Whodunnit? The case of midge swarms

TL;DR

This paper addresses the paradoxical observation that midge swarms exhibit small polarization yet strong, scale-free correlations with a correlation length that scales with swarm size in natural settings. It surveys existing swarm theories and introduces an anisotropic harmonically confined Vicsek model (HCVM) to reconcile laboratory and field data. Through numerical simulations, the authors show that HCVM yields static exponents $\nu$ and $\gamma$ and dynamic exponent $z$ that converge toward experimental values, particularly when confinement is anisotropic, producing elongated swarm shapes and improved data collapse behavior for the dynamic correlation function. The work proposes scale-free–chaos transitions as a unifying mechanism and suggests that anisotropic confinement provides a natural link between 2D-like and 3D-like critical behavior, with practical implications for predicting and controlling swarm dynamics in real environments.

Abstract

As collective states of animal groups go, swarms of midge insects pose a number of puzzling questions. Their ordering polarization parameter is quite small and the insects are weakly coupled among themselves but strongly coupled to the swarm. In laboratory studies (free of external perturbations), the correlation length is small, whereas midge swarms exhibit strong correlations, scale free behavior and power laws for correlation length, susceptibility and correlation time in field studies. Data for the dynamic correlation function versus time collapse to a single curve only for small values of time scaled with the correlation time. Is there a theory that explains these disparate observations? Among the existing theories, whodunnit? Here we review and discuss several models proposed in the literature and extend our own one, the harmonically confined Vicsek model, to anisotropic confinement. Numerical simulations of the latter produce elongated swarm shapes and values of the static critical exponents between those of the two dimensional and isotropic three dimensional models. The new values agree better with those measured in natural swarms.

Figures (4)

• Figure 1: Isotropic confinement. (a) Phase diagram of the confinement vs noise plane indicating regions of deterministic and noisy chaos, noisy quasiperiodic (NPQ) attractors, and mostly noise. (b) Regions [I] $(\beta_c(\eta;N),\beta_i(\eta;N))$ (M-cluster or multicluster chaos), [III] $(\beta_0(\eta;N),\beta_c(\eta;N))$ (S-cluster or single-cluster chaos), and line [II] $\beta=\beta_c(\eta;N)$. In (a) and (b), $N=500$. (c) Shrinking of the criticality region as $N$ increases. Adapted from Fig. 1 of Ref. gon24.
• Figure 2: Anisotropic confinement. Phase diagram in the $(\eta,\beta)$ plane. Chaotic and non-chaotic regions are separated by $\beta_0(\eta)$. The black curve is $\beta_0 = c_0 \eta^{m_0}$ with $m_0 = 3.10 \pm 0.13$ and $c_0 = 0.006 \pm 0.001$. Here $N=500$.
• Figure 3: (a) Correlation length and (b) susceptibility as functions of $\beta$ for different values of the noise on the critical curve $\beta_0$. (c) Determination of the dynamic critical exponent by LS fitting of correlation time versus correlation length. We find $\nu=0.35\pm 0.03$, $\gamma=0.86\pm 0.13$, $z=1.06\pm 0.03$ (LS fit). Here $N=500$.
• Figure 4: (a) Critical region between $\beta_0$ and $\beta_c$ for $N=500$. The blue curve is $\beta_c = c_c\eta^{m_c}$, with $m_c= 0.54\pm 0.02$ and $c_c= 0.0081 \pm 0.0003$. (b) LS and RMA fittings of the dynamic critical exponent for $\tau_k$ vs $\xi$ plane. We find $z_{LS}=1.17 \pm 0.15$, $z_{RMA}=1.33\pm0.15$. (c), (d) Partial collapse of the DCF. The best visual collapse is with $z\approx1.1$.