A non-Lévy random walk in chacma baboons: what does it mean?

A Non-Le´vy Random Walk in Chacma Baboons: What
Does It Mean?
Ce´dric Sueur1,2*
1 Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America, 2 Unit of Social Ecology, Free University of
Brussels, Brussels, Belgium
Abstract
The Le´vy walk is found from amoebas to humans and has been described as the optimal strategy for food research. Recent
results, however, have generated controversy about this conclusion since animals also display alternatives to the Le´vy walk
such as the Brownian walk or mental maps and because movement patterns found in some species only seem to depend
on food patches distribution. Here I show that movement patterns of chacma baboons do not follow a Le´vy walk but a
Brownian process. Moreover this Brownian walk is not the main process responsible for movement patterns of baboons.
Findings about their speed and trajectories show that baboons use metal maps and memory to find resources. Thus the
Brownian process found in this species appears to be more dependent on the environment or might be an alternative when
known food patches are depleted and when animals have to find new resources.
Citation: Sueur C (2011) A Non-Le´vy Random Walk in Chacma Baboons: What Does It Mean? PLoS ONE 6(1): e16131. doi:10.1371/journal.pone.0016131
Editor: Gonzalo Garcı´a de Polavieja, Cajal Institute, Spain
Received October 11, 2010; Accepted December 14, 2010; Published January 13, 2011
Copyright: � 2011 Ce´dric Sueur. 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: This study was funded by the Alsace region (http://www.region-alsace.eu/), the Franco-American Commission and the Fyssen Foundation. The funders
had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The author has declared that no competing interests exist.
* E-mail: csueur@ulb.ac.be
Introduction
Particles suspended in a fluid, air or water move in a random
way called Brownian motion [1–3]. This rule is also used,
however, to explain phenomena in geology, ecology or social
sciences. Several studies suggested that animals used random walks
as a strategy to find food or reproductive partners by increasing
the probability of encountering the respective item [4–7].
All random walks are composed of three basic measurements:
the waiting time to an area A, the step length between areas A and
B and the turning angle. Whereas Brownian random walks are
characterised by constant length of steps and waiting times, Le´vy
walks describe movement patterns characterised by many small
steps connected by rare long steps. In the first case, the probability
distribution of step length is exponential whereas in the second
case it is power-law. This Le´vy walk was defined as an optimal
strategy for a forager searching without information about its
heterogeneous environment with low density food patches [3,4,8].
There has been growing interest in the Le´vy walk and this strategy
is reported in many species such as soil amoebas [9], zooplankton
[5], jackals [10], albatrosses [11] and elephants [12]. This
similarity between these phylogenetically distant species suggests
that random walks are efficient and adaptive. Some studies cast
doubt on Le´vy walks as an optimal strategy existing in animals,
however, first because of methodological shortcomings in the
estimation of power-law exponents but also because of the impact
of resource distribution and the probability of species’ cognitive
abilities being sufficient to find this strategy [3,13–15].
Primates are known to use high cognitive processes in their
foraging and travel decisions [16]. The use of spatial memory to
remember patterns of resource availability and distribution was
shown in several species (red-tailed monkeys, Cercopithecus ascanius
[17], Japanese macaques, Macaca fuscata [18], long-tailed ma-
caques, M. fascicularis [19], white-faced capuchins, Cebus capucinus
[20], brown capuchins, C. apella nigritus [21]). Thus, it is expected
that primates would not walk ‘‘randomly’’ in their environment.
Studies about random walks in primates are scarce but recent
studies showed that spider monkeys (Ateles geoffroy [14,22]) and
hamadryas baboons (Papio hamadryas hamadryas [23]) used Le´vy
walks in their foraging movements. Probability distributions of
their step lengths as well as of their waiting times follow a power
law. Other studies also showed that baboons used the shortest
linear route to travel from one location to another and that they
speeded up as they approached a water or food source, indicating
goal-directed and mental map processes [24–27]. Consequently,
the definition provided by Viswanathan and colleagues [28] of the
Le´vy walk being an optimal search strategy without prior
information is questionable.
Here, I study the movements of a group of chacma baboons (P.
ursinus) in their natural environment. I will assess whether
distributions of step lengths and also times correspond to the
Brownian or Le´vy walk. I will also study trajectories and speed of
animals in order to assess whether they use more cognitive
processes.
Methods
Study site and subjects
Data on chacma baboons were scored at the Wildcliff Nature
Reserve, Western Cape, South Africa (33.959997uN,
21.034478uE) from May to August 2009. The reserve is a
mountain wilderness reserve consisting of deep ravines with afro-
mountain forest, rocky mountain tops and high meadows of
fynbos. An invasive plant, the black wattle, and a grassy meadow
PLoS ONE | www.plosone.org 1 January 2011 | Volume 6 | Issue 1 | e16131

are also present. Three groups of chacma baboons populate this
reserve and its surroundings. At the time of the study, the study
group consisted of between 90 and 100 individuals (about 9.1%
were males, 37% were females without babies (,1year), 5.6%
were females with babies, 16.5% were sub-adults (4-6 years old)
and 31.8% were juveniles (1–3 years old)). During this study,
animals were just observed. No animal handling or invasive
experiment was done on studied subjects. We declare that our
study is in full accordance with the ethical guidelines of our
institution with the approval of the latter (certificate number: 67-
339, French Republic, Bas-Rhin County Hall, French veterinary
services). Our experiments comply with European animal welfare
legislation.
Data collection
During the study period I, accompanied by a field assistant,
scored the location of the baboons’ group from dusk (about 7:00)
to dawn (about 17:00). The location of the baboon group was
defined as the geographical centre of the band [23,26] by means of
a GPS Sirf 3 Holux. I scored locations every ten minutes. The
GPS accuracy is inferior to two meters. Activity of baboons, the
number of individuals observed in each activity and the kind of
vegetation were also scored by means of Cyber Tracker 3.0 (Cyber
Tracker Conservation, Bellville, SA) with a PDA Asus 620. The
activities of baboons included moving (locomotion including
walking, running, climbing and jumping), foraging (reaching for,
picking, manipulating, masticating, or placing food in mouth, as
well as manipulating the contents of a cheek pouch), resting (body
stationary, usually sitting or lying down) and socialising (playing,
grooming, sexual and aggressive behaviour) [29,30]. I defined five
kind of vegetations by characterising the dominant species; fynbos,
black wattle, grass, pine, afro-mountain forest. This vegetation
study was done prior to the baboon study. The sleeping sites were
identified by me, my field assistant and Paula Pebsworth (personal
communication). The clay site was identified by Paula Pebsworth
(personal communication). I only kept data for which all samples
were obtained throughout the day. I obtained eighteen days’
worth of observations and 740 samples.
Data analysis
A step was defined as an interval in which any or both of the
coordinates in two consecutive samples differed. The length L of a
step was calculated with the formula:
L(i,j)~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(xj{xi)2z(yj{yi)2
q
x and y are respectively the latitude and longitude of points i and j.
The length expressed in decimal degree was then transformed in
meters.
I carried out survival analysis to study the inverse cumulative
distribution of step lengths. This distribution of step lengths was
compared with exponential, power and Weibull law [31].
I also calculated the mean squared displacement using the
following procedure: for each day the length of a line joining the
first recorded location (commonly the sleeping site) of the group
with its location at different steps (every ten minutes). Then, all-
day squared displacements were averaged at intervals of 30
minutes, from 6:30 to 18:00. I also calculated the mean path
length every 30 minutes. Waiting times were calculated from the
number of samples in which the group did not change position. I
carried out survival analysis to study the inverse cumulative
distribution of waiting times. This distribution was compared with
exponential and power law. All these distributions were analysed
with curve estimation tests. The results for Weibull distribution
were only indicated when the correlation coefficient between the
Weibull distribution and the tested variable was higher than the
one with exponential distribution. I also calculated the mean path
length for each kind of vegetation and correlated it with the
foraging time animals spent in each kind of vegetation. Differences
of mean path length between kinds of vegetation were analysed
with a median test (detection of differences in shape and location).
Only samples where animals changed the type of vegetation were
kept for this analysis. Correlation between path length and
foraging time was tested with a curve estimation test. All tests were
done with SPSS 10.0., a=0.05.
Results
Figure 1 shows four daily trajectories of the group. These
trajectories seem to be made up of steps of different lengths.
Moreover, animals seemed to go first to a waterhole (observed in
100% of cases), and then to go everyday to a clay site (observed in
83.3% of cases), to forage and eventually to come back at the end
of the day to their sleeping site, doing a kind of ellipse all along
their home range.
The distribution of step lengths for all observations fitted better
with exponential law (R2 = 0.98, F1,69 = 4060, p,0.00001,
y = 0.6384e20.004x, Figure 2) than to power law (R2= 0.81, F1,69 =
291, p,0.00001, y = 150.86x21.256, Figure 2). More specifically,
this distribution better fits with a Weibull function (R2 = 0.99,
F1,69 = 12481, p,0.00001, y~e{0:006�x
0:99
). The Weibull distri-
bution is a continuous probability distribution with a tail heavier
than the one of exponential function [31]. Similarly, analysis of
step lengths between 10:30 and 13:30 shows that the distribution
more followed an exponential law (R2= 0.97, F1,234 = 8867,
p,0.00001, y=1.4419e20.001x) that a power one (R2=0.61, F1,234 =
360, p,0.00001, y=33.501x20.639). This confirms the high variabil-
ity of step lengths. The walk, however, seems to be more Brownian
than a Le´vy one because of the exponential distribution.
The distribution of waiting times also shows high variability.
Chacma baboons were stationary for a minimum of about ten minutes
to a maximum of about two hours (Figure 3). The distribution of these
waiting times seems however to follow an exponential curve (R2=0.99,
F1,8=1597, p,0.00001, y=1.7668e
20.0005x, Figure 3) rather than a
power one (R2=0.87, F1,8=57, p,0.00001, y=113303x
21.589,
Figure 3). Here again, animals seem to walk according to a Brownian
process rather than a Le´vyesque one because of the exponential
distribution.
The squared displacement of the group shows that animals go
away from one of their sleeping sites in the morning but come back
to it for the evening (Figure 4). This parabolic shape is confirmed
by a curve estimation test (R2 = 0.65, F1,22 = 38, p,0.00001,
y =20.003x2 +7.186x23311.3). The maximum squared displace-
ment is at 11:30.
The distribution of the mean step length per hour showed that
animals speeded up when leaving their sleeping site in the morning
(maximum at 8:30 a.m.) and when coming back to it at the end of
the afternoon (maximum at 15:30 p.m.) (Figure 5). Indeed, the
curve from 8:30 a.m. to 15:30 p.m. follows a parabolic law
(R2= 0.36, F1,14 = 12, p= 0.004, y = 0.0008x
221.886x +1258.5).
Finally, the mean step length differs according to the kind of
vegetation animals are going to (N=490, x2= 13.7, p=0.032).
Moreover, this mean step length is correlated with the time animals
foraged in each kind of vegetation (R2=0.89, F1,4 =32, p=0.005,
y=0.00003e0.0493x). This shows that the more animals need to eat a
specific species (correlated with the foraging time in each type of
vegetation), the more they speed up to go to the respective area.
Random Walk in Baboons
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Discussion
This study shows that chacma baboons present evidence of
random walks in their daily trajectories. Distributions of step
lengths as well as waiting times follow exponential laws, suggesting
a Brownian process. These baboons, however, also seem to have a
routine: starting every day from their sleeping site, going first to a
waterhole, going in almost each case to a clay site and then coming
back to their sleeping site for the evening. This result suggests that
animals do not travel randomly in their environment but have a
mental map. These two apparent opposite results should be
discussed in order to understand what really happens in baboons’
minds.
Viswanathan and colleagues [4,6,28] suggest that the Le´vy walk
is an optimal food research strategy for animals and therefore
should be found in almost all species. This study shows, however,
that this group of chacma baboons does not use the Le´vy walk but
Figure 1. Four daily trajectories of the study group of chacma baboons. S represents the location where animals started their daily travel. E
represents the location where animals ended their travel. Circles represent sleeping sites. Squares are waterholes. The triangle is a clay site.
doi:10.1371/journal.pone.0016131.g001
Figure 2. Inverse cumulative distribution of step lengths. The
inset (b) shows the log plot of the same data. Circles are observed data.
The continuous line is the theoretical exponential curve representing
the Brownian random walk. The dotted line is the theoretical power
curve representing the Le´vy walk.
doi:10.1371/journal.pone.0016131.g002
Figure 3. Inverse cumulative distribution of waiting times. The
inset (b) shows the log plot of the same data. Circles are observed data.
The continuous line is the theoretical exponential curve representing
the Brownian random walk. The dotted line is the theoretical power
curve representing the Le´vy walk.
doi:10.1371/journal.pone.0016131.g003
Random Walk in Baboons
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rather a Brownian process. Actually, instead of being an optimal
strategy, the Le´vy walk may also be a pattern emerging from the
environment. Distribution of food patches or prey might lead
animals to change from one kind of random walk to another one,
from Brownian to Le´vy walk for instance. Humphries et al. [3]
found that movement patterns in marine predators depend on
environmental context. The Le´vy walk should be adopted when
resources are sparse and unpredictably distributed whereas
Brownian movement is efficient when resources are abundant
and homogeneously distributed. Consequently, Boyer et al. [15]
explained how the Le´vy walk found in spider monkeys is
reproduced by a simple model where animals forage in a spatially
disordered environment with patches of heterogeneous size
distribution. The previous findings and the results of this study
suggest that random walks by chacma baboons depend more on
resource distribution in their environment than on a cognitive
process. This should also explain why the Le´vy walk and not the
Brownian one was found in hamadryas baboons: patches in the
environment of the studied hamadryas baboons were sparser than
those in that of chacma baboons [23].
Another striking point is finding random walks in species known
for their high cognitive abilities [32]. Similar questions were also
raised in studies on spider monkeys [14] and hamadryas baboons
[23]. Some findings in spider monkeys do not point to a purely
Le´vy walk in this species but suggest that animals also use mental
maps. Spider monkeys come back almost every night to specific
sleeping sites. The shape of their home range looks like a circle.
They often use the same route when returning to the sleeping site.
These results suggest a knowledge base in their memory that
animals use. This is contradictory with Le´vy walk as an optimal
research strategy. This assumption was confirmed by a model
where animals using memory and a metal map to search for food
in a spatially disordered environment exhibited random walks
[15]. The studied chacma baboons show similar cognitive
processes. Their daily trajectory looks like an ellipse going through
specific locations such as waterholes and clay sites and with
sleeping sites as start and end points. Analyses of their step lengths
also show increasing speed when baboons leave or arrive at their
sleeping site. Baboons also speed up when going to important food
locations. This result suggests goal directness in animals: they
know where to go. Noser & Byrne [27] found similar results in
another group of chacma baboons. By studying travel speed and
route linearity of baboons, they found that animals seem to plan
their journeys and actively choose their out-of-sight resources,
reaching them in an efficient and goal-directed way. These
characteristics allow us reasonably to infer the presence of mental
maps and use of memory [26,33,34]. Chacma baboons in this
study should use the same cognitive processes to find their way but
it is also possible that sometimes animals use a random walk to find
new and unknown sources, switching between the two processes
[14].
This study confirms that random walks, whether Brownian or
Le´vyesque, should not be considered as the only food strategies
animals have. Existence of these processes in animals may depend
on food patch distribution but may also be used as an alternative
strategy to find new resources when known food patches are
depleted. Instead of only considering resources, we also need to
assess how predation risk influence movement patterns of animals.
Acknowledgments
I would like to thank Alexandre Brotz and Paula Pebsworth for their help
on the field and Marie Pele´ and Gonzalo G. de Polavieja for their
comments on this manuscript.
Author Contributions
Conceived and designed the experiments: CS. Performed the experiments:
CS. Analyzed the data: CS. Contributed reagents/materials/analysis tools:
CS. Wrote the paper: CS. Other: CS.
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