Influence of sensorial delay on clustering and swarming

Rafal Piwowarczyk, Martin Selin, Thomas Ihle, and Giovanni Volpe

Phys. Rev. E 100, 012607 – Published 15 July 2019

Abstract

We show that sensorial delay alters the collective motion of self-propelling agents with aligning interactions: In a two-dimensional Vicsek model, short delays enhance the emergence of clusters and swarms, while long or negative delays prevent their formation. In order to quantify this phenomenon, we introduce a global clustering parameter based on the Voronoi tessellation, which permits us to efficiently measure the formation of clusters. Thanks to its simplicity, sensorial delay might already play a role in the organization of living organisms and can provide a powerful tool to engineer and dynamically tune the behavior of large ensembles of autonomous robots.

Received 28 March 2018
Revised 4 May 2019

DOI:https://doi.org/10.1103/PhysRevE.100.012607

Physics Subject Headings (PhySH)

Collective behavior Swarming

Nonlinear time-delay systems

Vicsek model

Statistical Physics & Thermodynamics

Authors & Affiliations

Rafal Piwowarczyk^1,2, Martin Selin^1,2, Thomas Ihle³, and Giovanni Volpe^1,*

¹Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden
²Department of Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden
³Institute of Physics, University of Greifswald, DE-17489 Greifswald, Germany

^*giovanni.volpe@physics.gu.se

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Vol. 100, Iss. 1 — July 2019

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Images

Figure 1
The presence of sensorial delay $d$ influences the clustering and swarming of $N = 400$ self-propelling agents (circles, speed $ν = 3$ , detection radius $R = 20)$ in a 2D Vicsek model (arena size $2000 \times 2000$ with periodic boundary conditions): (a) $d = 0$ (standard Vicsek model); (b) $d = 1$ enhances the formation of swarms; (c) $d = 25$ prevents the emergence of swarms. The snapshots are taken once the system has reached the steady state (timestep 8000). In the top row only the agents are shown, while in the bottom row also the borders of the corresponding Voronoi tessellation are shown. The color of the agents denotes the area of the corresponding Voronoi cell going from black (smallest area) to white (largest area). See also Supplemental Material video 1 [25].
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Figure 2
Global alignment and clustering parameters. (a) The evolution of the global alignment parameter $ψ$ as a function of time does not show a clear difference between delays $d = 0$ , 1, and 25, and (b) the average steady-state (i.e., measured after 5000 timesteps) $ψ$ does not show statistically significant differences as a function of $d$ (the shaded area represents a standard deviation over ten independent runs). (c) The evolution of the global clustering parameter $c$ shows marked differences between delays $d = 0$ , 1, and 25, and (d) the average steady-state (i.e., measured after 5000 timesteps) $c$ clearly depends on $d$ (the shaded area represents a standard deviation over ten independent runs). In all cases, we consider $N = 400$ agents with $ν = 3$ and $R = 20$ in a square $2000 \times 2000$ arena with periodic boundary conditions.
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Figure 3
Dependence of clustering and swarming on the agents' speed $ν$ . (a) The dependence of the global clustering parameter $c$ on the delay $d$ varies as a function of $ν$ (the symbols are the averages in the steady state, and the shaded areas are the standard deviations over ten independent runs). The disruptive effect on the formation of the swarming clusters is more pronounced for large speeds (stars, $ν = 7$ , see also Supplemental Material video 2 [25]) than for small speeds (crosses, $ν = 1$ , see also Supplemental Material video 3 [25]). The dots correspond to the model explored in Figs. 1 and 2. (b) Corresponding plot of the values of $c$ as a function of the dimensionless delay $d ν / R$ . In all cases, we consider $N = 400$ agents with $R = 20$ in a square $2000 \times 2000$ arena with periodic boundary conditions.
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Figure 4
Negative and positive delays. (a) Global alignment parameter $ψ$ and (b) global clustering parameter $c$ (the blue symbols are the steady-state averages and the shaded areas are the standard deviations over ten independent runs) as a function of delay $d$ . The black dots in (a) and (b) represent the same data as in Figs. 2 and 2, respectively, provided for comparison. In all cases, we consider a Vicsek model with $N = 400$ agents $(ν = 3, R = 20)$ in a square arena $(2000 \times 2000)$ with periodic boundary conditions. See also Supplemental Material video 4 [25].
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