Detecting collective behaviour in animal relocation data ...

Detecting collective behaviour in animal relocation

data, with application tomigrating caribou

BenjaminD. Dalziel1,†,*, Mael LeCorre2, SteeveD. Cot�e2 andStephenP. Ellner1

1Department of Ecology andEvolutionary Biology, Cornell University, Ithaca, NY, USA; and 2Department of Biology, University

of Laval, Quebec, QC, Canada

Summary

1. Collective behaviour can allow populations to have emergent responses to uncertain environments, driven by

simple interactions among nearby individuals. High-throughput ethological studies, where individual behaviour

is closely observed in each member of a population (typically in the laboratory or by simulation), have revealed

that collective behaviour in populations requires only rudimentary cognitive abilities in individuals and could

therefore represent a widespread adaptation to life in an uncertain world. However, the ecological significance of

collective behaviour is not yet well understood, as most studies to date have been confined to specialized situa-

tions that allow intensivemonitoring of individual behaviour.

2. Here, we describe a way to screen for collective behaviour in ecological data that is sampled at a coarser reso-

lution than the underlying behavioural processes.We develop and test themethod in the context of a well-studied

model for collective movement in a noisy environmental gradient. The large-scale distribution patterns associ-

ated with collective behaviour are difficult to distinguish from the aggregated responses of independent individu-

als in this setting because independent individuals also align to track the gradient. However, we show that

collective idiosyncratic deviations from the mean gradient direction have high predictive value for detecting col-

lective behaviour. We describe a method of testing for these deviations using the average normalized velocity of

the population.

3. We demonstrate the method using data from satellite tracking collars on the migration patterns of caribou

(Rangifer tarandus), recovering evidence that collective behaviour is a key driver of caribou migration patterns.

We find moreover that the relative importance of collective behaviour fluctuates seasonally, concurrent with the

timing ofmigration and reproduction.

4. Collective behaviour is a potentially widespread dynamic property of populations that can, in some cases, be

detected in coarsely sampled ecological data.

Key-words: animal movement, caribou, collective behaviour, flocking, group dynamics, statistical

test for collective behaviour, swarming

Introduction

Many populations exhibit collective behaviour, where local-

ized interactions among neighbouring individuals lead to

broad-scale patterns in the behaviour of groups, as in the coor-

dinatedmovement of a flock of birds or a school of fish (Vicsek

2001; Vicsek & Zafeiris 2012). Populations influenced by col-

lective behaviour violate the assumption of mass action that

underlies standard models of population dynamics, in which

individuals are viewed as statistically independent (Ovaskainen

& Cornell 2006; Pascual, Roy & Laneri 2011). Through emer-

gent correlations in behaviour, collectives can track variable

resources more effectively than independent individuals

(Gr€unbaum1998; Simons 2004; Codling, Pitchford&Simpson

2007; Torney, Neufeld & Couzin 2009; Berdahl et al. 2013),

leading to increased fitness through population-level cognitive

responses to variable environments (Clark & Mangel 1986;

Handegard et al. 2012; Ioannou, Guttal &Couzin 2012; Olson

et al. 2013).

Research on collective decision-making has advanced by

identifying the underlying behavioural processes that govern

interactions among neighbouring individuals in a population

(Nagy et al. 2010; Gautrais et al. 2012; Berdahl et al. 2013;

Nagy et al. 2013). Analytical techniques have recently been

developed that use fine-scale ethological observations to infer

how strongly, and in what ways, individuals are influenced by

social interactions (Eriksson et al. 2010; Bode et al. 2012;Gau-

trais et al. 2012). The behavioural rules revealed by these anal-

yses are often simple, requiring only rudimentary cognitive

abilities.

Simulations show how simple rules can lead to the emer-

gence of collective behaviour. Individuals that aggregate can

*Correspondence author. E-mail: [email protected]†Current address: Department of Ecology and Evolutionary Biology,

PrincetonUniversity, 106AGuyotHall, Princeton,NJ 08544,USA

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society

Methods in Ecology and Evolution 2016, 7, 30–41 doi: 10.1111/2041-210X.12437

pool their estimates of a noisy environmental gradient, allow-

ing improved navigation for the group (Simons 2004). Align-

ing velocities with neighbours can improve navigation by

simulated groups (Gr€unbaum 1998; Codling, Pitchford &

Simpson 2007; Torney, Neufeld & Couzin 2009). A collective

response can occur even if only some individuals have informa-

tion about the environment, and the informed individuals are

not distinguishable by the others (Couzin, et al. 2005, 2011).

The theoretical and empirical research to date therefore sug-

gests that collective behaviour could be a widespread adapta-

tion to life in an uncertain world, with an important influence

on ecological and evolutionary dynamics (Franks et al. 2002;

Saigusa et al. 2008; Torney, Neufeld & Couzin 2009; Hande-

gard et al. 2012; Ioannou, Guttal & Couzin 2012; Aplin et al.

2014; Farine et al. 2014).

However, the role of collective behaviour in the ecological

and evolutionary dynamics of populations remains largely

unexplored, because few populations are intensively sampled

at the scale of individual behavioural decisions. Existing meth-

ods developed for high-throughput ethological data (Eriksson

et al. 2010; Nagy et al. 2010; Bode et al. 2012; Gautrais et al.

2012; Berdahl et al. 2013; Nagy et al. 2013) may in some cases

work on coarser scale data, but this remains largely untested.

In general, scaling up from fine-scale patterns to course scale

process is an enduring problem in ecology (Levin 1992).

In this paper, we suggest an approach to screen for collective

behaviour in ecological data – data on the distribution patternsof organisms collected at a coarser scale than the underlying

behavioural processes (Levin 1992) – to help identify systems

where more detailed studies of the role of collective behaviour

may be fruitful. Because the informationwe have onmost pop-

ulations is not resolved to the level of individual behavioural

interactions, this paper focus on detecting population-level fea-

tures of collective behaviour that are robust to changes in the

details of the underlying individual interactions. This addresses

the inverse problemof distinguishing collective behaviour from

the aggregated responses of independent individuals, when the

causal behaviours are not observed (Perony et al. 2012; Sump-

ter,Mann& Perna 2012). In general, canonical models of pop-

ulation dynamics assume that each individual’s response to the

environment is independent. To understand changes in real

populations over time, it may sometimes be useful to test

whether this assumption holds.

Potts, Mokross & Lewis (2014) have addressed the broader

goal of integrating collective behaviour into models of the spa-

tial dynamics of animal populations, for instance using cou-

pled step selection functions. In a complimentary paper,Mann

(2011) describes a Bayesian approach for determining the

interaction rules that generate collective behaviour from very

fine-scale data on animalmovements, finding that the rules can

often been uncovered with relatively small ethological data

sets. A related modelling approach, based on hidden Markov

models, has shown how the combined effects of environmental

variation and social cues may be disentangled when positional

data and environmental covariates are available for a single

animal group (Langrock et al. 2014). A recently described

approach for detecting the influence of conspecifics on popula-

tion distribution patterns tests the extent to which the proxim-

ity between individuals can be attributed to independent

randomdisplacements (Delgado et al. 2014).

Like Delgado et al. (2014), the approach we describe uses

a comparison with what independent individuals would do

as a basis for detecting collective behaviour. Instead of

attempting to parameterize a model of collective behaviour,

we focus on measuring emergent features of population

redistribution patterns that are fundamental to collective

behaviour, would be unlikely to occur in an ensemble of

independent individuals, and are robust to unobserved indi-

viduals, as well as to data observed at much coarser scale

than the underlying movement decisions. Our approach

focuses on animal relocation data, but it generalizes straight-

forwardly to other movement data and to other kinds of

changes in individual state.

To demonstrate the approach, we first use a well-studied

model of collective animal movement (Couzin et al. 2005)

implemented in an environment that has a gradient represent-

ing the population’s preferred direction of travel. The environ-

ment is noisy, so that at each place and time an individual’s

experience of the gradient varies. The appearance of broad-

scale order among gradient followers is not diagnostic of col-

lective behaviour. However, populations influenced by collec-

tive behaviour show collective idiosyncratic deviations from

the true gradient direction that are visible in sparsely sampled

data, and are statistically unlikely for independent individuals,

regardless of the behavioural rules that independently govern

each of their trajectories. That is, individuals all heading in the

right direction might be independently reading the environ-

ment accurately. But individuals all heading in roughly the

same wrong direction are probably not acting independently.

We show that these ‘collective mistakes’ represent a character-

istic feature of collective behaviour that can be detected in

sparsely sampled ecological data.

Finally, we demonstrate the method on a novel set of wild-

life data – observations of the migration patterns of caribou

(Rangifer tarandus) collected by satellite tracking collars. The

data encompass 9721 observations of 143 individuals from

2008 to 2011. Applying the method we describe to the caribou

data reveals a major role of collective behaviour in one of the

world’s longest terrestrial migrations.

Methods

ANALYTICAL APPROACH

The approach we propose considers a group of N individuals whose

locations change continuously in space and time. However, the data

are limited to observations of individual locations at M discrete time

points t1; t2; . . .; tM. Sampling periods sj ¼ tj � tj�1 measure the tem-

poral separation between ‘bouts’ of observation, wherein each individ-

ual’s location is recorded. Let xiðtjÞ represent the cartesian coordinates

of the ith individual observed at time tj. As a spatial location, x is a vec-

tor with length equal to the number of dimensions of the ‘landscape’ in

which individuals move. For example, in applications involving

terrestrial animals, the landscape will typically be two-dimensional, so

the vector x will have two elements (storing longitude and latitude, for

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 30–41

Testing for collective behaviour 31

instance). Aquatic or avian applications may include altitude or depth

as a third coordinate. The velocity vi of an individual i associated with a

pair of adjacent sampling times is also a vectorwith the same dimension

as the position vectors, which gives the magnitude jvij (the observed

‘speed’ of individual i) and direction vi=jvij of the change in position

over time. It is estimated from location data as

viðtj; sjÞ ¼ xiðtjÞ � xiðtj�1Þsj

; j[ 1 eqn 1

and thus can vary with different sampling periods, as well as over time

and among individuals.

Throughout the paper, equations will include addition and multipli-

cation of vectors (e.g. location, velocity) and scalars (e.g. time), without

explicit remark, because the type of operations (i.e. vector or scalar) will

be evident from which types of variables are involved. For readers not

accustomed to vector operations, a heuristic approach to the equations

is to imagine a one-dimensional landscape where individuals can travel

only left or right (for instance, towards or away from a goal). Then,

location and velocity vectors will only have one element, and all opera-

tions reduce to scalar arithmetic.

Awell-studiedmeasure of collectivemovement is the following order

parameter, which is calculated using individual observed velocities

wðtj; sjÞ ¼ 1

N

XN

i ¼ 1

viðtj; sjÞviðtj; sjÞ��

��

�� eqn 2

and represents the average normalized velocity of the population at

time tj. In the limit, as N becomes large, wðtj; sjÞ ranges from 0, when

individual velocities have uniform random directions, to 1, when indi-

vidual velocities all have the same direction (Vicsek & Zafeiris 2012).

The intuition, which is similar to the mean resultant length from circu-

lar statistics, is that independent velocities will tend to be evenly dis-

tributed among all possible directions, and will thus cancel each other

out when they are summed. Thus in the case of independent random

velocities, the summed vector will have a magnitude that approaches 0.

By contrast, if velocities are non-independent (because animals are

interacting), then they may tend to all point in a common direction.

Their sum will also point in that direction with significant magnitude,

resulting in a higher value forw.We note that animalsmay also interact

by adjusting their speed as well as the direction of their velocity

(Berdahl et al. 2013), so although intuitive and well studied, this statis-

tic does not necessarily capture all aspects collective behaviour.

The sampling period s is typically <1 second in ethological studies of

collective behaviour (Couzin et al. 2005; Nagy et al. 2010; Berdahl

et al. 2013). However in ecological data, swill typically be much larger,

as the spatial distribution of organisms is usually sampled at discrete

and relatively distant times in ecological studies, rather than in nearly

continuous time. Correspondingly, we now consider the behaviour of

the order parameter for finite populations whose positions are sampled

at discrete time points that can be arbitrarily distant.

We begin with the null case where individuals’ velocities are indepen-

dent of one another. Suppose individuals move in a two-dimensional

environment that has a gradient with direction vector / = (0,1) which

represents the preferred direction of travel for each individual in the

population. Because individuals move independently from one

another, the spatial distribution of the population will follow an advec-

tion–diffusion process(Skellam 1951; Patlak 1953; �Okubo & Levin

2001; Machta et al. 2013). If independent individuals’ movement

directions are temporally autocorrelated, such as would be the case

for correlated random walks, the population will converge to

advection–diffusion dynamics at time-scales longer than the range of

the temporal autocorrelation (Patlak 1953).

Advection–diffusion processes are governed by the relative values of

two parameters. The advection parameter ɛ represents the perceived

strength of the environmental gradient. The parameter d controls the

rate of diffusion, representing random movement not associated with

following the preferred direction.As ɛ increases while d is held constant,individual velocities become increasingly aligned in the preferred direc-

tion of travel, all else equal. Fixing ɛ, higher values of d can represent a

‘noisier’ environment, where each individual’s estimate of the preferred

direction of travel at a given time is increasingly uncertain. If a popula-

tion of independent individuals is released at the point (0,0) and

observed s seconds later, their spatial distribution will approximate a

bivariate Gaussian distribution, where the mean spatial location of an

individual (in cartesian coordinates) is given by ɛ/s = (0,ɛs) and vari-

ance and covariance in positions among individuals are given by the

variance–covariance matrix sdI, where I is the identity matrix. Note

that variance in location grows linearly with time in advection–diffu-

sion, which is why the variance–covariance matrix has a factor of s.

This process can be simulated by sampling from the appropriately

parameterizedGaussian distribution.

We calculated the order parameter wðtj; sjÞ on simulated advection–

diffusion data for a range of population sizes, strengths of environmen-

tal bias, and sampling periods (Fig. 1). This demonstrates the intuitive

result that ecological data on individual movements can display highly

ordered velocities even when individuals are moving independently.

This order can be generated by chance in smaller populations, where

there is a higher probability that velocities will be aligned by pure

chance (Fig. 1a), by a strong environmental gradient that dominates

the effect of noise, and by longer sampling intervals which reduce the

effect of noise by averaging it over a long time period (Fig. 1b). Detect-

ing collective behavior in ecological data therefore requires a different

statistic, that attains distinctly different values for independent indivi-

duals and collectives.

Define v0 as the expected value for vi if individuals were acting inde-

pendently. As a model for what independent individuals would do, v0

could be complex and we return to the issue of determining v0 below,

addressing in particular the case when the preferred direction, /, variesover space and time. To introduce our approach, we focus on the sim-

ple case where/ is constant over the period of observation. In that case,

if individuals are independent, the velocity of each individual at each

time is a random draw from the same probability distribution. We can

then estimate v0 by themean of all observed velocities and sowe set

v0 ¼ 1

MN

XM

j ¼ 1

XN

i ¼ 1

viðtj; sjÞ eqn 3

which includes all time points. In other words, if/ is constant, then the

overall mean velocity over time is a good model for the expected veloc-

ity of any one independent individual, at any time. This works because

independent individuals do not interact with one another and so, in

with a spatiotemporally invariant environmental gradient, their veloci-

ties are exchangeable across time points.

Now consider an adjusted velocity

�viðtj; sjÞ ¼ viðtj; sjÞ � v0ðtj; sjÞ eqn 4

and the corresponding adjusted order parameter

�wðtj; sjÞ ¼ 1

N

XN

i ¼ 1

�viðtj; sjÞ�viðtj; sjÞ��

��

�� eqn 5

which is identical to eqn 2 except that it is calculated with adjusted

velocities �vi instead of raw velocities vi. As with velocities vi and the

order parameterw, the adjusted order parameter �w is a function of time

t and sampling period s. However we will sometimes write these func-


32 B. D. Dalziel et al.

tions with their arguments suppressed for visual clarity. When we do

so, comparisons of the value of a function under different circum-

stances (e.g. ‘the value of �w is higher than . . .’) imply that the compar-

ison is being done at an arbitrary time point and sampling period,

unless otherwise indicated (Fig. 1c).

The value of the �w tends to be lower thanw for independent individu-

als, meaning that adjusted velocities, �vi, are less ordered than raw veloc-

ities vi for independent individuals. This is because the subtraction of v0

in eqn 4 removes some of the order that is due to individuals heading in

the same direction independently – either by random chance or because

of exposure to a common environmental gradient. As a result, �w is less

influenced by population size (Fig. 1c) and is unaffected by the strength

of the environmental gradient or the length of the sampling interval s

(Fig. 1d).

Whereas �w tends to be low for ecological data on independent indi-

viduals, we hypothesize that increasing levels of collective behaviour

lead to increases in the value of �w. Increased values of �w for populations

influenced by collective behaviour are not due to increases in their abil-

ity to travel in the true gradient direction (the effect of travelling further

up the gradient is removed by the subtraction of v0 in eqn 4), or

because collective behaviour increases alignment in the gradient direc-

tion (because an improved overall alignment is also captured by v0).

Rather, the proposed mechanism is that at any moment, a population

influenced by collective behaviour will have its own idiosyncratic devia-

tion from the true gradient direction, due to the propagation to larger

scales of interactions between nearby individuals.

These ‘collective mistakes’ cause adjusted velocities in collectives –

calculated using eqn 4) – to bemore polarized for groups of individuals

sampled at similar times, compared to individuals sampled at different

times. This leads naturally to a test for collective behaviour, based on

comparing �w to the value of a null statistic �w0, obtained by calculating�w on a randomized data set where the adjusted velocities for each indi-

vidual have been randomly shuffled in time. This breaks correlations

among individuals observed at the same (randomized) time. If such cor-

relations do not exist because individuals are moving independently,

the time randomization will not affect statistical properties of the data.

Ecological data on the distribution patterns of a single population over

time can be tested for collective behaviour by testing the hypothesis that�w [ �w0 (see Fig. 3a).

COLLECTIVE BEHAVIOUR MODEL

To demonstrate our method for detecting group decision-making in

ecological data, we simulated data from well-studied model for collec-

tive behaviour (Couzin et al. 2005). The model is implemented in an

unbounded environment and observed over a long period relative to

the time step of the simulation (the model steps forward 0�2 s at a time,

and we observe it for four simulation hours). As above, the environ-

ment has a gradient with a constant mean direction, and uniform noise

that is independent and identically distributed over space and time. The

environmental gradient represents the preferred direction of travel for

individuals when acting independently of social cues.

At each time step, individual velocities in themodel are given by

viðtþ hÞ ¼ hvðtÞii þ agi þ zi eqn 6

followed by rescaling to unit length

viðtþ hÞ ! viðtþ hÞjviðtþ hÞj eqn 7

where the vector viðtÞ is the velocity of the ith individual at time t, and

the time step of themodel is h. hvðtÞii represents the velocity chosen by iin response to the positions and velocities of its neighbours, as detailed

below. Individuals are constrained by a maximum turning angle, such

that the interior angle between viðtþ hÞ and viðtÞ can be atmost hmax.

The random variable gi represents the preferred direction of travel as

it is perceived by individual i at time t. As above, we assume a two-di-

mensional world in which the true preferred direction is the vector /.Each time step, an individual has access to a noisy estimate of the gradi-

ent that has unit magnitude and deviates from the true direction by an

angle hg, which is uniformly distributed on the interval ð�rg;rgÞ. Anindividual weighs gi in their final desired velocity according to the

gradient response parameter a. When a becomes large, individuals

move independently from one another.

Population size, N

Ord

er p

aram

eter

,

101 102 103

0·0

0·5

1·0

0·00 0·05 0·10 0·15 0·20

Ord

er p

aram

eter

, 0·

00·

51·

0

N = 16N = 256

4 h

10 min

Population size, N

Adj

uste

d or

der,

101 102 103

0·0

0·5

1·0

Environmental bias, (m s–1)

0·00 0·05 0·10 0·15 0·20Environmental bias, (m s–1)

Adj

uste

d or

der,

0·0

0·5

1·0

N = 16N = 256

(a) (b)

(c) (d)

Fig. 1. Order in the movements of indepen-

dent individuals exposed to environmental

gradients of varying strength with noise level

d = 1. Horizontal axes in each pane comprise

100 points, with 100 replicates per point.

Outer, lighter polygons extend vertically from

the 5th to the 95th percentile; inner darker

polygons encompass the interquartile range.

(a) Order parameter for populations of vary-

ing size in the absence of an environmental

gradient (ɛ = 0). (b) Order parameter for

populations subject to varying strengths of

environmental bias, for small (yellow) and

large (purple) populations, sampled after 10

minutes (lower curve) and four hours (upper

curve). Panes (c) and (d) show the same analy-

sis as (a) and (b) using the adjusted order

parameter. Variation in sampling time has no

effect on the value of the adjusted order

parameter for independent individuals, so the

polygons for each of the two sampling periods

are on top of one another.



The vector zi represents random error in velocity. zi is a randomly

chosen point on a circle centred at (0,0) with radius rz. The larger the

value of rz, the less an individual’s velocity is based on cognitive

responses to the environment or to the locations and velocities of its

neighbours. As rz becomes large, each individual performs a random

walk.

hvðtÞii is chosen based on the locations and velocities of i’s neigh-

bours as follows. Each time step, an individual’s first priority is collision

avoidance. If there are other individuals within the ball with radius ra,

representing the focal individual’s zone of avoidance, then hvðtÞiipoints away from themean direction to those individuals.

If there are no individuals within the focal individual’s zone of avoid-

ance, then hvðtÞii is based on the positions and velocities of neighbours

within the zone of social interaction, a ball with radius rs [ ra. hvðtÞiiis then the average of the vector towards the centroid of i’s neighbours,

and the vector representing the mean velocity of those neighbours.

hvðtÞii is always normalized to have unit magnitude.

An individual’s position changes over time according to

xiðtþ hÞ ¼ xiðtÞ þ fviðtþ hÞ eqn 8

where f is the speed of each individual. Note that spatial variation is

implicit in the model because at each time t, individual i is at a specific

location xiðtÞ. The preferred direction in the environment, and individ-

uals’ perceptions of their neighbours’ locations and velocities thus vary

spatially as well as temporally.

To summarize the model, simulated populations attempt to follow a

noisy environmental gradient with constant mean direction using a

mixture of individual- and group-level cognitive responses. The balance

between the two types of cognition is determined by the gradient

response parameter a, with increasing values of a representing increas-

ing independence among individuals. In each simulation, we obtained

ecological data at a coarse scale representing the net effect of many

movement decisions by recording the spatial distribution of the popula-

tion every 10 min for 4 h. The parameterizations we used are shown in

Table 1, and follow Couzin et al. (2005). The sensitivity of the emer-

gent dynamics of this model to variation in the parameter values has

been previously explored in Couzin et al. (2002). Although collective

behaviour can give rise to different complex patterns under different

parameterizations of the model and initial conditions, polarization

order in velocities among nearby individuals is a ubiquitous symptom

of collective movement. This polarization is the feature that is the focal

point for ourmethod.

In themodel presented here, the control parameter a is applied to theterm representing gradient response, gi. However, themodel could also

have also been formulated by applying the control parameter to the

term representing collective behaviour, hvðtÞii.We note that associating

a with gi in our formulation has a side effect, which is that as the rela-

tive strength of collective alignment increases, the relative contribution

of random error also increases, due to normalization of the velocities at

each simulation step. However, in all cases the magnitude of the ran-

dom error is small relative to the other terms in the equation. For exam-

ple, even when collective behaviour is very strong (alpha approaching

0), so that the relative contribution of noise is also strong, the magni-

tude of the noise vector is at most 2% of the overall magnitude of the

velocity. Moreover, increasing noise with increasing collective beha-

viour causes the residual velocities of nearby individuals to be less

polarized, reducing the value of our test statistic. Therefore, this side

effect of the structure of ourmodel renders the test more conservative.

In many applications, the preferred direction of travel / will vary

over space and time. In these cases, it will be necessary to learn how /changes and incorporate that into estimates of v0ðt; sÞ. This could be

done by straightforward estimates of the response of independent indi-

viduals to the state of the environment [for example, connecting gradi-

ents in light levels to the swimming speed of fish (Berdahl et al. 2013),

or to the velocity of phytoplankton (Mitbavkar & Anil 2004)]. As an

example of applying our approach in this context, we conducted a ser-

ies of two identical experiments using the collective behaviour model.

Each experiment consisted of running the collective behaviour simula-

tion with a gradient direction that changes deterministically over time.

Data from the first experiment were used to estimate the gradient direc-

tion over time, while data from the second experiment were used to test

for the strength of collective behaviour. From the first experiment, we

estimate the gradient direction over time by fitting a cubic spline to the

velocity data from all individuals. We then apply our approach for

detecting collective behaviour to the velocity data from the second

experiment using the spline-smoothed velocity from the first experi-

ment as v0ðt; sÞ. As above, this approach uses small but significant col-

lective deviations from the true gradient directions (which is estimated

independently) to detect collective behaviour. In some applications

(such as the cariboumigration analysis), one set of observations is suffi-

cient for both detecting the gradient and analysing collective deviations

from it.

DETECTING COLLECTIVE BEHAVIOUR IN CARIBOU

MIGRATION PATTERNS

We applied our method to data on the migration patterns of the Riv-

i�ere-aux-Feuilles caribou herd inNorthernQu�ebec, Canada (LeHenaff

1976). The herd has varied in size from approximately 56 000 individu-

als in 1975 to at least 628 000 in 2001, to approximately 430 000 in

2011 (Le Henaff 1976; Couturier et al. 2010; Taillon, Festa-Bianchet &

Cot�e 2012). These caribou usually overwinter in the boreal forest in the

southern Ungava peninsula. Each spring they migrate up to approxi-

mately 1200 km to calving grounds located on the northern part of the

peninsula, in tundra. Tundra is a highly seasonal environment, and the

arrival on the calving ground is synchronized with the peak of produc-

tivity of the vegetation at the onset of the short growing season (Post &

Forchhammer 2008). Almost all females return to the same calving

ground each year (Boulet et al. 2007).

The data consist of 9721 observations of the locations of 143 caribou

observed over three years (2008–2011). Caribou were captured using

net guns fired from a helicopter and handledwithout chemical immobi-

lization. The data were collected using Argos tracking collars (Service

Table 1. Simulation parameters

Parameter Interpretation Value

h Time step 0�2 s

ra Radius of avoidance 1 m

rs Radius of social interaction 6 m

f Speed 1 m s�1

hmax Maximum turning angle 2 rad

hg Environmental noise 2�5hz Individual noise 0�02N Population size 256

M Number of replicates 10

s Sampling period 10 min

a Gradient response

parameter

0�125, 0�25,0�5, 1, 2, 4, 8, 16, 32

Initial positions Uniformwithin

a 30 9 30 m square



Argos Inc., Largo, MD, USA) that record the locations of animals

every 5 days (120 h � 1�66 SD). The median observation period for a

single animal in the data is 320 days, with approximately 30 unique

individuals observed on average during any month. These data repre-

sent an unbiased subsample of the movement patterns of the herd that

is small relative to the size of the herd, but large relative to most empiri-

cal studies of animalmovement patterns to date.

We assumed caribou respond independently to environmental and

physiological migratory cues that vary spatially and seasonally, in addi-

tion to a potential influence of collective behaviour. We therefore esti-

mated v0ðx; tÞ – the expected velocity of independent caribou as a

function of space and time – by smoothing the observed velocities of

caribou observed in similar spatial locations at similar times of year,

using a weighted average. Based on preliminary analysis, the weighted

average for a given location and time of year was calculated using tricu-

bic kernel (Hastie, Tibshirani & Friedman 2008) with a spatial band-

width of 170 km and a temporal bandwidth of 30 days. The results of

this analysis are qualitatively identical using other kernel functions and

over awide range of other spatial and temporal bandwidths.

After calculating v0, the application of our method for detecting col-

lective behaviour consists of obtaining adjusted velocities by subtract-

ing velocities predicted by v0ðx; tÞ, from observed velocities for the

caribou. We then calculate �w using eqn 5. In order to test whether the

strength of collective behaviour varied seasonally and spatially, we cal-

culated the value of �w in a spatiotemporal neighbourhood surrounding

each observed caribou location, by including other caribou velocities

that were observed within 20 km and 24 h of that point. We chose

these inclusion criteria to minimize the spatial and temporal widths of

the ‘bins’ for �w (thus focusing on caribou whoweremore likely to actu-

ally interact), while simultaneously maintaining enough observations

at a particular location and time to apply the method. As above, the

results of this analysis are qualitatively identical using over awide range

of other spatial and temporal bandwidths. In general, the appropriate

smoothing function and bandwidths for ourmethodwill depend on the

sampling properties of the data. As the final step in the analysis, we cal-

culated the null statistic �w0 using a repeated randomization procedure

by sampling with replacement from the entire data set the same number

of velocities as were included in that calculation of �w for each caribou.

Results

DETECTING COLLECTIVE BEHAVIOUR IN SIMULATED

DATA

At the maximum value of the gradient response parameter we

examined (a = 32), individual velocities are nearly indepen-

dent. Correspondingly, populations at that level of a follow

the spatiotemporal patterns predicted by advection–diffusion(Fig. 2a,c,d). In particular, in multiple replicate runs of the

model, populations of independent individuals tend to the

same broad-scale spatial distribution in all replicates, because

X (m)

Y (

m)

X (m)

Y (

m)

X (

m)

Gradient response, α

Y (

m)

(a) (b)

(c)

(d)

Fig. 2. Contrasting collective behaviour with

the aggregated responses of independent indi-

viduals using ecological data. (a,b) Distribu-

tion over space and time for model individuals

released at the origin and heading upwards in

a noisy environmental gradient. The cool col-

ours (blues and greens) show 10 replicate

model runs for nearly independent gradient

followers (a = 32). The hot colours show 10

replicate runs for populations that exhibit col-

lective behaviour (a = 0�5). (c) Distribution

perpendicular to the gradient direction at the

end of four hours for varying levels of collec-

tive behaviour, with each replicate for a given

value of a shown side by side. Boxes enclose

the interquartile range, and lines enclose the

entire range of the data. Colours correspond

to those in the previous panes. (d)Distribution

parallel to the gradient direction.



the velocities of independent individuals are exchangeable over

space and time. In contrast, lower values of the gradient

response parameter lead to systematic differences in velocity

over time and among replicates (Fig. 2b). These systematic dif-

ferences are driven by social interactions among neighbouring

individuals that scale up to cause population-level idiosyn-

cratic deviations from the preferred direction of travel – ‘collec-tive mistakes’ (Fig. 2b,c). At the same time, scaling up local

conspecific interactions is what advantages collectives over

independent individuals in variable environments, enabling

populations with collective behaviour to travel more quickly

and precisely in the preferred direction of travel (Fig. 2d). This

effect persists until values of a become so low that individuals

cease to respondmuch to the gradient. In this case, the popula-

tion still ‘drifts’ in the direction of the gradient, while maintain-

ing a highly heterogeneous spatial distribution (e.g. a = 0�125in Fig. 2c,d).

Because populations of independent individuals also align

to follow the gradient, alignment of movements in the gradient

direction is not sufficient evidence for collective behaviour in

ecological data. However, alignments that involve broad-scale

group-level deviations from the mean gradient direction do

have a higher predictive value for identifying collective beha-

viour, particularly when the underlying behaviours are not

observed. Group idiosyncratic deviations from the mean pre-

ferred direction generate significantly more order in observed

velocities at a particular time, compared with average velocity

over time, leading to increased values for �w in populations

influenced by collective behaviour (Fig. 3). The collective mis-

takes that produce this difference are the result of patterns in

which a large portion of the population travels in a certain

common direction at a particular time, but where that direc-

tion varies randomly over time. These dynamics are highly

unlikely for populations of independent gradient followers,

where independent trajectories, by definition, are as likely to

show similarity within a given sampling period as among sam-

pling periods.

The adjusted order parameter �w is correlated with the gradi-

ent response parameter a (R2 ¼ 0 � 49, P < 0�0001, for linearregression of �w as a function of log a, with each observation

time in each replicate as a single data point; R2 ¼ 0 � 5,P < 0�0001 on average when the analysis is done on a single

replicate). While the value of �w fluctuates over time points and

replicates, populations with the strongest collective behaviour

(0�125 ≤ a ≤ 0�5) are clearly distinguishable from those with

low levels of collective behaviour (8 ≤ a ≤ 32) based only on

values of �w(Fig. 3a,b). In populations with intermediate levels

of collective behaviour, where the influence of the environmen-

tal gradient on individual velocities is at least as strong as that

of social interactions, but not overwhelming (1 ≤ a ≤ 4), �wattains intermediate values (Fig. 3a). In some of these interme-

diate cases, the value of �w varies systematically over time, due

to long transient patterns caused by the aggregation of the

population in the initial conditions (Fig. 3c). In particular,

when the influence of collective behaviour is weak, such as

when the environmental gradient is twice as strong as the influ-

ence of collective behaviour (a = 2), we found that long tran-

sient patterns caused by the aggregation of the population in

the initial conditions sometimes inflated the value of �w0 relative

to its value at other levels of a.We measured the statistical power of this test as follows.

First, for each replicate simulation at each level of a, we com-

puted the average value of �w over all times.Next, we calculated�w0 in each of M = 100 randomizations of the sampling times

for each individual. These were also averaged over all times,

yieldingM comparison average values of �w0 for each value of

0·125 0·25 0·5 1 2 4 8 16 32Gradient response,

Adj

uste

d or

der,

(t) o

r 0(t)

0·0

0·5

1·0

Time (min)

Adj

uste

d or

der,

(t)

0 60 120 180 240

0·0

0·5

1·0

Time (min)0 60 120 180 240

(a)

(b) (c)

Fig. 3. Detecting collective behaviour in eco-

logical data. (a) Adjusted order (�w) in the spa-

tial distribution of simulated populations with

different levels of collective behaviour. Filled

boxes enclose the interquartile ranges for the

distributions of �w across all sampling times

and replicates. Lines extend from the 5th to

95th percentiles of the distributions. Hollow

boxes show the analogous distribution for the

null parameter �w0. (b) Adjusted order over

time for populations strongly influenced by

collective behaviour (a from 0�125 to 0�5; tri-angles) and weakly influenced by collective

behaviour (a from 8 to 32; circles), averaged

over replicates. Colours correspond to those

in (a). (c) Average adjusted order over time for

populations with intermediate levels of collec-

tive behaviour (a = 1, dotted line and crosses;

a = 2, solid line and squares; a = 4, solid line

and diamonds).



the average �w calculated in the first step. For each replicate

simulation at a given level of a, we then compared the average

value of �w to the 95th percentile of the comparison average val-

ues of �w0, recording for each level of a the frequency with

which the average value of �w exceeded the 95th percentile of

the distribution of average �w0s over replicates. For all levels of

a except one (a = 32; i.e. the response to the environmental

gradient is 32 times stronger than collective behaviour), this

frequency was 1�0, indicating our test is very sensitive. Even

with almost no collective behaviour (a = 32), the frequency

with which �w exceeded the 95th percentile of the null distribu-

tion was 0�8. Thus, individuals must be moving almost per-

fectly independently for our test to fail detect the consequences

of collective behaviour.

Although individual movements may be not be perfectly

independent, the presence of biologically significant levels of

collective behaviour is a separate question. As is always the

case when assessing the biological significance of a statistical

test, appropriate criteria for biologically significant levels of

collective behaviour will depend on the system in question.We

suggest as a starting point that appropriate criteriamay be if (i)�w is high, and (ii) the interquartile ranges of �w and �w0 do not

overlap. These criteria are uniquely associated with strong col-

lective behaviour in the simulated data (Fig. 3a). Correspond-

ingly, we use these criteria in the analysis of the caribou

migration patterns, described below.

We tested the robustness of our method to unobserved indi-

viduals by repeating the analysis on randomly selected subsam-

ples of the simulation data (Fig. 4). Downsampling causes a

modest decrease in the specificity of our test for collective beha-

viour (decreasing the probability that independent populations

are correctly identified) but does not significantly affect the sen-

sitivity of the test (the rate at which populations with collective

behaviour are correctly identified). More specifically, sparsely

sampled populations of nearly independent individuals

(a = 32) show higher adjusted order as a result of sampling

effects as sample size becomes small, increasing the chance of

being falsely identified as strongly influenced by collective

behaviour. However, the decrease in the specificity due to

unobserved individuals is modest. For instance, in populations

of nearly independent individuals, mean adjust order remains

below 0�5 even if only a few individuals are observed on aver-

age per time point. By contrast, in populations exhibiting col-

lective behaviour, adjusted order remains higher than for

populations of independent individuals, even if only a few indi-

viduals in the population are observed.

Figure 5 shows the application of our approach to detecting

collective decision-making in a temporally shifting environ-

mental gradient, when two experimental runs are available –one to estimate the behavioural response to the environmental

gradient and one to test for the strength of collective beha-

viour. The velocities of individuals closely track the gradient as

it shifts direction over time (Fig. 5a). As a result, the average

velocity of the population over time can be used to reconstruct

the gradient. Then, in a second experiment we can test for

group-level deviations from expected velocity over time

(Fig. 5b), which can reliably distinguish between populations

influenced by collective behaviour and populations of indepen-

dent gradient followers.

THE DYNAMICS OF COLLECTIVE BEHAVIOUR IN

MIGRATORY CARIBOU

We validated the statistical model for the velocities of indepen-

dent caribou, v0ðx; tÞ, by releasing virtual non-interacting par-ticles into the velocity field given by v0ðx; tÞ, numerically

integrating their positions forward in time, and observing the

population’s redistribution patterns over the same range of

time covered by the caribou data. Particle positions were

updated daily in the simulations, which were run forward for

three years from initial conditions, allowing ample opportunity

for errors in predicted position to accumulate. The indepen-

dent particles nonetheless precisely reproduce the broad-scale

relocation patterns of the herd (Fig. 6). This means that

v0ðx; tÞ for the caribou is a reasonable model for the advec-

tion–diffusion component of caribou migration, not a ‘straw

man’: the model was fitted to velocity data, and in the simula-

tions, there is ample opportunity for errors to accumulate as

position is integrated forward in time.

We found strong evidence of collective behaviour in the

adjusted velocities of nearby caribou, once the expected effects

of environmental and/or physiological gradient had been sub-

tracted (Fig. 7). During the spring and fall migrations, and

shortly after calving, the distribution of the adjusted order

parameter �w in the caribou data was significantly higher than

the null statistic �w0, measured by examining the degree of over-

lap in the interquartile ranges. During these times of year, the

degree of alignment of in adjusted velocities of nearby caribou

is thus consistent with strong collective behaviour. Intrigu-

ingly, the dynamics of collective behaviour vary seasonally in

the caribou in a systematic way. Oneway of looking at it is that

Proportion remaining, pMea

n ad

just

ed o

rder

und

er d

owns

ampl

ing

0·0

0·5

1·0

100 10–1 10–2

256 26 3Number remaining

= 0·5

= 32

Fig. 4. Behaviour of the adjusted order parameter when some indi-

viduals are unobserved. Lines show the average value of �w across

replicates for populations showing collective behaviour (a = 0�5;warm colours) and populations of nearly independent gradient fol-

lowers (a = 32; cool colours), when a proportion of the individuals

are randomly removed from the analysis. Each of the 10 lines of a

given colour and style corresponds to a repetition of the analysis on

a different replicate simulation. Each line is composed of 30 points,

which are each derived from a different random sample of the

original simulation output.



at some times of year (for example, during migration, or fol-

lowing calving) if you ‘zoom in’ on nearby caribou, transition-

ing from the broader bandwidths associated with advection–diffusion the narrower interaction neighbourhoods associated

with collective behaviour, nearby caribou, are much more

ordered than expected for independent particles. Yet at other

times of year (for example, in February, when the herd tends to

be more sedentary), the level of order in their velocities does

not change much when zooming in on nearby individuals, sug-

gesting more independence in the trajectories of nearby indi-

viduals.

Discussion

Collective decisions emerge by the propagation of local beha-

vioural interactions to broader scales, influencing the spatial

and temporal dynamics of populations. As such, the approach

we propose for detecting collective behaviour in ecological

data – using broad-scale collective deviations from the mean

gradient direction that would be unlikely for independent indi-

viduals – rests on a fundamental property of collective beha-

viour (Vicsek et al. 1995; Couzin et al. 2002). Our

contribution is to identify specific quantitative features of this

process that are observable in ecological data, where only a

fraction of individuals are observed and the time between

observations is much longer than the time-scale of individual

behavioural decisions. Without recourse to fine-scale observa-

tions of individual behaviour, the approach we describe can,

under some conditions, reject the null hypothesis that the data

were generated by independent responses to a common envi-

ronment (such as a chemical gradient), or to physiological

stimuli operating independently among individuals (such as

physiological responses to photoperiod).

Ultimately, research on the causes and consequences of

collective behaviour requires identifying the underlying

mechanisms that drive its emergence, maintenance and

dynamics. However, discovering the ecology of collective

behaviour in nature also requires methods for learning about

its prevalence in populations that are not exhaustively sam-

pled at the resolution of individual behaviour. As in the

study of ecological competition, or evolutionary adaptation,

pattern-oriented ‘top-down’ approaches to studying collec-

tive behaviour can complement ‘bottom-up’ mechanistic

approaches, and the most exciting discoveries often involve a

combination of both (Sumpter, Mann & Perna 2012). To

complement high-throughput ethological approaches in labo-

ratory and wild populations, our approach has the power to

screen for collective behaviour where fine-scale behavioural

data have not yet been collected, with the potential to diver-

sify and enlarge the set of populations where collective beha-

viour is considered.

The migratory caribou we study display large-scale seasonal

variation in the level of order in their velocities that cannot be

parsimoniously explained by independent responses to season-

ally fluctuating physiological cues or a seasonally and spatially

fluctuating environment. Particularly during migration, cari-

bou velocities are significantly influenced by the velocities of

nearby individuals, in addition to the physiological/environ-

mentally driven advection field they are each exposed to. Col-

lective behaviour may therefore play an important and

dynamic role in animal migration patterns –more so than has

been previously shown.

–0·0

40·

000·

04

Time (min)

Adj

uste

d ve

loci

ty, v

(t)

10 30 50 70 90 110 140 170 200 230

0·0

0·4

0·8

Adj

uste

d or

der

0

0·0

0·2

0·4

0·6

0·8

1·0

Adj

uste

d or

der

050 100 150 200Time (min)

Dire

ctio

n2

02(a)

(c)

(b)

Fig. 5. Detecting collective decision-making

in a temporally shifting environmental gradi-

ent. (a) Direction of travel over time for a pop-

ulation with a = 0�5. Vertical lines encompass

the range of directions of travel for individuals

observed at a particular time. The thick grey

curves shows the true direction of the gradient.

The red line shows a cubic spline fit to the

velocity data. (b) Adjusted order and null

order over time in a second replicate realiza-

tion of the process, where velocities were

adjusted by subtracting the spline fit to the

velocity data in the first replicate (as shown in

a). The inset in (b) shows the analogous results

for a population of nearly independent gradi-

ent followers (a = 32). (c) shows the distribu-

tion of the horizontal component of the

adjusted velocities in the second experiment,

showing collective deviations from the true

gradient direction as estimated from the data

of the first experiment.



N

S

EW

1 km h–1

Day of year

Nor

thin

g (k

m)

0 100 200 300

060

012

00

Day of year

Eas

ting

(km

)

0 100 200 300

060

012

00

(a)

(b) (c)

Fig. 6. Migration patterns of the Rivi�ere-aux-Feuilles caribou herd. The inset globe shows location of the study area. The two larger maps show the

locations and velocities of GPS-collared caribou observed during the spring (May) and fall (October) migrations, pooled across years from 2008 to

2011. The style of the points shows each caribou’s velocity, according to the legend on the left. The bottom panels show time series of individual loca-

tions as a function of time of year. Points are median locations; vertical lines enclose the interquartile range, based on running quantiles calculated

using a 7-day non-overlapping window. Grey polygons enclose the analogous interquartile range for simulations of independent particles released

into the velocity field v0ðx; tÞ fitted to the data.

0 100 200 300Day of year

Adj

uste

d or

der i

n ca

ribou

vel

ocity

, 0·

00·

51·

0

Springmigration Calving

Fallmigration

Fig. 7. The dynamics of collective behaviour inmigratory caribou. Systematic seasonal variation in the adjusted order of caribou velocities, �w (upper

time series, in orange) showing elevated levels of collective behaviour concurrent with the timing of migration and immediately following reproduc-

tion. The shaded area shows the interquartile range, and the central line the median, of the observed adjusted order across individual caribou

observed at a given time of year, based on running quantiles calculated using a 7-day non-overlapping window. The lower time series, in blue, shows

the same information for �w0, which is calculated on time-randomized adjusted velocities, thus breaking correlations among nearby individuals that

could have been caused by collective behaviour. High values for �w and non-overlapping interquartile ranges of �w and �w0 (indicated by the horizontal

red lines) are consistent with strong collective behaviour (see also Fig. 3a).



Fluctuations in the level of order within groups of nearby

caribou indicate that the influence of collective behaviour on

caribou relocation patterns is dynamic. These fluctuations

coincide with reproduction, suggesting that collective beha-

viour is not just important for relocation patterns but can be a

dynamic part of the life history of animal populations. We do

not know what ecological processes cause the spike in collec-

tive behaviour after calving each year – perhaps it could be

related tomovement to the summer grounds following calving,

where the herd is led to by certain experienced females.

Further applications for our approach include understand-

ing the emergence and stability of seasonal migration patterns

in other systemswheremigrationmay originate fromamixture

of environmental/physiological stimuli and collective beha-

viour. For example, the migration patterns of herring (Clupea

harengus) may depend on ocean currents and food availability

(Jorgensen et al. 2005) as well as on juveniles learning the

migration routes by following older age classes (Huse 2002). In

humans, daily movement patterns in cities show the influence

of the external (built) environment as well as the effect of social

processes, such as when a significant fraction of people com-

mute to work in a few specialized areas of the city (Bettencourt

et al. 2007; Batty 2008; Dalziel, Pourbohloul & Ellner 2013).

The caribou migrations studied here may therefore represent

the tip of the iceberg of the ecological significance of collective

behaviour in wild populations. While a ‘bottom-up’ approach

to detecting collective behaviour in these populations is often

limited by the availability of fine-scale behavioural data, the

signal of collective decision-making is, under some conditions,

detectable in coarser scale ecological data.

Data accessibility

The authors of this paper do not own the data used, and permission to

archive the data was not granted. Requests to access the data may be

directed to Caribou Ungava: http://www.caribou-ungava.ulaval.ca/en/accueil/.

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Received 1 April 2015; accepted 9 June 2015

Handling Editor: JasonMatthiopoulos



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