Journal of Theoretical Biology 355 (2014) 151–159

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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi

Ordering dynamics in collectively swimming Surf Scoters Ryan Lukeman n Department of Mathematics, Statistics, and Computer Science, St. Francis Xavier University, Antigonish, NS, Canada

H I G H L I G H T S







A positive correlation between group alignment and average individual speed exists in surf scoter flocks. Based on observations, an individual-based model is constructed for the transient group dynamics of these flocks. Through parameter exploration, biologically realistic interactions which optimize repolarization of flocks are specified. Optimal repolarization parameters are consistent with parameters previously derived from matching model to data.

art ic l e i nf o

a b s t r a c t

Article history: Received 5 August 2013 Received in revised form 2 March 2014 Accepted 7 March 2014 Available online 25 March 2014

One striking feature of collective motion in animal groups is a high degree of alignment among individuals, generating polarized motion. When order is lost, the dynamic process of reorganization, directly resulting from the individual interaction rules, provides significant information about both the nature of the rules, and how these rules affect the functioning of the collective. By analyzing trajectories of collectively swimming Surf Scoters (Melanitta perspicillata) during transitions between order and disorder, I find that individual speed and polarization are positively correlated in time, such that individuals move more slowly in groups exhibiting lower alignment. A previously validated zone-based model framework is used to specify interactions that permit repolarization while maintaining group cohesion and avoiding collisions. Polarization efficiency is optimized under the constraints of cohesion and collision-avoidance for alignment-dominated propulsion (versus autonomous propulsion), and for repulsion an order of magnitude larger than attraction and alignment. The relative strengths of interactions that optimize polarization also quantitatively recover the speed-polarization dependence observed in the data. Parameters determined here through optimizing polarization efficiency are essentially the same as those determined previously from a different approach: a best-fit model for polarized Surf Scoter movement data. The rules governing these flocks are therefore robust, accounting for behavior across a range of order and structure, and also highly responsive to perturbation. Flexibility and efficient repolarization offers an adaptive explanation for why specific interactions in such animal groups are used. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Collective behavior Self-organization Polarization Self-propelled particles

1. Introduction Animals moving collectively to achieve a goal often must also simultaneously satisfy a number of constraints, such as collision avoidance and group cohesion, while maintaining order in the face of environmental noise (Parrish and Hamner, 1997; Guttal and Couzin, 2010). When order is jeopardized, efficient and robust recovery from disorder is advantageous to the collective. The reaction of the group to such perturbations is governed by the individual rules of interaction followed by group members. On a basic level, these rules are described by a combination of ‘social forces’ of attraction, repulsion,

n

Corresponding author. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jtbi.2014.03.014 0022-5193/& 2014 Elsevier Ltd. All rights reserved.

and alignment with neighboring individuals (Huth and Wissel, 1992; Grunbaum and Okubo, 1994; Parrish and Edelstein-Keshet, 1999). However, moving beyond this basic description, these rules are notoriously difficult to ascertain explicitly and specifically in real animal groups due to the inverse nature of the problem (Katz et al., 2011). That is, the group behavior resulting from a set of rules is easy to establish, but the exact set of rules governing an observed collection of trajectories is typically not easily obtained. Under the basic social forces, individuals can form cohesive, wellspaced groups, moving in a highly aligned fashion. However, since many different implementations of the basic forces can generate such behavior, a much more detailed analysis is required to specify how animals use these rules. This analysis mandates a close comparison with empirical data, now becoming possible due to the recent emergence of a number of high-quality trajectory-based

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datasets of moving animal groups. Interactions in starling flocks have been shown to be topological (based on a fixed number of neighbors, irrespective of density), rather than metric (Ballerini et al., 2008), and also, the extent of velocity correlations within the group were shown to scale with flock size (i.e., are scale-free) (Cavagna et al., 2010). Both have been reproduced in a computational zone-based model based on social forces (Hemelrijk and Hildenbrandt, 2011). Interactions in Surf Scoters, an aquatic duck, were quantified using a zone-based model, showing repulsion to be the dominant interaction over attraction and alignment (Lukeman et al., 2010). Furthermore, an additional frontal interaction was shown to account for a preference for neighbors directly ahead. Three recent studies of fish (Katz et al., 2011; Herbert-Read et al., 2011; Gautrais et al., 2012) identified key social interactions between individuals. In Katz et al. (2011) and Herbert-Read et al. (2011), evidence pointed to attraction and repulsion over alignment, and also to speed regulation as an important component of interaction, while in Gautrais et al. (2012), both positional and alignment effects were found, also with explicit dependence on speed. Polarized motion, wherein individuals are all highly aligned with one another, permits efficient group motion toward a goal, for a given individual speed of motion. However, order can be lost if the collective experiences conflicting directional information. This may be due to intrinsic factors such as noise (Yates et al., 2009; Grünbaum, 1998), changes in density (Vicsek et al., 1995; Buhl et al., 2006; Hemelrijk and Hildenbrandt, 2008; Kunz and Hemelrijk, 2003) or differences among group members, or extrinsic factors caused by spatially heterogeneous environmental influences such as predators (Bode et al., 2010; Hamilton, 1971; Inada and Kawachi, 2002; Zheng et al., 2005). In these cases, whether and how the group reorganizes are significant biological questions. Changes in the level of alignment in self-organized collectives have been observed across biological scales. In many experimental systems, these changes are akin to phase transitions, in which the long-term behavior of the group changes between low-alignment and high-alignment as some parameter is modified. Examples include fish (Becco et al., 2006; Makris et al., 2009; Tunstrøm et al., 2013), locusts (Buhl et al., 2006; Yates et al., 2009), bacteria (Zhang et al., 2010) and cells (Szabo et al., 2006). In most cases, the experimentally observed transition arises as a function of increasing group density (Becco et al., 2006; Makris et al., 2009; Buhl et al., 2006; Zhang et al., 2010; Szabo et al., 2006). This densitydependent transition is duplicated in a classical minimal selfpropelled particle model by Vicsek et al. (1995), providing a possible mechanistic basis for the observed group-level changes in state. The Vicsek et al. (1995) model also shows that alignment can be lost as the noise in the system increases. Biologically, sensitivity of group order to intrinsic noise is not necessarily disadvantageous: it can allow groups to amplify social information (Katz et al., 2011), such as quickly transferring information about predators or food (Sumpter et al., 2008). Tunstrøm et al. (2013) found that schooling fish in a tank readily transitioned between three states (polarized, swarming, and milling) for fixed group size, initiated by perturbations (both internal and external) to the system as opposed to density variation. Counterintuitively, at fixed densities, locusts were experimentally found to respond to a loss in alignment by increasing inherent noise (Yates et al., 2009). The flexibility afforded to readily transitioning collectives does, however, also carry a cost; noise (intrinsic or extrinsic) can break the high-alignment state, leading to a disorganized collective and hampering group motion toward a goal. In this case, it is important that the individual rules of interaction lead to repolarization, and that this occurs efficiently. Understanding how animals moving collectively are able to repolarize requires individual-based trajectory data of real groups, and forms a central focus of this work.

I ask what group properties vary during changes in the level of group order, and how our understanding of this process can inform the interaction rules followed by individuals. To address this question, field observations of collectively swimming Surf Scoters (Melanitta perspicillata) are linked to a model framework to characterize the mechanistic basis (i.e., social forces) governing how individuals interact, and transfer directional information in response to disorder. The dataset analyzed here contains individual trajectories of Surf Scoters, completely reconstructed during a series of transitions between high and lowalignment states during collective foraging of mollusks in shallow coastal waters near Vancouver, BC. Importantly, the disordered state analyzed here was clearly an ephemeral phenomenon – overwhelmingly, the group moved with clear spacing structure and high alignment during foraging bouts, and only occasionally became disordered. Unlike most previous studies of order transitions in animal groups, but similar to Tunstrøm et al. (2013), the onset of low alignment does not stem from changes in density, but from a perturbation. The specific perturbation in the flock appears to be different directional preferences among group members (for instance, merging subgroups), although the cause of disorder is not the focus of this study. The low-alignment state is also transient – a relatively short-lived state caused by perturbation, as opposed to a phase-like transition, where the long-time behavior of the system is changed. Inferring individual interactions from trajectory data typically involves averaging over time and individuals to overcome a low signal-to-noise ratio and obtain a clear pattern of interaction. In this work, the focus is on the development of order within the group, and so the temporal structure of the data is retained in much of the analysis. Previously, both in modeling studies and laboratory studies, modification of intrinsic group properties led to different steady-state group configurations (Vicsek et al., 1995; Couzin et al., 2002; Hemelrijk and Hildenbrandt, 2011); that is, the transition was parameter-dependent. In contrast, this work focuses on the transient dynamics of a group transitioning between order and disorder, under the assumption that individuals become disordered through a perturbation (i.e., conflicting directional information) but are governed by the same set of rules and parameters through time. Although there is evidence that the group can modify intrinsic parameters for distinct behaviors (e.g., foraging versus resting and immediate predator avoidance), I assume that the observations studied here occur within the same behavioral context of directed motion (which occasionally is perturbed), hence keeping behavioral parameters fixed. I first analyze properties of group motion in time-series during periods in which relatively large changes in alignment occur. These observations are then used to guide the development of a suitable model, which in turn is used to investigate which interactions lead to the most efficient repolarization while simultaneously satisfying the usual constraints on group motion, namely cohesion and collision avoidance. A zone-based model (Sakai, 1973; Reynolds, 1987; Huth and Wissel, 1992; Couzin et al., 2002) framework is used, where interactions are implemented through a series of layered ‘zones’ (repulsion, alignment, attraction) around an individual. The zone in which a neighbor is found determines which of the interactions take place. In previous work (Lukeman et al., 2010), this approach was shown to be an appropriate tool to study the collective behavior exhibited in the Surf Scoter dataset. For clarity, I explicitly define the terms used to describe transitioning collectives. The primary categorization is that of ! ! polarization, defined as pt ¼ jð1=NÞ∑N v i ðtÞ=j v i ðtÞjj, where N is !i ¼ 1 the number of individuals, and v i ðtÞ the velocity of the ith individual. Values of pt (which is unitless) range from 0 to 1. Groups exhibiting a low pt value (e.g., pt o 0:5) exhibit low-alignment or

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disorder. In contrast, groups with a high pt value are highly-aligned, or ordered. The terms order and alignment are used interchangeably. Polarization efficiency is a combined measure of the ability of groups to evolve from low-alignment to high-alignment, and the rate at which this progression occurs. It is calculated by the timeR t^ averaged value of polarization, 1=t 0 0 pt dt, where t^ is a suitably chosen time value over which the transition takes place (and where t^ is fixed in this study), and as such is also unitless. High values of polarization efficiency correspond to groups that most quickly evolve to a highly polarized state. Optimization of polarization efficiency refers to the set of model parameters that lead to the highest value of polarization efficiency, averaged across simulation trials.

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periods in which large changes in order could be observed, across six different sequences in total (size of dataset limited by the infrequent nature of the phenomenon). Length of recorded sequences ranged from 15 s to 32 s, and average group size in a given sequence ranged from 50 individuals to 114.4 individuals per frame. Snapshots of a scoter flock first (a) exhibiting low alignment (Fig. 1a) and then later (b) highly aligned (Fig. 1b) show clearly distinct group states. An example of reconstructed trajectories during low alignment is shown

2. Results 2.1. Observational results While overwintering in shallow coastal regions of North America, Surf Scoters gather in flocks of hundreds, and synchronously forage on mollusks (Savard et al., 1998; Schenkeveld and Ydenberg, 1985). In the observation zone of this study, scoters move collectively from open water to a dock area to dive primarily for mussels (Mytilus edulis), after which the group collectively moves back to open water. Groups are generally well-spaced, highly polarized, and cohesive, but occasionally become disordered due to conflicting directional preferences among group members, such as when a subgroup moving toward the foraging area merges with a subgroup moving away in the opposite direction. Individual trajectories were recorded (see methods) during

Movie S1. An example of raw data of a time series of a Surf Scoter flock. The flock surfaces, but then becomes disordered, which persists for a period of time before the group eventually re-orders, heading collectively again toward the dock. This corresponds to the data in Series 1 in Fig. S1. A video clip is available online. Supplementary material related to this article can be found online at http://dx.doi. org/10.1016/j.jtbi.2014.03.014

Fig. 1. Snapshots of a flock of Melanitta perspicillata (Surf Scoter) swimming on the water surface, exhibiting (a) low alignment. The same flock, 8 s later, exhibits (b) high alignment. Images of the flock were analyzed to extract positions and velocities of individuals, resulting in reconstruction of individual trajectories during changes in alignment. An example of trajectory reconstruction for 11 s of motion is shown in (c). Green and red circles indicate start and end positions, respectively. Units are in terms of transformed image pixels (1 BL  46 pixels) . See also Movie S1. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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in Fig. 1c, showing high variation in paths, in contrast to the highly aligned trajectories extracted from polarized motion. To track changes in alignment for the group, the order parameter pt (i.e., group polarization) is measured in time, and compared in ! time series to average individual speed st ¼ 1=N∑N i ¼ 1 j v i ðtÞj for each sequence. Example time series of pt and st are shown in Fig. 2. Group polarization is positively correlated with average individual speed in all 6 series (r ranging from 0.108 to 0.8165), although correlation was significant in only 5 of the 6 series (p-value r 0:01 for series 1–5; p-value¼0.3181 for series 6) (see Table 1). Thus, as a group becomes less aligned, individuals tend to reduce speed (and vice versa). To determine the best regression model to relate pt and st, a wide variety of transformations were tested, but none uniformly reduced sum of squares error beyond a linear regression. Thus, to quantify how changes in each measure are related, a linear model p^ t ¼ ks^ t þ ϵt is fitted to the time-series data for each series, where p^ t and s^ t are group polarization and average individual speed, each translated about the mean value (p^ t ¼ pt  pt , s^ t ¼ st  st ) to eliminate the constant term of the linear regression, and ϵt is error. Slope values (k) in all cases are positive, ranging from 0.21 to 0.92 (0.27–0.92 for sig. series), significant in all but Ser. 6 (see Table 1 for details and confidence intervals, and Fig. A1 for plots of regression fit). Averaging across the five significant series gives k ¼ 0:58; on average, an increase in group polarization of 0.58 is associated with an increase in speed of 1 body-length per second (BL/s). Cross-correlation of the polarization and individual speed data showed maximal correlation at zero lag for all 6 series. Across the sequences analyzed, average individual speed ranged from 0.56 BL/s to 2.1 BL/s. During low-alignment (pt o 0:5), individuals had an average nearest-neighbor distance (NND) of 1.84 BL with standard deviation of 0.4 BL, slightly less dense and more variable than individuals in highly aligned groups (e.g., pt 4 0:8; NND¼1.69 BL, SD¼0.31 BL). A frontal bias for relative neighbor location, observed in a previous study (Lukeman et al., 2010), was not observed in the data during the low-alignment motion, although concentric structure in relative positioning was maintained. Analysis of residuals indicated autocorrelation in all series; plots of the autocorrelation function, and partial autocorrelation

function indicated a first-order autoregressive (AR(1)) structure. Because autocorrelation can lead to erroneous claims of significant correlation, the regression model was modified to include an AR (1) residual term via ϵt ¼ ρϵt  1 þ νt , where νt is the (non-serially correlated) random error. A Durbin–Watson test (DW statistic o 0:32 in all 6 series) was significant for AR(1). Estimates for autocorrelation parameter ρ ranged from 0.83 to 0.96 across the 6 series. A Cochrane–Orcutt procedure was used to pre-whiten the data via the transformations pnt ¼ pt  ρpt  1 , snt ¼ st  ρst  1 where ρ is estimated from the data. Resulting adjusted regression slopes k ranged from 0.1267 to 0.4069, and were again significantly positive in 5 of 6 series, indicating that the positive relationship between group polarization and average individual speed persists when adjusted for serial correlation effects (see Table 1 for details). 2.2. Modeling results An individual-based zonal model was used to study the mechanistic basis for repolarization of an initially disordered group. In this model, space around an individual is partitioned into hierarchical circular zones of repulsion, alignment, and attraction. Concentric structure in spacing in the Surf Scoter data has been shown in previous work (Lukeman et al., 2010) and in data analyzed here, and this model framework was shown to be suitable to generate insight into the interaction rules governing Table 1 Regression statistics for pt and st time-series data for each of the 6 series analyzed. Significance columns correspond to values in the preceding column, and test for significance of positive values. Series

# Pts

r

Sig.

Slope k (95% CI)

Sig.

1 2 3 4 5 6

98 52 62 44 67 96

0.435 0.791 0.817 0.476 0.646 0.108

p o 0:001 p o 0:001 p o 0:001 p¼ 0.001 p o 0:001 p¼ 0.3181

0.271 (0.175 0.443) 0.467 (0.364 0.569) 0.923 (0.755 1.092) 0.687 (0.292 1.083) 0.530 (0.375 0.685) 0.271 (  0.18 0.588)

p o 0:001 p o 0:001 p o 0:001 p ¼0.001 p o 0:001 p ¼0.292

Fig. 2. Data examples: time series showing progression of average individual speed st (measured in body-lengths per second (BL/s)) and group polarization pt (unitless) for four (a–d) of the analyzed sequences.

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this group. Denoting distance to a neighbor, interactions are implemented as piecewise-linear and pairwise. Magnitudes are governed by a normalized function g(x) (such that jgðxÞj is maximally 1) and accounts for both attraction (where gðxÞ 4 0) and repulsion (where gðxÞ o 0), as a function of the relative distance x between neighbors. Interactions are separated into force (per unit mass) components fk and weighted by ωk, where k ¼repulsion, alignment, and attraction. For n individuals indexed ! ! by i, position x i and velocity v i is modeled by ! dxi ! ¼ v i; dt

ð1Þ

! ! ! dvi ! ¼ f i;aut þ f i;int þ ξ i ; ð2Þ dt ! ! where f i;aut denotes an autonomous self-propulsive force, f i;int denotes the interaction (attraction/repulsion/alignment) force, and ! ξ i denotes the noise. Individuals are constrained to a maximal turning angle of 1201/s, measured directly from the data (see Appendix). Units of velocities are BL/s, and units of forces per unit mass are BL=s2 . Unlike fixed-speed zonal models (e.g., Couzin et al., 2002), this model allows for acceleration/deceleration, necessary to capture the observed changes in individual speed described above. Previous work (Lukeman et al., 2010) studied only polarized motion toward an environmental cue, and so the propulsive ! component of f i;aut was a constant vector. I now modify the ! ! ! ! ! ! previous model via f i;aut ¼ ðα  β j v i j2 Þ u i ; where u i ¼ v i =j v i j is a unit direction vector. In the absence of interactions and noise, individuals modeled as described reach an equilibrium speed of pffiffiffiffiffiffiffiffiffi direction of motion. α=β, but do not have a predetermined ! ! Repulsive and attractive interactions ( f i;rep ; f i;att )average the influence of all individuals found in the repulsion, and attraction zones, respectively, and alignment and attraction occur only if the repulsion zone is empty (see Appendix for details of the model). ! ! ! al The alignment force, given by f i;al ¼ ð1=nal Þ∑nj ¼ 1 vj =j vj j (with nal ¼ number of individuals in the alignment zone) is suitable for the observed correlation between polarization and speed, since here neighbors exhibiting low alignment will result in a lower vectorial sum (and thus lower propulsion) than if neighbors are highly aligned. The total interaction force is taken as the sum of repulsion, alignment, and attraction, each weighted by relative strengths (ωrep ; ωal ; ωatt respectively). Simulations are performed across a range of these parameters to determine the effect of each parameter on the group transition from disorder to order. Accounting for these interactions, individuals within a polarized group will

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi reach an equilibrium speed of ðα þ ωal Þ=β . The drag coefficient β ¼ 1:25 is chosen based on a biomechanical estimation (see Appendix). A previous study (Lukeman et al., 2010) showed that Surf Scoters obtain an average speed of 2.0 BL/s while in the polarized phase, so while investigating a range of ωal values, the constraint α þ ωal ¼ 5 is imposed to maintain an equilibrium speed of 2.0 BL/s. Development of collective motion from an initially disordered state is characterized here by four measures: group polarization and average individual speed described above, together with cohesion and collisions. Cohesion is defined to be the average distance between individuals, scaled by the average distance in a circular group of n¼ 100 in hexagonal lattice (the minimal packing of radially symmetric interacting individuals) with NND ¼ r rep ; a value of 1 would indicate a perfectly circular, cohesive group. Collisions are defined, at a given time, as the count of nearestneighbor distances below 0.5 BL. I seek development of order that generally maintains cohesion among group members (does not lead to group fragmentation), and is characterized by few collisions. Neither are observed in the data, and in this system are biologically disadvantageous. Furthermore, the degree of correlation between speed and polarization is tracked dynamically for comparison to observations. In Fig. 3, an example of time-series of these measures is presented under two sets of parameters: In Fig. 3a, behavior is unlike that observed in the Surf Scoter dynamics: individual speed remains near equilibrium (2.0 BL/s) despite low alignment, there is minimal and slow transition to a polarized state, and collisions occur between individuals. This development differs from that shown in Fig. 3b, which shows significant positively correlated variation in polarization and speed, and a cohesive, stable collective emerging while essentially avoiding collisions between individuals, as in our observations in the dataset. To quantify the effect of parameters on the development of the group, the time-series of each of the four measures (such as those in Fig. 3) are averaged over time (t¼0 to t^ ¼ 10 s) and over 100 simulations to obtain a single value summarizing the temporal progression for a given set of parameters. Simulations are performed for each parameter combination starting from an initially cohesive, well-spaced, but low-polarization state (headings are randomly assigned). Base parameters, from which parameter variations are sequentially explored, as chosen as ωrep ¼ 50 and ωatt ¼ 5. This choice is based upon the relative magnitude of parameters that optimally matched data in the original study of polarized motion (Lukeman et al., 2010), scaled according to autonomous forcing α þ ωal (see Appendix for details).

Fig. 3. Simulation of Eqs. (1) and (2): time series examples showing development from an initially disordered group (n¼100) of average individual speed st (units BL/s), polarization pt (unitless), cohesion (scaled average neighbor distance, unitless) and collisions, for two sets of parameters (time axis in seconds). In (a), the simulated group develops unlike observations, featuring little increase in polarity and frequent collisions, whereas in [b], simulations exhibit transition from low-alignment to high alignment, while maintaining cohesion and avoiding collisions. Parameters used in (a): ωrep ¼ 10, ωatt ¼ 5, ωal ¼ 0:5, α ¼ 4:5; in (b): ωrep ¼ 50, ωatt ¼ 4, ωal ¼ 5, α ¼ 0.

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As seen in Fig. 4a, cohesive groups are obtained only for sufficiently large values of ωatt (approximately 4–5, after which little cohesive benefit is obtained by higher attractive force). Varying relative weight of autonomous and alignment force α; ωal has little effect on cohesion. Thus, a lower threshold on ωatt exists, under which group-splitting occurs frequently during group progression from the initial disordered state. Collisions during development of group order occur more frequently as attraction force ωatt increases relative to repulsion, and alignment ωal decreases (Fig. 4b). Decreasing alignment corresponds to increasing autonomous propulsion α, and so collisions occur more frequently in groups where autonomous propulsion dominates over interaction-based propulsion. In contrast, there is a non-monotonic response of collisions to increasing repulsion (Fig. 4c): collisions are fewest for intermediate values of ωrep (  40–60), becoming more frequent under weak repulsion, or very strong repulsion. Collisions also become more frequent as autonomous propulsion increases (i.e., as ωal decreases).

Simulated groups showed the highest polarization efficiency under alignment-dominated propulsion (Fig. 4d and e), values of ωatt  4–5 (Fig. 4d), and under intermediate repulsion (ωrep  40–60) (Fig. 4e). Stronger attraction and repulsion reduced the efficiency of polarization slightly, but low attraction and repulsion led to groups that either were unable to polarize, or that did so very inefficiently. Taken together, results indicate that using the zone-based model here, simulated groups are able to reorder while maintaining cohesion, and largely avoiding collisions, as observed in the Surf Scoter flocks. The set of parameters that permits cohesive, collision-free progression to order (ωrep  40–60, ωatt  4–5 and ωal  5) also leads to optimal polarization efficiency. Significantly, this set of parameters corresponds approximately to the scaled parameters (ωrep ¼ 34, ωatt ¼ 4:8, ωal ¼ 5) observed in Lukeman et al. (2010), obtained from optimally matching simulations to data taken from the polarized phase. Apart from the question of how quickly the group evolves to a highly polarized state, I also investigate whether the dynamics of

Fig. 4. Results of parameter exploration. Group measures of (a) cohesion, (b–c) collisions, and (d–e) polarization, averaged over 10 s of progression from an initially disordered state, averaged over 100 simulations for each data point. Vertical axis on (a)–(e) indicates the relative contribution of alignment-mediated propulsion ωal and autonomous propulsion α, where ωal þ α ¼ 5. Horizontal axis ranges over weights of attractive force ωatt in (a), (b), and (d), and repulsion force ωrep in (c) and (e). Collision values above 30 in (c) are truncated for scaling purposes.

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Fig. 5. Relationship between polarity and individual speed across parameters. Values of ksim (averaged over simulations as described in Fig. 4) are shown (where p^ t;sim ¼ ksim s^ t;sim ) across a range of (a) attraction and (b) repulsion values . Vertical axis is as in Fig. 4. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

polarization in simulations match the empirical observations reported here. As in the data, a linear model p^ t;sim ¼ ksim s^ t;sim is fitted for mean-shifted group polarization and average individual speed for each simulation, and averaged over 100 simulations for each set of parameters. In Fig. 5, average values of k are given across a range of forcing weights, as in Fig. 4. The region of parameter space which leads to cohesive, collision-free, efficient progression to order (as determined above) corresponds to values of k  0:5–0:7 (light blue in Fig. 5), consistent with slope values determined by the linear regression of the data. This contrasts with other regions of parameter space (e.g., low ωal, high ωrep) which lead to k values that are larger than those observed in these simulations, relatively small changes in speed occur as the collective reorders.

3. Conclusions In this paper, I report on a trajectory-based dataset for flocks of Surf Scoters transitioning between order and disorder during collective motion. The data revealed a tendency for individuals to move more slowly in disordered groups, and to speed up as the collective became more ordered. This mirrors findings in experimental studies of schooling fish (Viscido et al., 2004; Tunstrøm et al., 2013). Here, only correlation is established, versus causation. Recent work on fish (Katz et al., 2011; Herbert-Read et al., 2011) has shown that interaction magnitude is speed-dependent: faster individual speeds led to a greater response to neighbors. Yet whether speed is controlled by individuals as a response to disorder, or whether speed reduction causes disorder is not known. More recently, Tunstrøm et al. (2013) were not able to conclusively distinguish between these two cases, but suggest that both causal relationships likely exist. Evidence from modelling (Viscido et al., 2004; Tunstrøm et al., 2013; Gautrais et al., 2012; Hemelrijk and Hildenbrandt, 2008; Kunz and Hemelrijk, 2003; Hemelrijk and Kunz, 2005) shows that speed-induced transitions are possible. In the current study, disorder appeared to occur due to differences in directional preference, which would lead to conflicting alignment information. It seems plausible, then, that this conflicting information in turn causes a reduction in individual speed (although this cannot be conclusively determined with the dataset studied here). One explanation for this reduction in speed is that increased disorder leads to an increased risk of collision with neighbors, and so in response, individuals move more slowly to reduce this risk (Hemelrijk and Hildenbrandt, 2008). Slower moving individuals could also promote repolarization, since spatial information could be integrated with less error, reducing sensitivity to noise. Note however, that in some groups

(e.g., locusts Yates et al., 2009), noise can facilitate order. Alternatively, the reduction in speed due to disorder could be attributed to increased energy expenditure to turning, reducing linear velocity. In the case of increasing alignment (from disorder to order), it becomes even more difficult to hypothesize about the causal relationship. An individual-based zonal model for collective motion was used to probe more deeply the process of repolarization in groups, specifically to determine how individuals are able to efficiently reorder collectively, while satisfying the usual constraints on collective motion: collision avoidance and maintaining group cohesion. The contribution of specific interaction forces (attraction, repulsion, alignment) significantly determined how effectively the group organized under these constraints. By weighting each force via a parameter, a region of parameter space emerged under which groups simultaneously maintained cohesion and avoided collisions. This parameter set also led to groups which organized most efficiently. This result is intriguing, as it suggests that here, efficient polarization does not come at the expense of spacing and cohesion, but in fact the two coincide: individual rules which lead to regulated spacing and cohesion also lead to quick recovery from disorder to order. The alignment interaction used here is based on summing the influence of neighbors within an alignment zone. If neighbors within this zone are not aligned, the vectorial sum is small, and a weak propulsive signal is given to the focal individual. As the group (and thus neighbors in the alignment zone) becomes more aligned, the propulsive signal is stronger. In this way, speed and group polarization become correlated: individuals within disordered groups move more slowly, speeding up as alignment increases. This is not the case with another common implementation of alignment in which individuals respond to relative differ! ! ! ! ! al ences in heading, via f i;al ¼ ð1=nal Þ∑nj ¼ 1 ð vj  vi Þ=ðj vj  vi jÞ. Here, propulsive force decreases as alignment increases, and so is unsuitable for modelling the Surf Scoter observations. The speed dependence on polarization strengthens as the ratio of alignment versus autonomous propulsion increases (i.e., ωal increases and α decreases). The relationship between speed and polarization during development of order was investigated over a range of parameters. In the region of parameter space that leads to cohesive, collision-free dynamics, while also leading to the most efficient polarization, the speed-alignment relationship that emerges from simulations is consistent with empirical observations. There are therefore two sources of evidence that support the biological relevance of the individual rules suggested by the model. First, the ratio of forces lead to biologically realistic behavior, in that individuals are able to maintain a safe distance from other neighbors,

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avoiding potentially costly collisions, yet remain cohesive, as maintenance of the group is integral to the existence, and therefore functioning of collective. Furthermore, the speed-alignment correlation in simulations was shown to be consistent with observations. Second, the ratio of interaction forces suggested in previous work on flocks of Surf Scoters in the polarized phase (Lukeman et al., 2010) are essentially the same as those established in this paper. Whereas previous work established these forces by fitting a model to data, this work obtained similar results from an adaptive standpoint: optimizing the efficiency of group organization under constraints of group motion. That similar interaction forces describe distinct behavioral phases suggests that these interactions permit flexible group behaviors in response to varying environmental or intrinsic factors. Further, under these behavioral rules, alignment is able to be regained efficiently, should disorder arise due to a perturbation. From a model standpoint, the agreement between the two studies provides evidence that the relative force strengths established in Lukeman et al. (2010) are robust to ! changes in model implementation (at least with respect to f aut ). By asking not just how animals interact within a group, but also what function is obtained through such rules, we can begin to construct an ecological context for an observed collective behavior. Why aggregation occurs has been broadly addressed for a variety of species and contexts (information sharing, predator protection and vigilance, confusion effects, and so on), but here we can go further to test what benefits are conferred by specific combinations of interaction forces. In this study, regaining order following a disturbance was the focus, but other benefits could potentially be evaluated, such as communication, energy savings, or foraging efficiency. Because field studies of collective motion feature groups in their native environment, such data are subject to the ecological pressures that shape the observed behavior. As mechanistic studies are more closely linked to the ecological function of the collective, the imperative for additional data sets drawn from the field will increase.

4. Methods Time series of 2D flocks at Burrard Inlet, Vancouver, BC, were recorded by oblique overhead photography, gathered over March 1– 12, 2008, by photography from an elevated promenade at Canada Place, overlooking the inlet where overwintering Surf Scoters were foraging. I used a Nikon D70s DSLR camera, and Nikon AF-S Nikkor 18–70 mm ED lens fixed at the maximal focal length (70 mm). Images were taken in continuous autofocus mode, at 3 frames per second (fps) at a resolution of 1000  1504 pixels with aperture at f4.5, and exposure times of 1/8000 to 1/250 s. Six separate sequences of group motion featuring polarization transitions were reconstructed. Each sequence was composed of between 45 and 101 frames at 3 frames per second, with up to 150 individuals per frame. Frames were digitized, and trajectories reconstructed until individuals left the frame using customized particle-tracking software in MATLAB. Images were corrected for perspective distortion. For each series, water currents were measured using intrinsic fluid markers (Surf Scoter excreta), and tracked manually through time in successive images, repeated and averaged over a number of independent measurements. Individual velocities were then adjusted for water currents by subtracting the current component from the observed motion. A complete description of experimental technique is contained in the supplementary information of the original study (Lukeman et al., 2010). This description includes the experimental set-up, calibration, perspective correction, and removal of water current effects. The analysis of images and tracking procedure is also described. Consult that document for full details.

Acknowledgments R.L. is funded by an NSERC Discovery Grant no. 386638. I am grateful to Dr. L. Edelstein-Keshet and Dr. X Wang for helpful comments. Computational facilities are provided by ACEnet, the regional high-performance computing consortium for universities in Atlantic Canada.

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