Intrinsic Fluctuations and Driven Response of Insect Swarms

Rui Ni, James G. Puckett, Eric R. Dufresne, and Nicholas T. Ouellette

Phys. Rev. Lett. 115, 118104 – Published 10 September 2015

Abstract

Animals of all sizes form groups, as acting together can convey advantages over acting alone; thus, collective animal behavior has been identified as a promising template for designing engineered systems. However, models and observations have focused predominantly on characterizing the overall group morphology, and often focus on highly ordered groups such as bird flocks. We instead study a disorganized aggregation (an insect mating swarm), and compare its natural fluctuations with the group-level response to an external stimulus. We quantify the swarm’s frequency-dependent linear response and its spectrum of intrinsic fluctuations, and show that the ratio of these two quantities has a simple scaling with frequency. Our results provide a new way of comparing models of collective behavior with experimental data.

Received 17 March 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.118104

Authors & Affiliations

Rui Ni^1,*, James G. Puckett², Eric R. Dufresne^1,3, and Nicholas T. Ouellette^1,†

¹Department of Mechanical Engineering & Materials Science, Yale University, New Haven, Connecticut 06520, USA
²Department of Physics, Gettysburg College, Gettysburg, Pennsylvania 17325, USA
³Departments of Physics, Applied Physics, Chemical and Environmental Engineering, and Cell Biology, Yale University, New Haven, Connecticut 06520, USA

^*Present address: Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.
^†Present address: Department of Civil and Environmental Engineering, Stanford University, Stanford, California 94305, USA. nto@stanford.edu.

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Vol. 115, Iss. 11 — 11 September 2015

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Images

Figure 1
(a) Power spectra of one component of the velocity for an individual midge in a swarm (dashed lines) and the center of mass of the swarm (solid lines). Data are shown for the undriven case (black) and for swarms excited by the sound of a male midge sinusoidally modulated at a frequency of $ω_{d} / 2 π = 1 Hz$ and a maximum intensity of $h_{0} = 75 dB$ (red). (b) Phase-averaged velocities and trajectories of the center of mass [see Eq. (1) for the definition of phase averaging] for swarms driven at $ω_{d} / 2 π = 1 Hz$ with $h_{0} = 0$ [i.e., undriven (blue)], 63 dB (green), 68 dB (red), and 75 dB (black). The length of each arrow shows the instantaneous magnitude of the center-of-mass velocity normalized by the maximum observed value for that data set (green, $19 mm / s$ ; red, $35 mm / s$ ; black, $44 mm / s$ ). The sound source lies along the $y$ axis and points in the positive $y$ direction. (c) Probability density functions (PDFs) of the relative phase of the component of individual midges’ motion at the driving frequency for the same cases as in (b). The driving signal is defined to have a phase of 0.
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Figure 2
(a) Component of the center-of-mass velocity $V_{c . m .}$ projected along the direction toward the speaker for a sound signal with $h_{0} = 75 dB$ (4 arb. units) and a modulation frequency of $ω_{d} / 2 π = 0.5 Hz$ (left axes). The measured sound intensity is shown as a dashed line (right axes). Once the sound is initiated, the variation in $V_{c . m .}$ tracks the modulation of the acoustic driving. (b) Temporal power spectra of $V_{c . m .}$ for a range of driving frequencies. At all frequencies we tested, the peak response of the swarm was at the driving frequency. The solid line shows $ω_{d} = ω$ .
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Figure 3
(a)–(c) The amplitude $U$ (a), magnitude of susceptibility $| χ (ω_{d}) |$ (b), and phase $ϕ$ (c) of the response as a function of $h_{0}$ for a fixed driving frequency of $ω_{d} / 2 π = 1 Hz$ . $h_{0}$ is shown both in dB (top axes) and in arbitrary linear units (bottom axes). (d)–(f) $U$ , $| χ (ω_{d}) |$ , and $ϕ$ as a function of $ω_{d}$ for a fixed $h_{0} = 4$ arb. units (75 dB). Error bars show the standard error computed from measurements of several swarming events.
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Figure 4
(a) The power spectral density $ω \tilde{C} (ω)$ of the velocity fluctuations for undriven swarms (left axes). Data are shown for both the center of mass (circles) and individuals (diamonds). Also shown is $χ^{″} (ω)$ , the imaginary part of the susceptibility (triangles, right axes). $ω \tilde{C} (ω)$ and $χ^{″} (ω)$ disagree, particularly at low frequencies, violating the fluctuation-dissipation theorem. (b) The ratio of $ω \tilde{C} (ω)$ and $χ^{″} (ω)$ as a state variable $Θ$ that characterizes the swarm. $Θ$ decays roughly as $ω_{d}^{- 3 / 2}$ .
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