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Intermittent collective motion in sheep results from alternating the role of leader and follower

Abstract

Flocking behaviour is often presented as an example of a self-organized process, where individuals continuously negotiate on the direction of travel and compromise by moving along a local average velocity until the group reaches a consensus. Such a collective behaviour does not take advantage of the benefits of hierarchical organizational strategies that confer the leader of the group full control over it with a reduced information flow overhead. Here we study the spontaneous behaviour of small sheep flocks and find that sheep exhibit a collective behaviour that consists of a series of collective motion episodes interrupted by grazing phases. Each motion episode has a temporal leader that guides the group in line formation. Combining experiments and a data-driven model, we provide evidence that group coordination in these episodes results from the propagation of positional information of the temporal leader to all group members through a strongly hierarchical, directed interaction network. Furthermore, we show that group members alternate the role of leader and follower by a random process, which is independent of the navigation mechanism that regulates collective motion episodes. Our analysis suggests that it is possible to conceive intermittent collective strategies that take advantage of both hierarchical and democratic organizational schemes.

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Fig. 1: Intermittent collective motion and leader selection.
Fig. 2: Inconsistency with previous models.
Fig. 3: Guiding mechanism and interaction network (IN) during a CMP.
Fig. 4: Efficient group guidance and information processing.
Fig. 5: Generalization of the model to larger groups.

Data availability

The raw data in this work are available in the public repository Zenodo (https://doi.org/10.5281/zenodo.6905807). Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

Code availability

The custom codes written in Matlab (Matlab 2019a) to analyse the data presented in this paper are available from the corresponding author upon request.

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Acknowledgements

We thank A. Dusstour, A. Kacelnik and A. Perna for discussions and comments on the text. We also thank Montpellier Supagro Research Station (Domaine du Merle) and M.-H. Pillot for technical support in herd management and data acquisition. L.G.-N. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2002/1 ‘Science of Intelligence’—project 390523135. F.P. acknowledges financial support from CY Initiative of Excellence (grant ‘Investissements d’Avenir’ ANR-16-IDEX-0008), INEX 2021 Ambition project ‘Collective Intelligence’ (CollInt) [2021-008, CYIn-AAP2021-AmbEm-0000000031] and Labex MME-DII, projects 2021-258 and 2021-297.

Author information

Authors and Affiliations

Authors

Contributions

L.G.-N., R.B. and F.P. designed the study, performed statistical analysis of the data and derived the mathematical models used to interpret the data.

Corresponding author

Correspondence to Fernando Peruani.

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The authors declare no competing interests.

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Nature Physics thanks Cristian Huepe and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Experimental setup and characteristic times.

a, Scheme of the experimental setup, consisting of 4 adjacent arenas (area of 80 × 80 meters each one), with a 7-meter-high tower placed in the middle. The tower was used to acquire the videos, that would later be used to get the tracking of the individuals. b, Probability distribution of the duration times tCMP of the Collective Motion Phases (CMPs) observed in the experiments. c, Probability distribution of the duration times tGP of the Grazing Phases (GPs) observed in the experiments.

Extended Data Fig. 2 Quantifying the speed of individuals during CMPs.

a, Probability distribution P(v) of the individual velocities. This distribution allows us to define a threshold velocity vth = 0.45 m/s, which is the local minimum of the fit, depicted with a solid black line. b, Instantaneous velocities vi(t) of all the individuals of the group during one CMP. c, Average individual velocity 〈v〉CMP for all CMPs. The red line is the mean value over all CMPs, and the light red region represents the standard deviation. Figures ac correspond to experiments with groups of N = 4 individuals, but are representative for all group sizes.

Extended Data Fig. 3 Characterization of CMPs.

ac, Probability of finding a neighbour in a given direction as function of the rank of individuals within the line for group sizes N = 2, 3 and 4. di, Quantification of the orientational and spatial order by the average order parameter values 〈P〉 and 〈ψ〉 for all group sizes and for all CMPs. Data are presented as mean values (blue circles) ± SD computed over the whole duration of the corresponding CMP for each group size N = 2, 3 and 4. The red line depicts the mean value of the order parameters -〈P〉 and 〈ψ〉 - across all CMPs.

Extended Data Fig. 4 Quantification of leadership over many CMPs.

Probability distributions of playing the role of leader during a CMP, using m = 4. The blue bars represent the experimental results, and the red dots correspond to the binomial distribution B(n, m) that is associated to the null hypothesis that assumes that all individuals are equally probable to become CMP leader.

Extended Data Fig. 5 Quantification of leadership based on the statistics of the individual that acted as CMP leader most times.

The red histograms correspond to the χ2 distributions obtained from numerical simulations. The blue vertical line corresponds to the experimental measured χ2 value.

Extended Data Fig. 6 Exit-times distributions.

Histograms of the exit times of uninformed individuals obtained with the two models. a, The results for model P and IN3 are shown. b, The results for model V and IN3 are shown. c, Comparison between the average exit times obtained by using model P and three different interaction networks, namely IN1, IN2 and IN3.The results between IN3 and the other networks (IN1 and IN2) are significantly different. We used the Welch’s two-sided unequal variances t-test to quantify this hypothesis. In both comparisons (between IN3 and IN1, and between IN3 and IN2), the resulting p-value resulted to be p < 0.005. This confirms that the results between IN3 and the other two interaction networks - IN1 and IN2 - are significantly different. This is highlighted with the label ’***’. No adjustments were made for multiple comparisons. Data are presented as mean values ± SD.

Extended Data Fig. 7 Quantification of navigation accuracy.

Scheme of a sequence of CMPs of a group of N individuals. The light-grey trajectory is meant to follow the centre of mass of the group xg. The target is set on the x-axis at a distant position from the origin. The positions labelled as xg(tn) are the positions in space of the centre of mass of the group after n CMPs. Let us recall that each CMP possesses a leader.

Extended Data Fig. 8 Effect of the interaction network for multiple target scenario.

The figure shows the superposition of individual trajectories of 300 simulations using interaction network IN1, IN2, and IN3 and model P.

Supplementary information

Supplementary Information

The Supplementary Information contains mainly details on the experimental set-up, data acquisition and fitting of the experimental data using the models explained in the main text (model P, model V and model V + P).

Reporting Summary

Supplementary Video 1

Tracking of a sequence of four CMPs extracted from a trial with a group of size N = 4, where each individual adopts the leading position one time.

Supplementary Video 2

Alternating collective motion phases (CMPs) and grazing phases (GPs) observed in the experiments (raw data) for all group sizes (N = 2, 3 and 4).

Supplementary Video 3

Tracking of CMPs for group size N = 4.

Supplementary Video 4

Numerical simulations of the exploration of the maze shown in Fig. 4c in the main text for a group of size N = 4 using model P.

Supplementary Video 5

Numerical simulations of the exploration of the maze shown in Fig. 4c in the main text for a group of size N = 4 using model V.

Supplementary Video 6

Video taken by Richard Bon (Université Paul Sabatier, CRCA, Toulouse, France) of a large group of moving sheep. The video was taken in the Alps.

Supplementary Video 7

Numerical simulations implemented for the generalized model using a generalized network IN3 and a group of N = 40 individuals. The probability of following a neighbour from the back is p0 = 0.8 (Methods).

Supplementary Video 8

Numerical simulations implemented using our model (model P + IN3) and a group of N = 4 individuals. For aesthetic reasons, we included a more detailed implementation for the transition of the individuals between CMP and GP. For the realization of the video, we used a constant time interval between transitions of the individuals in such a way that the leader (rank K = 1) makes the transition at a given time t1, then the first follower (rank K = 2) makes the transition at time t2 = t1 + dt, the third follower (K = 3) makes the transition at time t3 = t2 + dt and so on. This implementation was used for both transitions: CMP → GP and GP → CMP. For the video, we used dt = 5 time units.

Supplementary Video 9

Numerical simulations implemented using a generalization of the model presented in Barberis et al.5, which is a generic model that considers restricted vision angles of the particles.

Source data

Source Data Fig. 1

Data used for the plots in Figure 1b–e.

Source Data Fig. 2

Data used for the plots in Figure 2c,d.

Source Data Fig. 4

Data used for the plots in Figure 4b,d,e,g.

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Gómez-Nava, L., Bon, R. & Peruani, F. Intermittent collective motion in sheep results from alternating the role of leader and follower. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01769-8

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