Silent Flocks: Constraints on Signal Propagation Across Biological Groups

Experiments find coherent information transfer through biological groups on length and time scales distinctly below those on which asymptotically correct hydrodynamic theories apply. We present here a new continuum theory of collective motion coupling the velocity and density fields of Toner and Tu to the inertial spin field recently introduced to describe information propagation in natural flocks of birds. The long-wavelength limit of the new equations reproduces the Toner-Tu theory, while at shorter wavelengths (or, equivalently, smaller damping), spin fluctuations dominate over density fluctuations, and second-sound propagation of the kind observed in real flocks emerges. We study the dispersion relation of the new theory and find that when the speed of second sound is large, a gap in momentum space sharply separates first- from second-sound modes. This gap implies the existence of silent flocks, namely, of medium-sized systems across which information cannot propagate in a linear and underdamped way, either under the form of orientational fluctuations or under that of density fluctuations, making it hard for the group to achieve coordination.

Figure 1

Left: Small speed ratio, $\epsilon <{\epsilon }_c$. In the overdamped—nonpropagating—zone (blue), we have $\mathrm{Re}\tilde{\omega }=0$, while in the propagating zone (white) $\mathrm{Re}\tilde{\omega }\ne 0$. The dotted lines indicate two paths along which $k$ may be changed. The lower path characterized by $\sin \theta <\epsilon /{\epsilon }_c$ crosses the nonpropagating region, giving rise to a gap (Fig. 2, top, red line). The overdamped region just touches the dashed lines marking $\sin \theta =\epsilon /{\epsilon }_c$. Right: Large speed ratio, $\epsilon >{\epsilon }_c$. The two propagating zones $\mathrm{Re}\tilde{\omega }\ne 0$ are now completely separated by an overdamped ring (blue). Any path in momentum space connecting the region of first-sound propagation (yellow) with that of second-sound propagation (white) must cross the overdamped ring, giving rise to a gap between first and second sound (Fig. 2, bottom).

Figure 2

Top: Small speed ratio, $\epsilon <{\epsilon }_c$. Real part of the frequency vs $|\tilde{k}|$ along two different directions (dotted lines of Fig. 1, left). For $\sin \theta >\epsilon /{\epsilon }_c$ (blue), there is no gap, and a propagating mode exists at all wavelengths. On the other hand, for $\sin \theta <\epsilon /{\epsilon }_c$ (red), an overdamped gap emerges, separating first sound (low $\tilde{k}$) from second sound (large $\tilde{k}$). Bottom: Large speed ratio, $\epsilon >{\epsilon }_c$. Real part of the frequency vs $|\tilde{k}|$ at one given value of $\theta$ (dotted line in Fig. 1, right). A nonpropagating gap between first and second sound is found at any $\theta$. For $\theta =0$, the gap extends down to $k=0$. The lower band edge of second sound occurs at $\tilde{k}\sim 1/2\epsilon$, that is, $k={\eta }_0/2c_2=k_2$, consistent with the second-sound limit (8).