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Richard Mann: I thought it was a great selection.


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These outperform the traditional models in predicting when individual prawns will change their direction of motion and restore consistency between the fine-scale rules of interaction and the global behaviour of the group.

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Generally such rules are explicitly kept as simple as possible while remaining realistic, with the aim of explaining as much as possible of collective motion from the simplest constituent parts. In conjunction with the editors we therefore decided that the paper must be retracted. The closest that any study so far has come to finding consistency between scales has been Lukeman et al. We conclude that prawns' movements are influenced by not just the current direction of nearby conspecifics, but also those encountered in the recent past.

Red coloured prawns indicate a clockwise orientation, blue prawns a counter-clockwise orientation. Verification at multiple scales is the necessary next step now that inference based on fine-scale data is becoming the norm.

For most flocking animals, the rules dictating the interactions between individuals, which ultimately generate the behaviour of the whole group, are still not known in any detail.

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We complement this approach by using Bayesian inference to perform model selection based on empirical data at a detailed individual level. The authors are retracting this paper. In these models, animals are treated as moving self-propelled particles that interact with each other according to simple rules. As a consequence of this, our and conclusions were based on only one experimental study, rather than the reported in the paper.

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Together these models help explain what aspects of inter-individual interactions are most important for creating emergent patterns of coherent group motion. With this proliferation of putative interaction rules has come the recognition that some patterns of group behaviour are common to many models, and that different models can have large areas of overlapping behaviour depending on the choice of parameters [3].

More recently Katz et al.

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Just as simulations of large-scale phenomena can appear consistent size observations of group behaviour without closely matching the local rules of interaction, so can fine-scale inferred rules be inconsistent with large-scale phenomena if these rules of inferred from too limited a set of possible models or from correlations between the wrong behavioural measurements. The mann striking features of the collective motion of animal groups are the large-scale patterns produced by flocks, schools and other groups.

This is an important trend in the field of collective motion as we move from a theoretical basis, centred around simulation studies, to a more data-driven approach. The most frequent approach to inferring these rules has been to find correlations between important measurable aspects of the behaviour of a focal individual and its neighbours.

Competing interests: The authors have declared that no competing richards exist. These patterns can extend over scales that exceed the interaction ranges of the individuals within the group [1] — [4].

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In all these systems, as density of these species is increased there is a sudden transition from random disordered motion to ordered motion with the group collectively moving in the same direction. The collective movement of animals in a group is an impressive phenomenon whereby large scale spatio-temporal patterns emerge from simple interactions between individuals. ificant anisotropy in the position of the -th nearest neighbour, averaged over all individuals, was regarded as evidence for an interaction between each bird and that neighbour.

The empirically observed phase transition and collective behaviour remain, as does the observation that individuals are more likely to change direction when in close proximity to each other. Recently, researchers have shown greater interest in using experimental data to verify which rules are actually implemented by a particular animal species.

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When faced with alternative hypothesised interaction rules, model-based parametric inference provides the best means of quantitatively selecting size them. I am grateful to Michael Osborne University of Oxford and David Duvenaud University of Cambridge who spotted this error when I passed the code and data on to them, while aiming to replicate our for their own project. This provides evidence not only for the existence of an interaction between neighbours but also estimates the rules that determine that interaction.

For example, when comparing the evolution of social behavior across different species, it is important to know if the same rules evolved independently in multiple instances, or whether each species evolved a different solution to the problem of behaving coherently as a group [1]. Each of the models in the literature is capable of reproducing key aspects of the large-scale behaviour of one or more biological systems of interest.

We show that the mean-field model fits the large scale behaviour of the system, but does not capture fine scale rules of interaction, which are primarily mediated by physical contact. Although typically found in large feeding aggregations, it does not appear to form social aggregations and has not been reported to exhibit collective behaviour patterns in the wild. Typically these models implement a simple form of behavioural mann, such as aligning the focal individual's velocity in the average direction of its neighbours or attraction towards the position of those neighbours.

Prawns moving within an annulus of mm external diameter and 70 mm internal diameter.

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Comparison of the marginal likelihood, the probability of the data conditioned on the model, integrating over the uncertain parameter richards, is a well developed and robust means of model selection that forms the core of the Bayesian methodology [25] — [28]. An size model-based approach that does fit self-propelled particle and similar models to data is mann by Eriksson et al. PLoS Comput Biol 8 1 : e This is an open-access article distributed under the terms of the Creative Commons Attributionwhich permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Common patterns of collective behaviour are also observed empirically across a diverse range of animal and biological systems. The first author explains the reasons below: A bug was found in the Matlab code used in this study, which resulted in only a small proportion of the full data set being analysed. However, none are able to reproduce both levels of description at the same time. The responsibility for this coding error is entirely mine Richard Mann.

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We find that the classic theoretical models can accurately capture either the fine-scale behaviour or the large-scale collective patterns of movement of the prawns. Paratya australiensis is an atyid prawn which is widepsread throughout Australia [24].

Recently researchers in the field have become interested in using tracking data from real systems on the fine scale to infer what precise rules of motion each individual uses and how they interact with the other individuals in the group [14] — [19]. Either fine scale individual-level behaviour is observed without explicit fitting of a model [18][19] or global properties, such as direction switches [11][20]speed distributions [21][22] or group decision outcome [23] have been compared between model and data.

Supplementary material

In these studies the rules of size are presented non-parametrically and cannot be immediately translated into a specific self-propelled particle model. Where each of experiments should have been down-sampled to half the original size for computational efficiency, instead the of experiments in the data set was repeatedly halved richards, until only one remained.

After correcting the mann and reanalyzing the full data set we found that our had changed ificantly, and some of our conclusions were no longer valid. We tracked, using semi-automated software, the position of each prawn through the duration of the experiments. In their study the local spatial distribution of neighbouring individuals in a group of scoter ducks was used to propose parametric rules of interaction, with some parameters measured from the fine-scale observables, but with others left free to be fitted using large-scale data. Theoretically, much of our understanding of animal group motion comes from models inspired by statistical physics.

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Instead we use reproduction of the large scale dynamics through simulation as a necessary but not sufficient condition of the correct model. Under this approach, the recorded fine-scale movements of individuals are used to fit the parameters of, and select between, these models in terms of relative likelihood or quality-of-fit.

Correction

We suggest that if group behaviour emerges from individual interactions, then the form of these interactions should be inferable solely from fine-scale data without additional fitting at the large-scale. The funders had no role in study de, data collection and analysis, decision to publish, or preparation of the manuscript. In this instance the total of prawnsof clockwise-moving oriented prawnsthe polarisationand the excess polarisation.

Here, we use a Bayesian methodology to compare a variety of models to the collective motion of glass prawns Paratya australiensis. We select between models by calculating the probability of the fine scale motions using a Bayesian framework specifically to allow fair comparison between competing models of varying complexity.

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In many contexts however the rules of interaction are of more interest than the group behaviour they lead to. An inability to replicate the group behaviour using a selected model demonstrates that the model space has been insufficiently explored. What all empirical studies have lacked is a simultaneous verification of a model at both the individual and collective level. For example, Ballerini et al. In our study, we present a rigorous selection between alternative models inspired by the literature for a system of glass prawns.

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Conversely, the Markovian self-propelled particle model captures the fine scale rules of interaction but fails to reproduce global dynamics. Models are tuned through repeated simulation until they match the observed behaviour. Given the simplicity of prawns as a study system our research suggests that self-propelled particle models of collective motion should, if they are to be realistic at multiple biological scales, include memory of interactions and other non-Markovian effects.

We study the positions and directions of co-moving prawns in a confined annular arena See Methods and Materials and Figure 1. My coauthors were not involved in coding this stage of the analysis. However, the likelihood ordering of the different models for interactions between individuals is changed, and there is no longer a failure to reproduce large-scale by simulation of the Markovian spatial models.

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In adopting this approach, we reject the dichotomy of model inference based on either fine scale behaviour of the individuals or the motion of the group. For example, a form of phase transition from disorder to order has been described in species as diverse as fish [9]ants [10]locusts [11]down to cells [12] and bacteria [13]. These studies indicate that a great deal can be understood about collective behaviour without reduction to the precise rules of interaction.

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Nor are these models validated in terms of the global schooling patterns produced by the fish. We will be assessing the conclusions to be drawn from our reanalysis of the data and submitting a revised paper for publication in the future.

In this paper we study the collective motion of small groups of the glass prawn, Paratya australiensis. More recent work has used the fine scale motions of animals to validate and fit the rules of interaction of animals in groups.

Retraction

The most sophisticated model, the non-Markovian model, provides a good match to the data at both the fine scale and in terms of reproducing global dynamics. Inference of interaction rules of animals moving in groups usually relies on an analysis of large scale system behaviour. We pre-processed those raw tracking data by using a Hidden Markov Model to classify the movements of each prawn into a binary sequence of clockwise CW and anti-clockwise orientation see Methods and Materials.