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.
For animals of such modest size, swarms of midge insects pose a number of puzzling questions. Unlike bird [1] and sheep [2] flocks, fish schools [3] or human crowds [4], the polarization alignment of midge swarms is always quite small and the associated patterns rather simple [5].
Despite this, natural swarms exhibit strong correlations and scale free behavior: the correlation length is proportional to the size of the swarm [5,6]. However, swarms formed in the laboratory under controlled conditions in the absence of external perturbations are weakly coupled and have a small correlation length. Adding perturbations restores the scale free property [7]. Having scale free behavior, magnitudes related to correlation functions exhibit power laws [6,8].
Power laws seem to be ubiquitous in nature [9,10] but how can be sure they are for real?
Stumpf et al have proposed that a power law has to exhibit a linear behavior in a log-log plot for at least two orders of magnitude, there should be a theory that generates it, and there should be ample and uncontroversial empirical support for it [11]. Equilibrium second order phase transitions (e.g., paramagnetic to ferromagnetic phases at the Curie temperature [12,13]) satisfy all these requirements and Wilson’s renormalization group ideas about universality have been enormously fruitful to explain many phenomena beyond the original discoveries of power laws [14][15][16]. How useful are these ideas to explain midge swarming?
Here we explore midge swarming seeking to explain the known empirical facts that we take as clues. Similar to popular murder cases “Whodunnit?”, we examine different theories (the suspects) that fit some of the facts and discard those that fail to explain them all, thereby pointing out the most likely suspect. Section II presents known features of swarms from experiments in the lab and from field studies. Four possible theories are briefly presented and commented upon in Section III. Section IV discusses the theory that fits most clues and extends it to cover the anisotropic case hence providing new results. Then Section V discusses our findings, gives an overall picture of the subject and suggests possible future research.
Observation of insect swarming occur in nature and in the laboratory under controlled conditions. In nature, male midges form swarms at dawn or dusk over distinctive spots on the ground (wet areas, cow dung, man-made objects, etc) called markers [17,18]. Their purpose is to attract ties under controlled laboratory conditions [19]. Environmental illumination levels provide a behavioral cue to individual insects for when to swarm. Controlled illumination show that swarm formation and dissolution is clearly reflected in the rapid establishment and disappearance of an emergent central potential that binds individuals to the swarm [20]. Swarms consist of a condensed core and a vapor of insects that leave or enter it [21]. Midges acoustically interact when their distances are sufficiently small [5,6] and react collectively to external acoustic signals [22].
The distribution of speeds is peaked about some value and exhibits heavy tails for large swarms (perhaps due to the formation of clusters) [23]. The statistics of accelerations of individual midges in a swarm is consistent with postulating a linear spring force (therefore a harmonic potential) that binds insects together [23]. The trajectories of single midges follow Lévy flights [24]. The polarization order parameter of the swarm, W ∈ [0, 1]:
is very small [5]. Here v j (t) is the velocity of the midge j, j = 1, . . . , N , at time t and W is the time average over the duration of the film taken from the observation of the swarm.
In nature, midge velocities within swarms are strongly correlated and scale free, i.e., the correlation length is proportional to the size of the swarm [5,6,8]. In contrast to field studies, swarms exhibit only short range velocity correlations in the laboratory, which are due to controlled temperature and absence of external environmental effects (breeze, temperature gradients, fluid flow, etc) [25]. When perturbations are added, the correlation length is proportional to swarm size [7].
Field studies by Cavagna and coworkers have shown that the static and dynamic velocity correlation functions have peculiar qualitative features [8]. In critical dynamics about equilibrium second order phase transitions [16], curves of the dynamic correlation function versus time collapse to a single curve as they are written as functions of time scaled with the correlation time (or, equivalently, with correlation length to a power given by the dynamic critical exponent). Field data show that the collapse of the dynamic correlation function with rescaled time occurs only for short times [8]. In addition, the dynamic correlation function versus scaled time is flat at the origin in the sense defined in Ref. [8]. Furthermore, quantities such as the susceptibility, the corr
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