Finite-Size Scaling as a Way to Probe Near-Criticality in Natural Swarms

Collective behavior in biological systems is often accompanied by strong correlations. The question has therefore arisen of whether correlation is amplified by the vicinity to some critical point in the parameters space. Biological systems, though, are typically quite far from the thermodynamic limit, so that the value of the control parameter at which correlation and susceptibility peak depend on size. Hence, a system would need to readjust its control parameter according to its size in order to be maximally correlated. This readjustment, though, has never been observed experimentally. By gathering three-dimensional data on swarms of midges in the field we find that swarms tune their control parameter and size so as to maintain a scaling behavior of the correlation function. As a consequence, correlation length and susceptibility scale with the system’s size and swarms exhibit a near-maximal degree of correlation at all sizes.

Figure 1

(a) 3D trajectories for swarm ${20120907}_{-}\mathrm{A}1$, $N=169$. (b) Velocity correlation function. The correlation length, $\xi \sim r_0$, is much larger than the nearest-neighbor distance. The correlation is averaged over the whole time acquisition. (c) Alignment event between two midges (real trajectories).

Figure 2

(a) Vicsek model in $3d$. Susceptibility $\chi$ as a function of the rescaled nearest-neighbor distance, $x=r_1/\lambda$ for different swarm sizes, $N$. The maximum of $\chi$ occurs at the finite-size critical point, $x_{\max }(N)$, marked by the black line. Inset: rescaled susceptibility $\chi N^{-\gamma /3\nu }$ vs scaling variable $y=(x-x_c)N^{1/3\nu }$. (b) Susceptibility as a function of $N$ at fixed $x$. (c) Correlation length as a function of the linear system size, $L$, at fixed $x$. By increasing $N$ (and $L$) at fixed value of $x$ we are moving along the red path in panel (a), so that we end up being further away from the position of the maximum of $\chi$. Simulations have been performed using the Vicsek update rule in $3d$: ${\overrightarrow{v}}_i(t+1)=v_0{\mathcal{R}}_{\eta }(\sum _{r_{ij}<\lambda }{\overrightarrow{v}}_j(t))/|\sum _{r_{ij}<\lambda }{\overrightarrow{v}}_j(t))|$; ${\overrightarrow{r}}_i(t+1)={\overrightarrow{r}}_i(t)+{\overrightarrow{v}}_i(t)$, where ${\mathcal{R}}_{\eta }$ is a random uniform rotation in $[-2\pi \eta ,2\pi \eta ]$. $v_0=0.05$, $\lambda =1$, $\eta =0.45$ (see the Supplemental Material [12]).

Figure 3

Top: natural swarms data. Bottom: $3d$ Vicsek model in the critical region. (a) Susceptibility as a function of the number of midges, $N$ ($P$ value $=3.0\times 10^{-6}$). (b) Correlation length, $\xi$, as a function of the linear system size, $L$ ($P$ value $=1.0\times 10^{-7}$). Both susceptibility and correlation length show no saturation for large systems. (c) Control parameter $x$ as a function of $N$ ($P$ value $=1.4\times 10^{-3}$). (d) Susceptibility as a function of the control parameter, $x$ ($P$ value $=6.9\times 10^{-5}$). Each point corresponds to a different swarm averaged over time (error bars are standard deviations). Lower panels (e),(f),(g),(h): same quantities as in the upper panels, but calculated for the Vicsek model in the critical region, defined by a fixed value of the scaling variable $y$. This means that, unlike in Figs. 2 and 2, in panels 3 and 3 we are changing both $N$ and $x$ according to Eq. (3) [blue path in Fig. 2]. Lines are fit to Eqs. (3)–(6). For $3d$ Vicsek we obtain: $\nu =0.75\pm 0.02$, $\gamma =1.6\pm 0.1$, and $x_c=0.421\pm 0.002$, not far from the $3d$ Heisenberg exponents [37]. For natural swarms we obtain, $\nu =0.35\pm 0.1$, $\gamma =0.9\pm 0.2$, and $x_c=12.5\pm 1.0$. In natural swarms $L$ and $\xi$ are expressed in meters, while both $\chi$ and $x$ are dimensionless.