; collective behavior; active matter; coarse-grained dynamics; Traveling waves; collective motion; fish; DYNAMICS; BEHAVIOR

摘要

The complex formations exhibited by schooling fish have long been the object of fascination for biologists and physicists. However, the physical and sensory mechanisms leading to organized collective behavior remain elusive. On the physical side in particular, it is unknown how the flows generated by individual fish influence the collective patterns that emerge in large schools. To address this question, we here present a continuum theory for a school of swimmers in an inline formation. The swimmers are modeled as flapping wings that interact through temporally nonlocal hydrodynamic forces, as arise when one swimmer moves through the lingering vortex wakes shed by the others, leading to a system of time-delay-differential equations. Through coarse-graining, we derive a system of partial differential equations for the evolution of swimmer density and collective vorticity-induced hydrodynamic force. Linear stability analysis of the governing equations shows that there is a range of swimmer densities for which the uniform (constant-density) state is unstable to perturbations. Numerical simulations in periodic domains reveal families of stable traveling wave solutions, where a uniform school destabilizes into a collection of densely populated subschools separated by relatively sparse regions that move as a propagating wave. We find that distinct propagating waves may be stable for the same set of kinematic parameters. We also find that finite schools can evolve into packets of coarsening traveling waves whose overall spreading is described by a rarefaction fan moving upstream and a terminating downstream shock. Generally, our results show that temporally nonlocal hydrodynamic interactions can lead to rich collective behavior in schools of swimmers.