Collective behaviour without collective order in wild swarms of midges

Abstract

Collective behaviour is a widespread phenomenon in biology, cutting through a huge span of scales, from cell colonies up to bird flocks and fish schools. The most prominent trait of collective behaviour is the emergence of global order: individuals synchronize their states, giving the stunning impression that the group behaves as one. In many biological systems, though, it is unclear whether global order is present. A paradigmatic case is that of insect swarms, whose erratic movements seem to suggest that group formation is a mere epiphenomenon of the independent interaction of each individual with an external landmark. In these cases, whether or not the group behaves truly collectively is debated. Here, we experimentally study swarms of midges in the field and measure how much the change of direction of one midge affects that of other individuals. We discover that, despite the lack of collective order, swarms display very strong correlations, totally incompatible with models of non-interacting particles. We find that correlation increases sharply with the swarm's density, indicating that the interaction between midges is based on a metric perception mechanism. By means of numerical simulations we demonstrate that such growing correlation is typical of a system close to an ordering transition. Our findings suggest that correlation, rather than order, is the true hallmark of collective behaviour in biological systems.

Figure 1. Experiment.

a: A natural swarm of midges (Cladotanytarsus atridorsum, Diptera:Chironomidae), in Villa Ada, Rome. The digital image of each midges is, on average, a pixels light object against a dark background. b: The trajectories reconstructed for the same swarm as in a. Individual trajectories are visualized for a short time (roughly frames sec), to avoid visual overcrowding (see also Video S1 and S2). c: A microscope image of an adult male of Cladotanytarsus atridorsum. d: A detailed view of the hypopygium, used for species identification (see Methods); the same midge as in c. e: Scheme of the experimental set-up. Three synchronized cameras recording at frames per second are used. Two cameras m apart are used as the stereoscopic pair for the three dimensional reconstruction. The third one is used to reduce tracking ambiguities and resolve optical occlusions. Three dimensional trajectories are reconstructed in the reference frame of the right stereoscopic camera. f: The mutual geometric positions and orientations of the cameras are retrieved by taking several pictures of a known target. The accuracy we achieve in the determination of the mutual camera orientation is of the order of radians.

Figure 2. Natural swarms lack global order.

Order parameters in a typical natural swarm. In all panels the grey band around zero is the expected amplitude of the fluctuations in a completely uncorrelated system. In the left panels we report the time series of the order parameters, in the right panels their probability distributions. Top: The alignment order parameter, known as polarization, In red we report the reference value of the polarization in a flock of starlings. Middle: Rotational order parameter, Bottom: Dilatational order parameter,

Figure 3. Swarms correlation.

Black lines and symbols refer to natural swarms, red lines to simulations of ‘swarms’ of non-interacting particles (NHS). Each column refers to a different midge species. Top: Correlation function as a function of the distance at one instant of time. The dashed vertical line marks the average nearest neighbour distance, for that swarm. The correlation length, is the first zero of the correlation function. Red: correlation function in the NHS case. The value of for the NHS has been rescaled to appear on the same scale as natural distances. Each natural swarm is compared to a NHS with the same number of particles. Middle: Cumulative correlation, This function reaches a maximum The value of the integrated correlation at its maximum, is the susceptibility Bottom: Numerical values of the susceptibility in all analysed swarms. For each swarm the value of is a time average over the whole acquisition; error bars are standard deviations. Red: the average susceptibility in the non-interacting case.

Figure 4. Swarms susceptibility. Left

: Susceptibility as a function of the rescaled nearest neighbour distance, where is the body length. Each point represents a single time frame of a swarming event, and all events are reported on the same plot (symbols are the same for all species). The solid line is the best fit to equation (4). Right: Logarithm of the average susceptibility as a function of Dasyhelea flavifrons - blue squares; Corynoneura scutellata - green circles; Cladotanytarsus atridorsum - red triangles. The solid line represents the best fit to equation (4). Each data point represents the time average over the entire acquisition of one swarming event. Error bars indicate standard deviations.

Figure 5. Vicsek model.

Three-dimensional Vicsek model in a central potential. Left: Correlation function in the disordered phase, but close to the ordering transition. The dashed line is the nearest neighbour distance. Inset: polarization as a function of time. For this value of the system is disordered. Right: Logarithm of the susceptibility as a function of the rescaled nearest neighbour distance, where is the metric interaction range. The solid line represents the best fit to equation (4). Error bars are smaller than symbols' size.

Figure 6. Percolation threshold.

Percolation threshold as a function of the nearest-neighbour distance in natural swarms. The linear fit (black line) gives, Inset: Fraction of midges belonging to the largest cluster as a function of the clustering threshold In correspondence of the percolation threshold there is the formation of a giant cluster. We define as the point where (red dashed line). Because of the sharp nature of the percolation transition, the value of does not depend greatly on the threshold used.