Intermittent motion in desert locusts: behavioural complexity in simple environments.

Abstract

Animals can exhibit complex movement patterns that may be the result of interactions with their environment or may be directly the mechanism by which their behaviour is governed. In order to understand the drivers of these patterns we examine the movement behaviour of individual desert locusts in a homogenous experimental arena with minimal external cues. Locust motion is intermittent and we reveal that as pauses become longer, the probability that a locust changes direction from its previous direction of travel increases. Long pauses (of greater than 100 s) can be considered reorientation bouts, while shorter pauses (of less than 6 s) appear to act as periods of resting between displacements. We observe power-law behaviour in the distribution of move and pause lengths of over 1.5 orders of magnitude. While Lévy features do exist, locusts' movement patterns are more fully described by considering moves, pauses and turns in combination. Further analysis reveals that these combinations give rise to two behavioural modes that are organized in time: local search behaviour (long exploratory pauses with short moves) and relocation behaviour (long displacement moves with shorter resting pauses). These findings offer a new perspective on how complex animal movement patterns emerge in nature.

Full text

Page 1

Intermittent Motion in Desert Locusts: Behavioural
Complexity in Simple Environments
Sepideh Bazazi1*, Frederic Bartumeus2, Joseph J. Hale1, Iain D. Couzin3
1Department of Zoology, University of Oxford, Oxford, United Kingdom, 2Center for Advanced Studies of Blanes CEAB-CSIC, Girona, Spain, 3Department of Ecology and
Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America
Abstract
Animals can exhibit complex movement patterns that may be the result of interactions with their environment or may be
directly the mechanism by which their behaviour is governed. In order to understand the drivers of these patterns we
examine the movement behaviour of individual desert locusts in a homogenous experimental arena with minimal external
cues. Locust motion is intermittent and we reveal that as pauses become longer, the probability that a locust changes
direction from its previous direction of travel increases. Long pauses (of greater than 100 s) can be considered reorientation
bouts, while shorter pauses (of less than 6 s) appear to act as periods of resting between displacements. We observe power-
law behaviour in the distribution of move and pause lengths of over 1.5 orders of magnitude. While Le ´vy features do exist,
locusts’ movement patterns are more fully described by considering moves, pauses and turns in combination. Further
analysis reveals that these combinations give rise to two behavioural modes that are organized in time: local search
behaviour (long exploratory pauses with short moves) and relocation behaviour (long displacement moves with shorter
resting pauses). These findings offer a new perspective on how complex animal movement patterns emerge in nature.
Citation: Bazazi S, Bartumeus F, Hale JJ, Couzin ID (2012) Intermittent Motion in Desert Locusts: Behavioural Complexity in Simple Environments. PLoS Comput
Biol 8(5): e1002498. doi:10.1371/journal.pcbi.1002498
Editor: John M. Fryxell, University of Guelph, Canada
Received October 21, 2011; Accepted March 13, 2012; Published May 10, 2012
Copyright: ? 2012 Bazazi et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors acknowledge support from the Natural Environment Research Council (S.B.), the Spanish Ministry of Science and Innovation: MICINN-RyC
2009-04133 and BFU2010-22337 (F.B.) Searle Scholars Award 08-SPP-201 (I.D.C.), National Science Foundation Award PHY-0848755 (I.D.C.), Office of Naval
Research Award N00014-09-1-1074 (I.D.C.) and a DARPA Grant No. HR0011-09-1-0055 (to Princeton University) and an Army Research Office Grant W911NG-11-1-
0385 (I.D.C.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: sbazazi@gmail.com
Introduction
An essential focus of experimental and theoretical studies of
animal movement is to reveal the underlying drivers (internal and
external) of the complex statistical patterns of animal motion that
appear in nature [1]. Such patterns can be considered to emerge
as a result of interactions between organisms and their environ-
ment [1], or they may be directly the mechanism by which
behavioural processes are governed [2,3,4]. Examining how
animals move and the properties of their movement at different
scales is critical in understanding the drivers of the complex
patterns found in animal movement, and one of the main goals of
ecological research [5].
Observations of animal locomotion have shown that intermit-
tent movements are a common feature [6,7,8]. Movement is not
constant or continuous but rather intrinsically discrete, interrupted
by accelerations, decelerations or pauses [9]. Intermittent loco-
motion can be found in terrestrial, aquatic and aerial environ-
ments and occur in a range of ecological contexts, such as search
behaviour, habitat assessment or the pursuit of prey [6]. It implies
that animals can discretize their movement behaviourally in a
series of move lengths, pauses, and turns in response to certain
cues of the changing environment [2,6,8,10]. Behavioural
intermittence may perhaps be due to an animal’s energetic
restrictions, to allow an animal to recover from fatigue, for prey
detection [6], or for navigation, such as path integration [11].
Interruptions to continuous motion are also thought to be
adaptive in search processes, resulting in increased search efficiency
[9,10,12]. They can facilitate sharp reorientations that may break
the animal persistence of its previous directional motion and,
depending on the temporal pattern, can thereby allow it to explore
effectively an area [2,9,10]. Such a process may be beneficial to
animals living in dynamic and fluctuating environments, where
situations are likely to change as time progresses [13]. This idea that
interruptions might be adaptive by enhancing behavioural plasticity
is not new [14], but is yet to be explored empirically for animal
movement behaviour. In addition, if there is alternation of scanning
and non-scanning phases, the search process itself becomes
intermittent. Theoretical models have shown that random searches
withoptimalproportionsofscanning/non-scanningphasesenhance
encounter success [15,16,17].
Behavioural intermittence appears as an essential characteristic
of the movement patterns exhibited by many animals [8] and has
long since been documented [18], and detailed experiments on
intermittency are becoming increasingly common [7,19,20,21,22,
23,24,25]. Therefore there is limited knowledge about the causes
of movement: is an organism’s motion internally governed or a
reflection of their external environment? Thus far, few studies
have examined long-term animal motion under limited external
cues and there is a limited understanding of the null movement
patterns of motion without contributing external influences
[26,27].
PLoS Computational Biology | www.ploscompbiol.org1May 2012 | Volume 8 | Issue 5 | e1002498

Page 2

One of the greatest challenges of movement ecology is linking
the statistical properties of movement to specific behaviours [28]
and identifying behavioural transitions in the movement patterns.
To achieve this, an elemental view of the movement path is
needed, with identification of all displacements and pauses in a
trajectory and associating these to the behaviour of the organism
[1]. Getz and Saltz (2008) suggested identifying the potential
determinants of movement using canonical activity modes (CAMs)
consisting of shorter duration fundamental movement elements
(FMEs) [29]. Behavioural modes have previously been identified in
elk, defined by relocation distances and turning behaviour [30],
and switching between different behavioural modes has been
observed at various spatiotemporal scales [31]. Such studies
highlight the importance of examining high temporal resolution
data over different scales in order to identify the mechanistic
determinants of movement.
Previous ecological studies often involve short length (spatial or
temporal) empirical data, which makes it difficult to assess the
statistical properties of movement behaviour [32,33,34,35] or,
more recently, contain high resolution long length movement data
of animals in their natural environment but under uncontrolled
conditions [19,22,24,25,36,37,38]. Unravelling the drivers of the
complex statistical patterns (including intermittence) observed in
animal movement requires high resolution data on animal
movement over large spatiotemporal scales [4,12], and controlled
conditions (for example, in the absence of strong environmental
fluctuations or interactions with other individuals).
Here we examine the movement of isolated individual juvenile
desert locusts, Schistocerca gregaria, in a homogeneous experimental
arena, thus minimizing environmental fluctuations that may
influence motion. Desert locusts are typically found in relatively
barren land where the location of resources may be scarce and/or
unpredictable, hence, the experiment depicts a common ecological
situation of the species. We record locusts’ movements by locating
them at a fine temporal resolution (every 0.2 s) for 8 h. Under
such simple conditions, we consider in detail the nature of
behavioural intermittency and provide a comprehensive view of
the complex structure and the long-term variability of the
intermittent patterns observed in locusts. We quantify the role of
pauses as a turning (reorientation) mechanism, and we examine
the distribution of move and pause lengths. We also quantify short
and long-term correlation properties of moves and pauses,
unveiling the overall organization of move and pause sequences.
The analysis allows us to determine the relationship between the
key features of locusts’ movement: moves, pauses and reorienta-
tions, and therefore to understand how complex search patterns
are generated by organisms under minimal external sensory
stimuli.
Materials and Methods
Experiments
Healthy, intact freshly moulted gregarious desert locusts
(Schistocerca gregaria) in the 5th(final nymphal) instar, reared under
conditions described in Roessingh et al. [39], were placed in groups
of 20 individuals per plastic cage (30620610 cm), each with a
mesh roof, containing sawdust, an expanded aluminium perch and
a water supply. These were fed one of three dry, granular synthetic
diets ad libitum for 48 h, as described in [40]. We found no
significant differences among diets on the frequency distribution of
moves and pauses (comparing the power-law scaling exponent, m
among diets; ANOVA: F(2,90)=0.041, p=0.959, data were log
transformed to achieve normality; and ANOVA: F(2,90)=1.906,
p=0.154, respectively). Furthermore, Bazazi et al. (2011) previ-
ously found that nutritional state has minimal influence on the
proportion of time spent moving and on the speed of isolated
locusts. Marching and feeding behaviour are low and irregular
24 h post moult [18] but by 48 h locusts have high and uniform
marching and maintain a high food intake.
After 48 h a single locust was placed in a ring-shaped
experimental arena (80 cm diameter, walls 52.5 cm high and a
central dome 35 cm diameter [41]). 40 W fluorescent lamps
illuminate the arena and reduce visual stimuli available to locusts
above the arena. This setup effectively simulates a large featureless
environment within a reasonable space both for experimental
tractability and for the purposes of tracking, which has inherent
restrictions due to resolution constraints. The motion of the locusts
in the arena was then filmed for 8 h using a digital video camera
(Canon XM2). Automated digital tracking software [41,42], which
captured images at a rate of 5 times per s, was used to analyse the
video footage and obtain information regarding the position, speed
and direction of an individual between successive frames. Each
trial was started in the morning between 9:00AM–10:00AM. We
carried out a total of 93 experimental trials (93 individuals). A
video clip of an experimental trial is available in Supporting
Information (Video S1). No individual was used more than once.
Data analysis
Intermittent movement: moves and pauses.
served motion of individual locusts was made up of moves and
pausing bouts of variable length (see Figure 1). Thus the motion of
individuals can be discretized into a series of moves and pauses
with ‘‘moving’’ defined as displacement greater than 0.3 cm
between successive frames (0.2 s) and a pause as displacement less
than or equal to 0.3 cm (during which a locust can show resting or
fidgeting behaviour [43]). The threshold for moving was calculated
by plotting histograms of locusts’ speeds between successive frames
and selecting the speed just below the second peak in the
distribution (the first peak was at speed=0). This threshold is
similar to that used in Bazazi et al. [40,41] and Buhl et al. [42].
Using these criteria we determined whether a locust was moving in
each frame, and therefore the duration of moves and pausing
bouts. Data from individuals that were found within 3 cm of the
outer wall and central dome were excluded from the analysis to
remove edge effects (analysis with the inclusion of data from
The ob-
Author Summary
The movement of organisms is an essential feature of life
and is fundamental to almost all ecological and evolution-
ary processes. The motion of animals can have a significant
impact on the environment, for example on the distribu-
tion of resources, habitat fragmentation or the spread of
pests and diseases. Locusts exhibit one of the most
devastating examples of animal movement, where locust
swarms are a significant global pest. Therefore identifying
the mechanisms of such movements is critical in under-
standing a range of ecological processes. An important
challenge in studying animal motion is identifying the
drivers of the complex movement patterns generated by
organisms. Movement patterns may be the result of
interactions between animals and their environment or
may be directly the mechanism by which their behaviour is
governed. Here we examine the movement behaviour of
individual desert locusts in a homogenous experimental
arena with minimal external cues. These findings offer a
new perspective on how complex animal movement
patterns emerge in nature.
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org2May 2012 | Volume 8 | Issue 5 | e1002498

Page 3

individuals found within 3 cm of the arena wall show similar
qualitative results for the distribution of move and pause lengths
and suggest that move length truncation might be an intrinsic
property of locusts’ movement rather than simply an artefact of the
experimental design- see Figure S1).
In order to understand the behaviour of an individual during
pauses, we examined the number of changes in direction during a
pause for all pauses, for all locusts. Our experimental setup,
consisting of a circular arena and central dome, meant locusts
were able to move continuously around the arena, and allowed us
to reduce the system to a one dimensional representation of locust
movement (as in Buhl et al. [42]). Thus we determined whether a
locust showed a change in direction (turn) by examining whether it
switched its head direction from clockwise (CW) to anti-clockwise
(ACW) or vice versa between time steps. To do this we calculated
the change in the sign of the cross product of its positions between
successive frames. Thus we examined whether or not there had
been a change of direction, from CW-ACW movement, rather
than measuring the turning angle. We defined a turn as a change
from CW to ACW movement or vice versa. We quantified the
CW-ACW switching behaviour within pauses and moves (see also
Figure 2A) and the proportion of CW-ACW switches within a
pause/move. This information allowed us to compare the
probability of turning and the proportion of turns within moves
and pauses, and therefore determine whether a pause can be
considered a reorientation bout. Furthermore we determined
whether there had been a change from CW to ACW movement or
vice versa between the time steps immediately before and after a
pause in order to see how the duration of a pause affects this
probability.
Behavioural mode analysis.
relationship between moves and pauses, we carried out correlation
analyses for move lengths, pause lengths and between moves and
pauses (see Text S1 for details). In addition we classified local
search and relocation behavioural modes based on a partial sums
In order to understand the
(PS) approach [44]. The PS algorithm is a form of the Cumulative
Sum Analysis [45], widely used in many disciplines (e.g. industrial
engineering, economics, and medicine) to analyse the deviations of
a process from a target or reference value. This method uses a
cumulative sum equation to generate a sequence of observations
(time series), which is then analysed to identify the main transitions
between different phases/modes/regimes in the variable of
interest. The PS algorithm can allocate sequences of moves and
pauses into two behavioural modes: i) local search, i.e. sequences
of long pauses and short moves, and ii) relocations, i.e. sequences
of long moves and short (non-turning) pauses.
We have adapted Knell and Codling (2011)’s algorithm [44] in
the following way. For each experiment, we modified the move
and pause length time series by assigning negative signs to pauses
and positive signs to moves. We used the value T=0 as the
reference value in order to unambiguously distinguish the
contribution of moves and pauses to the cumulative sum equation
Figure 1. The intermittent nature of movement. Individual
motion can be discretized into a series of move, blue, and pause, red,
lengths. The black lines indicate switches between these states. The
pattern of movement is shown for an individual with Brownian motion
(A) and for individual locusts observed in experiments (B–E) for 40 s.
We calculated a total of 44,710 move lengths and 60,103 pause lengths
for all individuals. Since our measurements of locusts’ movements were
recorded per frame, we treated move and pause length durations as
pre-binned (discrete) data, rather than continuous (following Edwards
et al. [68]).
doi:10.1371/journal.pcbi.1002498.g001
Figure 2. Individual behaviour after and within a pause.
(A) showing our calculation of locust turning behaviour within moves
or pauses, or after a pause. We define a turn as a change from CW to
ACW movement or vice versa. Arrows indicate the time steps for which
the switch between CW to ACW was considered. Within a move or
pause only consecutive time steps were examined (dotted arrows). For
turning after a pause, the time steps immediately before and after the
pause were considered (solid arrow). (B) shows the mean probability of
changing direction after a pause for observed pause lengths (s), using
log-binned averages. The left and right dashed lines show 6 s and
100 s, respectively. (C) shows the mean probability of changing
direction after a pause for pause lengths of up to 20 s on a normal
scale. (D) shows the mean probability of turning within a pause for
different pause lengths. We have presented pause lengths up to 6 s as
pause lengths greater than 6 s show a probability of one. For (B–D)
error bars show 95% confidence intervals of the mean. (E) shows the
relationship between the mean proportion of turns within a pause and
the probability of changing direction after a pause for pause lengths of:
less than 6 s (blue squares); between 6 s and 100 s (red triangles); and
greater than 100 s (black circles). Each data point is a mean calculated
from data within logged bin classes for pause length.
doi:10.1371/journal.pcbi.1002498.g002
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org3May 2012 | Volume 8 | Issue 5 | e1002498

Page 4

[44,45]. In the PS algorithm one assumes minimum time duration
for a behavioural mode to exist, i.e., the time threshold parameter
(e). The latter allows the allocation of breakpoints in the
cumulative time series, distinguishing distinct modes. We chose
e=5 minutes (other threshold values were also tested but did not
qualitatively change the results- see Figure S2). In addition, we also
identify fast and slow moves by measuring the angular speed, a
measure of how fast the locust moves around the arena (see Text
S1 for details).
Detection of power-law distributions.
stand the statistical properties of locusts’ movement we examined
the frequency distribution of move lengths and pausing bout
lengths for each locust. We fitted several simple probabilistic
models, often observed in dispersal or movement data [46,47] to
our data: a bounded (truncated) power-law model, a pure power-
law model, and a bounded exponential model. A mathematical
description of these models can be found in Text S1. In order to
determine which probabilistic model best fits the distribution of
moves and pauses for each locust, we carried out sequential
pointwise model comparison (SPWMC) tests. The SPWMC
analysis consisted of conducting point-wise maximum likelihood
estimates, and based on Akaike weight computations (wAIC)
examining the relative likelihood of each model compared with the
likelihood of the best-fit model [36]. The value of the wAIC gives
the weight of evidence in favour of a model, where wAIC=1 is the
maximum weight of evidence. The analysis also explored whether
different models could fit different regimes of the data.
In order to determine accurately the scaling exponent m of the
bounded power-law behaviour observed in the data, we used
maximum likelihood techniques and fitted two general models
consisting of: (i) a power-law model with a stretched exponential
function for the tail (i.e. large moves/pause lengths), and [48] the
same model as (i) but including an exponential distribution for the
small moves/pause lengths (for more details on the models see
Text S1). Despite some variability at shorter moves/pause lengths,
which show both power-law and exponential variability, most of
the individual locusts behaved similarly in statistical terms above
certain move/pause length values. A power-law model with a
stretched exponential tail function could be well fitted to data from
all individuals (for individual locust data analyses see Text S1,
Figure S3, Figure S4, Table S1, Table S2). The stretched
exponential distribution is an exponential distribution with a
parameter, b (where 0,b,1), which accounts for deviations from
exponential behaviour at the tail (b=1 represents pure exponen-
tial behaviour, and the smaller the b value, the fatter the tail). We
pooled the data for all locusts together to get more statistical power
on our analysis. We computed the distributions of move/pause
lengths to represent the behaviour of an ‘‘average’’ locust and
fitted a power-law with a stretched exponential tail model to these
data.
We computed the empirical complementary cumulative distri-
bution functions (CCDFs) by plotting for a variable x (here either
move or pause lengths in seconds) the proportion of observations
that were equal to or larger than x, i.e., P(X$x) on a logarithmic
scale [38,49]. We also computed the empirical probability density
functions (PDFs) for the move and pause lengths (see Text S1 for
calculations). We excluded pause lengths greater than 1000 s
(which account for 0.02% of all pause lengths) to remove the
effects of those locusts considered to be exhibiting atypical
behaviour. Once we performed a fit of our model to the empirical
data, we carried out model criticism on our analysis by visually
examining how our observed distributions deviate from the
expected distribution +/22 SD [38,50].
In order to under-
Results
Pauses as reorientation bouts or rests
Our quantification of locusts’ turning behaviour both within
pauses and within moves demonstrates that a change in direction
is more likely to be found in a pause than during a move. The
mean proportion of pauses with changes in direction from total
bouts (moves and pauses) with changes in direction is 0.8609 (+/
20.0966, one SD). By contrast the mean proportion of moves with
changes in direction is 0.098 (+/20.0726, one SD). Therefore we
can consider moves as displacements without reorientations, and
pauses as opportunities for turns.
We then considered whether the duration of the pause
influences the mean probability that a locust changes direction
after a pause (Figure 2B). This probability shows a strong positive
relationship with pause length for pauses lasting up to 6 s
(Figure 2B and Figure 2C). For pauses between 6 s and 100 s,
very little correlation appears with the probability of changing
direction, remaining between 0.2 and 0.3 (Figure 2B). Increasing
pause length beyond 100 s results in a further increase in the mean
probability of changing direction (Figure 2B). Our data also show
that the mean probability of turning within a pause, reflecting the
fidgeting behaviour of locusts, increases as the pause duration
increases, and plateaus to one at 6 s (Figure 2D).
The mean proportion of turns within a pause is significantly
higher for pause lengths of 100 s or greater (0.6505+/20.0191,
one SD) than for pause lengths between 6 s and 100 s (0.5771+/
20.1049, one SD; T-test: p,0.0001, T-statistic=210.7966,
Df=4830). There also exists a positive relationship between
turning behaviour within a pause and the probability of changing
direction after a pause (Figure 2E) for pauses shorter than 100 s.
For pauses longer than 100 s the probability of turning within a
pause remains just above 0.6, and does not affect the probability of
changing direction after the pause. Therefore increasing turning
within a pause, that is, increasing fidgeting while paused, increases
the likelihood that after the pause the locust changes direction, but
only for pauses lasting less than 100 s (Figure 2E).
We also calculated the most influential pause length for changes
in direction after a pause, which emerges from the combination of
the distribution of pause lengths and the probability of changing
direction for a given pause length (Figure S5). We find that even
though the probability of turning is very low in the smaller pause
lengths, the latter contribute more to turning behaviour, when
considering the overall locust trajectory motion, because they are
overwhelmingly abundant.
Detecting power-laws in move and pause length
distributions
The results of the SPWMC tests for the moves and pauses
averaged for all individuals are shown in Figure 3A–B. For shorter
moves we see an overlap between the exponential and bounded
power-law models. This indicates that shorter moves follow a
mixture of probabilistic models. However for longer moves (the
tails), the bounded power-law model is more dominant (Figure 3A).
The dominance of the bounded power-law model in Figure 3B
strongly favours a very heterogeneous (fat-tailed) distribution of
pauses. We note that SPWMC tests do not show actual fits, but
instead are meant to be a first explorative analysis to compare
among a reasonable set of models.
Figure 3C–D shows the CCDFs and the PDFs, respectively, for
the moves and pauses once the data have been pooled (for
individual locust analyses results see Text S1, Figure S3, Figure S4,
Table S1, Table S2). The value of the Le ´vy exponent m in the
negative power-law equation fitted to the data is 1.49 for moves
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org4May 2012 | Volume 8 | Issue 5 | e1002498

Page 5

and 1.67 for pauses. For moves, h, the cut-off where the stretched
exponential tail begins (also determined from the model fit), is 8 s,
and for pauses h is 15.27 s.
Figure 3E–F also allows us to check visually how well our model
fits our observed data. We find that most deviations exist at the
beginning and at the tail of the move and pauses length
distributions, with pauses showing larger deviations. However
the deviations are small and do not show a systematic pattern.
Error accumulations in the smallest and largest regimes of our
variables are responsible for the spurious results when using
standard goodness-of-fit tests (see Text S1 for further details).
However, Figure 3 demonstrates that our model (a power-law with
a stretched exponential tail) provides a reasonably good fit to the
data for moves and pause distributions.
Defining behavioural modes: combining moves and
pauses
When we examined the relationship between moves and pauses
we observed strong, negative, first-order correlations (Figure 4A
inset and Figure S6). Short pauses are associated with moves of all
lengths. Longer pauses however are more likely to be associated
with shorter moves. Furthermore locusts tend not to exhibit large
move lengths and large pauses together. We therefore carried out
more complete correlation analyses on moves and pauses. Our
partial autocorrelation results reveal that moves show much
stronger local correlations than pauses (Figure 4A). Cross-
correlation analysis between moves and pauses reinforce the idea
that there is a negative correlation, particularly at local scales
(Figure 4B). Thus long moves tend to be associated with short
Figure 3. Detailed analyses of moves and pauses. The SPWMC analysis with bounded power-law, red, pure power-law, green, and bounded
exponential, blue, for moves (A) and pauses (B). A value of weighted Akaike information criteria (wAIC) of one gives the maximum weight of
evidence in favour of the models. The results here are means for all individuals. Error bars indicate +/2 one SD. The probability density functions (C)
and the complementary cumulative distribution plots (D) and for moves, blue, and pauses, red, showing the empirical data and the model fits for the
power law with a stretched exponential tail model (black line). m is the scaling parameter of the power-law, b is a parameter that tells us the deviation
of the tail from an exponential. For moves: m=1.49, b=0.55; for pauses: m=1.67, b=0.23. (E–F) shows the observed and expected distributions for
moves and pauses. Log-log plot of the frequency distribution of different move (E) and pause lengths (F). Open circles show the observed
distribution from our data and dots show the expected distribution from the model fit (a power-law with a stretched exponential tail model). We
assume a Poisson distribution for the deviations from the expected values for each bin. The black error bars show +/22 SD from our expected value.
doi:10.1371/journal.pcbi.1002498.g003
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org5May 2012 | Volume 8 | Issue 5 | e1002498

Page 6

pauses and vice versa. In addition, long moves account for faster,
and more energetic, circling in the arena. Thus, non-turning short
pauses could allow for some energy recovery. In particular we
observed that the average angular speed increased as move lengths
increased, reaching a saturation of 15 degrees/s, at moves of 15 s
in duration (see Figure S7).
For each experiment, the PS algorithm allocates sequences of
moves and pauses into two behavioural modes: i) local search, i.e.
sequences of long pauses and short moves, and ii) relocations, i.e.
sequences of long moves and short (resting and non-turning)
pauses. At the population-level, that is, averaging over all the
individuals, we can obtain the probability that a locust is in one
mode or another over time. The data show that the probability
that a locust performs relocation behaviour at the beginning of the
experiment is small (0.2) but increases as the experiment
progresses to approximately 0.6. Conversely a locust performs
local search at the beginning of the experiment at a probability
fluctuating around 0.8 within the first 1.5 hrs but this decreases as
the experiment progresses. After almost 5 hours there is a shift and
both modes stabilize around 0.5, with the probability of relocation
being slightly larger than the probability of local search.
Discussion
We have carried out a thorough statistical description of isolated
locusts’ motion using a large data set (93 experimental trials, each
lasting 8 h, with positions and orientations acquired every 0.2 s).
Controlled laboratory conditions were used to study movement,
pausing and turning behaviours, thereby minimizing the amount
of interference from external cues to individual motion. This is not
to neglect the influence of environmental factors, but rather to
help elucidate whether complex statistical properties of movement
Figure 4. Correlation analysis for moves and pauses and behavioural modes. Partial autocorrelation analysis results (A) for moves, blue
circles, and pauses, red triangles, reveals that moves are positively correlated and pauses show a slight positive, if any, correlation, therefore moves
show much stronger local correlations than pauses. Inset shows first order correlation between moves and pauses (see Figure S6). A cross correlation
of moves and pauses (B) shows a high negative correlation at a local scale, circles, which persists for larger time scales. The shuffled data are also
shown (triangles) for comparison. (C) shows the probability that a locust performs two different behaviours (local search and relocation) over the
course of the 8 hr experiment. Relocation behaviour (blue asterisks) is defined as long moves preceded or followed by short pauses. Local search
behaviour (red circles) is defined as long pauses preceded or followed by short moves. Behavioural modes are classified using the Partial Sums
algorithm with a minimum time threshold of 5 min.
doi:10.1371/journal.pcbi.1002498.g004
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org6 May 2012 | Volume 8 | Issue 5 | e1002498

Page 7

may be generated in their absence. Our results show that
intermittence with Le ´vy signatures can be considered the
behavioural template for search in individual locusts. It is however
also clear from our analysis that intermittent patterns are
constrained, perhaps by biomechanical, physiological or neurobi-
ological factors, and in addition, can be modulated by the animal’s
internal state.
Our examination of turning behaviour during pauses suggests
that pauses serve different functions, depending on their duration.
Longer pauses (beyond 100 s) appear to act as reorientation bouts,
which may serve to interrupt the persistence of previous
directional motion since they are most likely to result in a
complete change in direction of movement after the pause
(Figure 2B). In addition longer pauses involve more turning within
the pause itself, resulting from body rotation without displacement
(Figure 2D–E). Such fidgeting behaviour could perhaps be the
stochastic effect resulting from a locust scanning (and barely
moving from) its surroundings. As locusts use optic flow to gather
information from their environment, they are able to ‘‘see’’ their
surroundings both when in motion and during fidgeting behaviour
that involves head movements [51]. The shortest pauses (of less
than 6 s) appear to act as periods of resting between displace-
ments, as they do not involve a high probability of changing
direction after the pause (Figure 2B–C) or greater turning within
the pause (Figure 2D–E), and are associated with moves of variable
length (Figure S6).
Our correlation analyses reveal that move and pause lengths are
negatively correlated with one another (Figure 4B and Figure S6).
The correlations of short (slow)/long (fast) moves with long
(reorientation)/short (resting) pauses lead to different relative
proportions of local search and relocation behaviour, which are
organized in time (Figure 4).
Local search is characteristic of ‘pottering’ behaviour, during
which a locust moves then stops at intervals to test its environment
with its palps and antennae, resulting in frequent changes in
direction [18]. Relocation behaviour may be associated with
‘marching’ activity, consisting of continuous locomotion in a
persistent direction [18,52]. Our results suggest that the two
behavioural modes identified are steadily decreasing (local search)
or increasing (relocations) up to approximately 6 hours, after
which both behavioural modes show stationary fluctuations. We
acknowledge the statistical issue of the degree of independence in
sequential time series data of this sort. However this is an inherent
problem with all such analyses, including that of this work.
The motivation resulting in an increase in relocation activity
towards the end of our experiments is not explicitly explored here.
However previous studies on locusts have revealed that increased
marching may be associated with hunger; as the amount of food in
their gut decreases, marching activity increases [18,40]. Locusts
may be investing more time in local explorations at the start of the
experiment. As the cumulative information of non-available
resources becomes stronger, local search and relocations appear
to happen with more similar proportions: a stationary exploratory
behaviour seems to emerge.
In our analysis of the distribution of moving and pausing step-
lengths we observe that the probability of very long moves or very
long pauses is small but not negligible (far beyond the Gaussian tail
expectation), and that locusts’ movements are better described by
means of a general class of random walks known as Le ´vy walks
[53,54]. Our results show that power-law behaviour is naturally
bounded to some range of scales [7,55,56], meaning that the time
over which an individual can move or pause in a single bout is
limited, perhaps owing to some physical constraints or to some
strategic advantage [38].
When we pool our data for all individuals together (to obtain the
average), we observe power-law behaviour of over 1.5 orders of
magnitude for moves and pauses after which there is a cut-off and
the stretched exponential tail begins (Figure 3). These results
suggest power-law behaviour with additional complexity. The
scaling exponents obtained from our data (m=1.49 for moves,
m=1.67 for pauses) lie within the range expected from a Le ´vy walk
(1,m#3) [54]. We find that locust behaviour shows movement
patterns that are not entirely Ballistic (with exponent of m<1).
Ballistic motion is useful to a foraging animal if targets are
homogeneously located far away with respect to the initial
searching position. Le ´vy patterns with m<2 [57,58,59,60] become
optimal in patchy landscapes, where far away and nearby targets
exist [9,61]. Recent results show the impact of landscape
heterogeneity in optimal random search strategies, and suggest
that the over dispersed and highly heterogeneous nature of desert
vegetation [62,63] could have promoted intermittent motion
within the Le ´vy range: 1,m#2 [61], which we observe here in
locusts.
The presence of power-law regimes in empirical distributions of
animalmovementdatahas
[2,4,10,49,64,65,66,67,68], however, there is strong empirical
evidence for power laws in animal movement within natural
habitats [36,37,38,69,70] and under experimental conditions
[26,27]. Our results suggest that while these patterns may result
from interactions with the environment, they can also be
generated internally. However Le ´vy distributions do not fully
characterize locusts’ movements. The behavioural template of
locusts in the absence of environmental cues results from the
relationship between moving, pausing and turning and involves
both some physical constraints and some higher-order movement
structure. In our experiments, internal state behavioural modula-
tion may exist in association with a ‘‘starvation/satiation state’’, or
a ‘‘present/absent food memory’’ [9,71]. The switch from local
search behaviour at the beginning of the experiment towards
relocation behaviour may be due to the general effects of food
deprivation, which is known to result in increased marching
[18,40], either as food memory is lost or starvation levels increase.
We may understand complex intermittence as the interweaving
of different behavioural modes [26] that are likely to be
constrained by species-specific physical and biological factors.
For example, fidgeting is physically impossible at small pauses but
is constant at large pauses and large moves need to be interspersed
with small (resting but not turning) pauses so that locusts can make
large scale-free displacements in random directions. Future
experiments should be designed to determine whether such
behavioural constraints are driven at the biomechanical, the
physiological or the neurobiological level. The idea of a null scale-
free (Le ´vy-like) behavioural template may be in concordance with
neuronal activity patterns, which in desert locusts also show a Le ´vy
distribution with an exponent of approximately 1.5 [72,73].
Overall, our results add upon the random paradigm debate in
movementecology[28]on whether internal states or externalstimuli
drive behavioural variability. Our findings suggest that the complex
intermittent patterns observed are mainly internally shaped and
governed. Therefore spontaneous and/or internally driven variabil-
ity should be considered in order to achieve a comprehensive
understanding of animal motor reactions to the environment, which
is the ultimate goal in the field of movement ecology.
generatedmuchdebate
Supporting Information
Figure
functions including and excluding boundary data. The
S1
Complementary cumulativedistribution
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org7May 2012 | Volume 8 | Issue 5 | e1002498

Page 8

complementary cumulative probability distribution plots for moves
(A) and pauses (B). The plots show the empirical data either
including, blue, or excluding, red, data from individuals within
3 cm of arena walls. As we define move lengths based on
rotational direction (CW/ACW), contact with the border could be
included in on our definition of move length. Nevertheless, such
contact can introduce new behavioural components that we have
avoided to include in our main analysis. Analysis with the inclusion
of data from individuals found within 3 cm of the arena wall show
similar qualitative results for the distribution of move lengths (A);
the power-law spans over slightly longer time scales and the
stretched exponential tail starts later (fitted parameters: m=1.33,
h=14 s, b=0.75). For the distribution of pauses (B), the inclusion
of data from the borders results in the stretched exponential tail
being less pronounced such that the power-law spans over much
longer time scales (fitted parameters: m=1.45, h=89.9 s,
b=0.579).
(DOC)
Figure S2
of being in local search or relocation modes over time for different
time duration thresholds. In the PS algorithm used to detect the
two behavioural modes, the time duration threshold parameter
represents the minimum time threshold for a behavioural mode to
be sustained in order to be considered different from the previous
mode. It is a constant numerical value that allows the allocation of
breakpoints in the cumulative deviation series (Knell and Codling
2011) thereby distinguishing distinct modes, according to this
minimum behavioural mode time duration threshold. Time
thresholds of 1 min (top panel), 5 min (middle panel) and
10 min (bottom panel) were tested, meaning that a mode should
last at least 1, 5, or 10 minutes. All thresholds tested yield the same
qualitative results.
(DOC)
Behavioural mode thresholds. The probability
Figure
functions for move lengths. Each subplot shows a log-log
plot of the complementary cumulative distribution function of the
empirical data (blue) of different move lengths, (x) exhibited by
each individual locust and the best model fit, either the
exponential followed by power-law with exponential tail model
fit (red) or the power-law with exponential tail model fit (black).
Numbers in each subplot indicate individual number (1 to 93).
(DOC)
S3
Complementary cumulativedistribution
Figure
functions for pause lengths. Each subplot shows a log-log
plot of the complementary cumulative distribution function of the
empirical data (blue) of different pause lengths (x) exhibited by
each individual locust and the best model fit, either the
exponential followed by power-law with exponential tail model
fit (red) or the power-law with exponential tail model fit (black).
Numbers in each subplot indicate individual number (1 to 93).
(DOC)
S4
Complementary cumulativedistribution
Figure S5
bility of turning after a pause. Bar chart showing, on the left
axis, the proportion of data points (white) and the mean
probability of turning after a pause (black) for each pause length
bin class (log binned classes). The influence (grey) of the data
points on the mean probability of turning after a pause for each
pause length is also shown (right axis). This was calculated by
multiplying the mean probability of turning by the proportion of
data points within each pause length bin class.
(DOC)
The influence of pause length on the proba-
Figure S6
or following pauses. The relationship between pause length (s)
and move length (s) for moves immediately preceding (A) or
following (B) pauses for all locusts. The relationship between pause
length and the following move length, or pause length and the
preceding move length show a similar pattern (since moves after
one pause are before another).
(DOC)
Pause lengths and move lengths proceeding
Figure S7
angular speed (in degrees per s), v, is measured as dh/dt, where dh
is the angle (in degrees) moved between the first and last frame of
the move, and dt is the move length in s. The black line shows a
non-linear least squares fit (of the type: a ? (1{exp({x=c)),
where a=3; c=18, in Matlab 2010b) to the data. The mean
angular speed saturates to 15 degrees/s at move lengths of 15 s.
(DOC)
Angular speed for different move lengths. The
Table S1
distribution of moves of each individual locust we calculated: the
best fit model to the data, either model 1 or model 2, Model; the
total number of move lengths, N; the maximum move length, Max
(in s), for each individual (the minimum move length for all
individuals is 0.2 s); the Le ´vy exponent, m; the parameter that tells
us the deviation of the tail from an exponential, b (where b=1 is
an exponential tail, and b=0 is a power-law tail); for model 1 the
mean lifetime (or characteristic) move length (in s), h (1) ; for model
2 the move length value (in s) delimiting the beginning of the
power law regime, x2(2); move length value (in s) delimiting the
end of the power law regime, x3, (the dots in column x3 show
where model 1 was a better fit than model2); the negative log-
likelihood function, NegLogLik; the goodness of fit test statistic
value, GOF; the p-value from a goodness-of-fit test telling us
whether the model is reliable or not (from Edwards et al. (2007)),
where p-values.0.1 are a good fit of the model to the data
(highlighted).
(DOC)
Individual model fit results for moves. For the
Table S2
distribution of pauses of each individual locust we calculated: the
best fit model to the data, either model 1 or model 2, Model; the
total number of pause lengths, N; the maximum pause length, Max
(in s), for each individual (the minimum pause length for all
individuals is0.2 s);theLe ´vyexponent,m;theparameterthattellsus
the deviation of the tail from an exponential, b (where b=1 is an
exponential tail, and b=0 is a power-law tail); for model 1 the mean
lifetime (or characteristic) pause length (in s), h (1); for model 2 the
pause length value (in s) delimiting the beginning of the power law
regime, x2(2); pause length value (in s) delimiting the end of the
power law regime, x3, (the dots in column x3 show where model 1
was a better fit than model2); the negative log-likelihood function,
NegLogLik; the goodness of fit test statistic value, GOF; the p-value
from a goodness-of-fit test telling us whether the model is reliable or
not(fromEdwardsetal.(2007)),wherep-values.0.1areagoodfitof
the model to the data (highlighted).
(DOC)
Individual model fit results for pauses. For the
Text S1
the materials and methods used. These include correlation
analyses between moves and pauses, the behavioural modes
analysis, and the quantification of move speeds. In addition, the
supporting text also shows the mathematical description of the
probabilistic models used in SPWMC, details of individual locust
data analyses, and calculations of the different models used in our
investigation.
(DOC)
The supporting text provides further details of
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org8May 2012 | Volume 8 | Issue 5 | e1002498

Page 9

Video S1
the movements of a single locust in the experimental
arena.
(AVI)
A video clip of an experimental trial showing
Acknowledgments
We thank the help of E. Raposo for the development of the equations for
the PDFs, CCDFs, and the likelihood functions of the two models fitted to
the locust data, and Joan Garriga for help with the behavioural modes
analysis. The authors also thank Susana Bernal, Vishwesha Guttal,
Christos Ioannou, and Colin Torney for helpful discussions.
Author Contributions
Conceived and designed the experiments: SB IDC. Performed the
experiments: SB. Analyzed the data: SB FB. Contributed reagents/
materials/analysis tools: FB JJH. Wrote the paper: SB FB IDC. Designed
the software used in analysis: JJH.
References
1. Nathan R, Getz WM, Revilla E, Holyoak M, Kadmon R, et al. (2008) A
movement ecology paradigm for unifying organismal movement research. Proc
Natl Acad Sci U S A 105: 19052–19059.
Bartumeus F (2009) Behavioral intermittence, Le ´vy patterns, and randomness in
animal movement. Oikos 118: 488–494.
Stephens GJ, Johnson-Kerner B, Bialek W, Ryu WS (2008) Dimensionality and
Dynamics in the Behavior of C. elegans. PLoS Comp Biol 4: e1000028.
Plank MJ, James A (2008) Optimal foraging: Le ´vy pattern or process? J R Soc
Interface 5: 1077–1086.
Levin SA (1992) The problem of pattern and scale in ecology. Ecology 73:
1943–1967.
Kramer DL, McLaughlin RL (2001) The behavioural ecology of intermittent
locomotion. Am Zool 41: 137–153.
Mashanova A, Oliver TH, Jansen VA (2009) Evidence for intermittency and a
truncated power law from highly resolved aphid movement data. J R Soc
Interface 7: 199–208.
O’Brien WJ, Browman HI, Evans BI (1990) Search strategies of foraging
animals. Am Sci 78: 152–160.
Bartumeus F, Levin SA (2008) Fractal intermittence in locomotion: linking
animal behavior to statistical patterns of search. Proc Natl Acad Sci U S A 150:
19072–19077.
10. Bartumeus F (2007) Le ´vy processes in animal movement: an evolutionary
hypothesis. Fractals 15: 151–162.
11. Mittelstaedt H, Mittelstaedt ML (1982) Homing by path integration. In: Papi F,
Wallraff HG, eds. Avian Navigation. Berlin: Springer-Verlag. pp 290–297.
12. Bartumeus F, da Luz MGE, Viswanathan GM, Catalan J (2005) Animal search
strategies: a quantitative random-walk analysis. Ecology 86: 3078–3087.
13. Hendersen RW (1985) Fearful memories: The motivational significance of
forgetting. In: Brush FR, Overmier JB, eds. Affect, conditioning, and cognition:
Essays on the determinants of behavior. Hillsdale, NJ: Erlbaum. pp 43–53.
14. Kraemer PJ, Golding JM (1997) Adaptive forgetting in animals. Psychon B Rev
4: 480–491.
15. Be ´nichou O, Coppey M, Suet P-H, Voituriez R (2005) Optimal search strategies
for hidden targets. Phys Rev Lett 94: 198101.
16. Be ´nichou O, Loverdo C, Moreau M, Voituriez R (2011) Intermittent search
strategies. Rev Mod Phys 83: 81–130.
17. Be ´nichou O, Loverdo C, Moreau M, Voituriez R (2006) Two-dimensional
intermittent search processes: An alternative to Le ´vy flight strategies. Phys Rev E
74: 020102.
18. Ellis PE (1951) Anti-Locust Bulletin. 7: The marching behaviour of hoppers of
the African Migratory Locust (Locusta migratoria migratorioides R. & F.) in the
laboratory. London: Anti-Locust Research Centre.
19. Boettiger AN, Wittemyer G, Starfield R, Volrath F, Douglas-Hamilton I, et al.
(2011) Inferring ecological and behavioral drivers of African elephant movement
using a linear filtering approach. Ecology 92: 1648–1657.
20. Martin RA, Pfennig DW (2009) Disruptive Selection in Natural Populations:
The Roles of Ecological Specialization and Resource Competition. Am Nat 174:
268–281.
21. Owen-Smith N, Fryxell JM, Merrill EH (2010) Foraging theory upscaled: the
behavioural ecology of herbivore movement. Philos T Roy Soc B 365:
2267–2278.
22. Patterson TA, Basson M, Bravington MV, Gunn JS (2009) Classifying
movement behaviour in relation to environmental conditions using hidden
Markov models. J Anim Ecol 78: 1113–1123.
23. Reynolds AM, Frye MA (2007) Free-Flight Odor Tracking in Drosophila Is
Consistent with an Optimal Intermittent Scale-Free Search. PLoS One 2: e354.
24. Vanak AT, Thaker M, Slotow R (2010) Do fences create an edge-effect on the
movement patterns of a highly mobile mega-herbivore? Biol Conserv 143:
2631–2637.
25. Yott A, Rosatte R, Schaefer JA, Hamr J, Fryxell J (2011) Movement and Spread
of a Founding Population of Reintroduced Elk (Cervus elaphus) in Ontario,
Canada. Restor Ecol 19: 70–77.
26. Maye A, Hsieh C-h, Sugihara G, Brembs B (2007) Order in Spontaneous
Behavior. PLoS One 2: e443.
27. Sorribes A, Armendariz BG, Lopez-Pigozzi D, Murga C, de Polavieja GG
(2011) The Origin of Behavioral Bursts in Decision-Making Circuitry. PLoS
Comp Biol 7: e1002075.
2.
3.
4.
5.
6.
7.
8.
9.
28. Nathan R (2008) An emerging movement ecology paradigm. Proc Natl Acad
Sci U S A 105: 19050–19051.
29. Getz WM, Saltz D (2008) A framework for generating and analyzing movement
paths on ecological landscapes. Proc Natl Acad Sci U S A 105: 19066–19071.
30. Morales JM, Haydon DT, Frair J, Holsinger KE, Fryxell JM (2004) Extracting
more out of relocation data: building movement models as mixtures of random
walks. Ecology 85: 2436–2445.
31. Fryxell JM, Hazell M, Borger L, Dalziel BD, Haydon DT, et al. (2008) Multiple
movement modes by large herbivores at multiple spatiotemporal scales. Proc
Natl Acad Sci U S A 105: 19114–19119.
32. Bartumeus F, Catalan J (2009) Optimal search behavior and classic foraging
theory. J Phys A- Math Theor 42: 434002–434013.
33. Bovet P, Benhamou S (1988) Spatial analysis of animals’ movements using a
correlated random walk model. J Theor Biol 131: 419–433.
34. Crist TO, Guertin DS, Wiens JA, Milne BT (1992) Animal movement in
heterogeneous landscapes: an experiment with Elodes beetles in shortgrass
prairie. Funct Ecol 6: 536–544.
35. Turchin P (1991) Translating foraging movements in heterogeneous environ-
ments into the spatial distribution of foragers. Ecology 72: 1253–1266.
36. Bartumeus F, Giuggioli L, Louzao M, Bretagnolle V, Oro D, et al. (2010)
Fishery Discards Impacts on Seabird Movement Patterns at Regional Scales.
Curr Biol 20: 1–8.
37. Sims DW, Southall EJ, Humphries NE, Hays GC, Bradshaw CJA, et al. (2008)
Scaling laws of marine predator search behaviour. Nature 451: 1098–1102.
38. Humphries NE, Queiroz N, Dyer JRM, Pade NG, Musyl MK, et al. (2010)
Environmental context explains Le ´vy and Brownian movement patterns of
marine predators. Nature 465: 1066–1069.
39. Roessingh P, Simpson SJ, James S (1993) Analysis of phase-related changes in
behaviour of desert locust nymphs. P Roy Soc B-Biol Sci 252: 43–49.
40. Bazazi S, Romanczuk P, Thomas S, Schimansky-Geier L, Hale JJ, et al. (2011)
Nutritional state and collective motion: from individuals to mass migration.
P Roy Soc B-Biol Sci 278: 356–363.
41. Bazazi S, Buhl J, Hale JJ, Anstey ML, Sword GA, et al. (2008) Collective motion
and cannibalism in locust migratory bands. Curr Biol 18: 735–739.
42. Buhl J, Sumpter DJT, Couzin ID, Hale JJ, Despland E, et al. (2006) From
Disorder to Order in Marching Locusts. Science 312: 1402–1406.
43. Ellis PE (1963) Changes in marching of locusts with rearing conditions.
Behaviour 23: 177–192.
44. Knell AS, Codling EA (2011) Classifying area-restricted search (ARS) using a
partial sum approach. Theor Ecol in press: DOI 10.1007/s12080-12011-10130-
12084.
45. Hawkins DM, Olwell DH (1998) Cumulative sum charts and charting for
quality improvement. New York: Springer-Verlag.
46. Bullock JM, Kenward RE, Hails R (2002) Dispersal Ecology; Bullock JM,
Kenward RE, Hails R, eds. Cambridge: Cambridge University Press.
47. Turchin P (1998) Quantitative analysis of movement: measuring and modelling
population redistribution in animal and plants. Sunderland, MA: Sinauer
Associates Inc.
48. Radakov DV (1973) Schooling in the ecology of fish. New York: J. Wiley.
49. Sims DW, Righton D, Pitchford JW (2007) Minimizing errors in identifying
Le ´vy flight behaviour of organisms. J Anim Ecol 76: 222–229.
50. Bolker BM (2008) Ecological Models and Data in R. Princeton, New Jersey,
USA: Princeton University Press.
51. Collett TS (2002) Insect Vision: Controlling Actions through Optic Flow. Curr
Biol 12: R615–R617.
52. Kennedy JS (1951) The Migration of the Desert Locust (Schistocerca gregaria
Forsk.). I. The Behaviour of Swarms. II. A Theory of Long-Range Migrations.
Philos T Roy Soc B 235: 163–290.
53. Metzler R, Klafter J (2004) The Restaurant at the End of the Random Walk:
Recent Developments in Fractional Dynamics of Anomalous Transport
Processes. J Phys A-Math Theor 37: R161–R208.
54. Shlesinger MF, Zaslavsky GM, Klafter J (1993) Strange kinetics. Nature 363:
31–37.
55. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in
empirical data. SIAM Rev 51: 661–703.
56. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp
Phys 46: 323–351.
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org9May 2012 | Volume 8 | Issue 5 | e1002498

Page 10

57. da Luz MGE, Buldyrev SV, Havlin S, Raposo EP, Stanley HE, et al. (2001)
Improvements in the statistical approach to random Le ´vy flight searches.
Physica A 295: 89–92.
58. Viswanathan GM, Afanasyev V, Buldyrev SV, Havlin S, da Luz MGE, et al.
(2000) Le ´vy flights in random searches. Physica A 282: 1–12.
59. Viswanathan GM, Afanasyev V, Buldyrev SV, Havlin S, da Luz MGE, et al.
(2001) Le ´vy flight search patterns of biological organisms. Physica A 295: 85–88.
60. Viswanathan GM, Buldyrev SV, Havlin S, da Luz MGE, Raposo EP, et al.
(1999) Optimizing the success of random searches. Nature 401: 911–914.
61. Raposo EP, Bartumeus F, da Luz MGE, Lima TA, Ribeiro-Neto PJ, et al. (2011)
How landscape heterogeneity frames optimal diffusivity in searching processes.
PLoS Comput Biol 7: e1002233.
62. Klausmeier C (1999) Regular and Irregular Patterns in Semiarid Vegetation.
Science 284: 1826–1828.
63. Sole ´ R (2007) Scaling laws in the drier. Nature 449: 151–153.
64. Buchanan M (2008) The mathematical mirror to animal behaviour. Nature 453:
714–716.
65. Reynolds AM (2008) How many animals really do the Le ´vy walk? Comment.
Ecology 89: 2347–2351.
66. Travis J (2007) Do wandering albatrosses care about math? Science 318:
742–743.
67. Turchin P (1996) Fractal analyses of animal movement: a critique. Ecology 77:
2086–2090.
68. Edwards AM, Phillips RA, Watkins NW, Freeman MP, Murphy EJ, et al. (2007)
Revisiting Le ´vy flight search patterns of wandering albatrosses, bumblebees and
deer. Nature 449: 1044–1048.
69. Hays GC, Bastian T, Doyle TK, Fossette S, Gleiss AC, et al. (2011) High activity
and Le ´vy searches: jellyfish can search the water column like fish. Proc Biol Sci
279: 465–73.
70. Sims DW, Humphries NE, Bradford RW, Bruce BD (2012) Le ´vy flight and
Brownian search patterns of a free-ranging predator reflect different prey field
characteristics. J Anim Ecol 81: 432–42.
71. Pierce-Shimomura JT, Morse TM, Lockery SR (1999) The Fundamental Role
of Pirouettes in Caenorhabditis elegans Chemotaxis. J Neurosci 19: 9557–9569.
72. Ayali A, Fuchs E, Zilberstein Y, Robinson A, Shefi O, et al. (2004) Contextual
regularity and complexity of neuronal activity: from stand-alone cultures to task-
performing animals. Complexity 9: 25–32.
73. Fuchs E, Ayali A, Robinson A, Hulata E, Ben-Jacob E (2007) Coemergence of
Regularity and Complexity During Neural Network Development. Dev
Neurobiol 67: 1802–1814.
Intermittent Motion in Desert Locusts
PLoS Computational Biology | www.ploscompbiol.org 10May 2012 | Volume 8 | Issue 5 | e1002498