Emergent dynamics of laboratory insect swarms

Abstract

Collective animal behaviour occurs at nearly every biological size scale, from single-celled organisms to the largest animals on earth. It has long been known that models with simple interaction rules can reproduce qualitative features of this complex behaviour. But determining whether these models accurately capture the biology requires data from real animals, which has historically been difficult to obtain. Here, we report three-dimensional, time-resolved measurements of the positions, velocities, and accelerations of individual insects in laboratory swarms of the midge Chironomus riparius. Even though the swarms do not show an overall polarisation, we find statistical evidence for local clusters of correlated motion. We also show that the swarms display an effective large-scale potential that keeps individuals bound together, and we characterize the shape of this potential. Our results provide quantitative data against which the emergent characteristics of animal aggregation models can be benchmarked.

Figure 1. Snapshot of a swarm and experimental arrangement.

(a–c) Snapshots from each of the three synchronized cameras (left, center, and right, respectively) focused on a common volume near the center of the swarm. Regions identified as midges are coloured red. (d) The experimental arrangement, seen from above and drawn to scale. Swarming midges remain far from container boundaries. (e) The corresponding three-dimensional snapshot. An arrow indicates the location of each tracked midge; the arrow lengths are proportional to speed and their orientations indicate flight direction. (f) The same snapshot, with each individual's current position indicated by a dot and past flight path indicated by a curve.

Figure 2. Swarm shape statistics.

(a) Distribution of the distances r of each individual from the swarm centre, normalised by the swarm size R_s. Each curve shows data for one swarm, and these volumetric probabilities P satisfy . (b) Swarm size as a function of mean swarm population. Each data point is computed as the average over the entire time of observation, and the ellipses show the standard deviation. Note that the number of individuals in each swarm is not fixed, since midges may enter or leave the swarm during the measurement period. Marker colours correspond to curve colours in (a). The dashed curve is a R_s ∝ 〈N〉^1/3 fit, as would be expected if the number density were independent of the swarm size. For the largest swarms, some of the midges flew outside the region imaged by the cameras; in these cases, the markers are outlined in grey. (c) Swarm aspect ratio as a function of swarm size. (d) Bulk swarm orientation. One axis of each swarm nearly aligns with gravity; for large swarms, it is the axis along which the swarm is longest, e₁. (e) Spatial variation of swarm density. Slices through the three-dimensional probability density function of midge position are shown in colour on a logarithmic scale for the swarm marked with a black arrow in (b).

Figure 3. Statistics of individual midge velocities.

(a) Standard deviations of swarm velocity components; note that the mean velocity in all directions is nearly zero. Typical horizontal velocities, as measured by these standard deviations, exceed vertical velocities, perhaps improving flight efficiency. (b) Polarisation p is near zero for all swarms observed, distinguishing swarming behaviour from flocking and schooling. (c, d) Standardised velocity distributions along (c) the vertical direction and (d) one horizontal direction. The distributions are nearly Gaussian (a reference Gaussian curve is shown in grey), with slight deviations in the tails. (e) Standardised speed distributions, with the standardised Maxwell–Boltzmann distribution shown in grey for comparison. A heavy, nearly exponential tail develops for large swarms, which may indicate the formation of clusters.

Figure 4. Distance to nearest neighbours.

(a) Root-mean-square nearest-neighbour distance for midge swarms and randomly distributed particles, shown as circles and squares, respectively. The data follow the same trend for each data set, but is always larger for real swarms. (b) Standardised PDFs of d_nn for measured swarms (upper) and randomly positioned particles (lower); note that the lower curves have been vertically offset for clarity. Each curve shows data for one swarm. The most probable d_nn is smaller for the swarms than for the random particles, (see also (c), which shows the same plots on linear axes), but the swarms also show a much longer tail, indicating larger voids.

Figure 5. Statistics of individual midge accelerations.

(a) Standard deviations of acceleration components as a function of swarm size. Horizontal and vertical accelerations are nearly identical in magnitude for all the observed swarms. (b,c) Standardised acceleration PDFs, for (b) the vertical and (c) horizontal directions. (d) Standard deviations of acceleration parallel and perpendicular to the instantaneous direction of flight. The two components are again nearly identical. (e,f) Standardised PDFs of acceleration parallel (e) and perpendicular (f) to the direction of flight.

Figure 6. Mean acceleration as a function of position.

(a,b) Mean acceleration conditioned on position in (a) the vertical direction, and (b) one horizontal direction. The conditional acceleration varies linearly with a negative slope as a function of position. (c,d) Effective “spring constants” extracted from linear fits to the data in (a) and (b). The effective elastic potentials are stiffer in both directions for smaller swarms. (e) Mean-squared displacement as a function of time averaged over all midges. The dashed line indicates a t² power law, and the data (solid) follow that trend, implying ballistic flight. This power law breaks down only at the swarm edges.