Propagating speed waves in flocks: A mathematical model

Andrea Cavagna, Daniele Conti, Irene Giardina, and Tomas S. Grigera

Phys. Rev. E 98, 052404 – Published 8 November 2018

Abstract

An efficient collective response to external perturbations is one of the most striking abilities of a biological system. One of the crucial aspect of this phenomenon is given by the information transfer, and resulting propagation of signals, within the group. In this respect the existence of density waves that propagate linearly on a flock of birds is well known. However, most aspects of this phenomenon are still not fully captured by theoretical models. In this work we present a model for the propagation of speed fluctuations inside a flock, which is able to reproduce the observed density waves. We study the full solution of the model in $d = 1$ , and we find a line in the parameter space along which the system relaxes as fast as possible without oscillating, resembling a generalized critical damping condition. By analyzing the parameters plane we show that this “critical damping” line indeed represents an attractor for a steepest descent dynamics of the return time of the system. Finally we propose a method to test the validity of the model through future experiments.

Received 17 November 2017
Revised 18 June 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052404

Physics Subject Headings (PhySH)

Animal behavior, social interactions & emergent phenomena Emergence of patterns Flocking

Birds

Collective dynamics Green's function methods Langevin equation Linear response theory Vicsek model

Statistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Andrea Cavagna¹, Daniele Conti^1,2,3,*, Irene Giardina^1,2,4, and Tomas S. Grigera^5,6

¹Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185 Rome, Italy
²Dipartimento di Fisica, Università Sapienza, 00185 Rome, Italy
³Laboratoire de physique statistique, CNRS, UPMC and École normale supérieure, 75005 Paris, France
⁴INFN, Unità di Roma 1, 00185 Rome, Italy
⁵Instituto de Física de Líquidos y Sistemas Biológicos CONICET, Universidad Nacional de La Plata, La Plata, Argentina
⁶CCT CONICET La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata, Argentina

^*Corresponding author: daniele.conti@uniroma1.it, dconti@lps.ens.fr

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Issue

Vol. 98, Iss. 5 — November 2018

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Images

Figure 1
Sketch of a chain of oscillator. Each oscillator, besides being connected to its first neighbors with strength $J$ (red spring), is also tied to a base with strength $g$ (black spring), which forces it to have a determined position.
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Figure 2
Sketch of the dispersion law. (a) It is drawn the real part of the frequency $ω$ for different cases. If $ɛ^{2} < 0$ (blue line), we are in the oscillating phase and there is propagation for every value of $k$ ; if $ɛ^{2} > 0$ (orange line), the system is nonoscillating and there is propagation only for $k > k_{0} = \frac{ɛ}{c}$ ; if $ɛ^{2} = 0$ (green line), we are at the critical damping and there is always linear propagation; for Langevin dynamics (fuchsia line) the real part of the frequency is zero. (b) The imaginary part of the frequency $ω$ for different cases. The green line represents the oscillating and critically damped situations in which $Im (ω)$ is constant; the orange line represents the nonoscillating regime in which the $Im (ω$ ) is constant only for $k > k_{0}$ and grows quadratically with $k$ for small values of $k$ .
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Figure 3
Different solutions of the speed equation in $d = 1$ . Depending on the value of $ɛ^{2}$ the solution will go to zero differently: at $ɛ^{2} = 0$ (green line) it reaches zero in the fastest way without oscillating; in the nonoscillating regime (orange line), $ɛ^{2} > 0$ the solution goes to zero more slowly, while in the oscillating case (blue line) the field displays oscillations before going to the original value.
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Figure 4
(a) Level lines of $τ (γ, ω_{0})$ are drawn with colors from blue for lower values to green for the higher ones. The vector field is given by the negative gradient of the function; here as well the color is the intensity of the field, ranging from yellow for lower values to dark red for the highest ones. The violet line represents the critical line. (b) Zoom of the gradient flow near the saddle point dividing the transparent zone from the intermediate zone [the region inside the black rectangle of panel (a)]. The green line represents the separatrix, which divides the steepest descent dynamics.
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Figure 5
(a) $C_{k} (t)$ in the nonoscillating regime $(ɛ^{2} > 0)$ of the inertial dynamics for $k < k_{0}$ and for every value of $k$ in the Langevin dynamics. The colors represent different values of $k$ , ranging from dark red for small values of $k$ , to yellow for high $k$ . (b) $C_{k} (t)$ in the inertial case for every value of $k$ if $ɛ^{2} \geq 0$ and only for $k > k_{0}$ if $ɛ^{2} > 0$ . The colors represent different values of $k$ , ranging from dark blue for small values of $k$ to aquamarine for high $k$ . (c) To better quantify the difference between inertial and noninertial system we define the function $h (x)$ (see Ref. [71]), which has a different form depending on whether the function is exponential-like or has a vanishing first derivative.
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