Traveling waves in a continuum model of schooling swimmers

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 "sub-schools" 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.