3078
Ecology,
86(11), 2005, pp. 3078–3087
q
2005 by the Ecological Society of America
ANIMAL SEARCH STRATEGIES: A QUANTITATIVE
RANDOM-WALK ANALYSIS
F
REDERIC
B
ARTUMEUS
,
1,2,5
M. G. E.
DA
L
UZ
,
3
G. M. V
ISWANATHAN
,
4
AND
J. C
ATALAN
1
1
Centre d’Estudis Avanc¸ats de Blanes (CEAB), CSIC, Acce´s a la Cala Sant Francesc, 17300 Blanes (Girona), Spain
2
ICREA-Complex Systems Research Laboratory, Universitat Pompeu Fabra C/ Dr. Aiguader 80 08003 Barcelona, Spain
3
Departamento de Fı´sica, Universidade Federal do Parana´, 81531–990 Curitiba-PR, Brazil
4
Departamento de Fı´sica, Universidade Federal de Alagoas, 57072–970, Maceio´-AL, Brazil
Abstract.
Recent advances in spatial ecology have improved our understanding of the
role of large-scale animal movements. However, an unsolved problem concerns the inherent
stochasticity involved in many animal search displacements and its possible adaptive value.
When animals have no information about where targets (i.e., resource patches, mates, etc.)
are located, different random search strategies may provide different chances to find them.
Assuming random-walk models as a necessary tool to understand how animals face such
environmental uncertainty, we analyze the statistical differences between two random-walk
models commonly used to fit animal movement data, the Le´vy walks and the correlated
random walks, and we quantify their efficiencies (i.e., the number of targets found in relation
to total displacement) within a random search context. Correlated random-walk properties
(i.e., scale-finite correlations) may be interpreted as the by-product of locally scanning
mechanisms. Le´vy walks, instead, have fundamental properties (i.e., super-diffusivity and
scale invariance) that allow a higher efficiency in random search scenarios. Specific bio-
logical mechanisms related to how animals punctuate their movement with sudden reori-
entations in a random search would be sufficient to sustain Le´vy walk properties. Fur-
thermore, we investigate a new model (the Le´vy-modulated correlated random walk) that
combines the properties of correlated and Le´vy walks. This model shows that Le´vy walk
properties are robust to any behavioral mechanism providing short-range correlations in
the walk. We propose that some animals may have evolved the ability of performing Le´vy
walks as adaptive strategies in order to face search uncertainties.
Key words: correlated random walks; foraging theory; Le´vy walks; random search strategies.
I
NTRODUCTION
Standard methods in spatial ecology consider
Brownian motion and Fickian diffusion as two basic
properties of animal movement at the long-term limit
(i.e., large spatial scales and long temporal scales).
Thus, it is assumed that animal movements can be mod-
eled (at the long-term limit) as uncorrelated random
walks (Okubo 1980, Berg 1983). The problem of un-
correlated random walks is that they do not account
for directional persistence in the movement (i.e., the
tendency by animals to continue moving in the same
direction). Such limitation was overcome with two dif-
ferent types of random walks, correlated random walks
(CRWs) and Le´vy walks (LWs).
CRWs appeared in ecology from the analysis of short
and middle-scaled animal movement data. Experiments
with ants, beetles, and butterflies were performed in
less than 25-m
2
arenas, or otherwise, in their natural
environments, and usually last less than an hour (e.g.,
Bovet and Benhamou 1988, Turchin 1991, Crist et al.
1992). From these studies, ecologists promptly became
Manuscript received 2 December 2004; revised 20 April 2005;
accepted 27 April 2005. Corresponding Editor: J. Huisman.
5
E-mail: fbartu@ceab.csic.es
aware of the necessity of adding directional persistence
into pure random walks to reproduce realistic animal
movements (Kareiva and Shigesada 1983, Bovet and
Benhamou 1988). More recently, the mathematical
properties of CRWs were used to explore the link be-
tween individual animal movements and population-
level spatial patterns (Turchin 1991, 1998). Further
studies have considered the relative straightness of the
CRW (i.e., degree of directionality; [Haefner and Crist
1994], or sinuosity, [Bovet and Benhamou 1991, Bovet
and Bovet 1993, Benhamou 2004]) as relevant prop-
erties characterizing animal movement.
The analysis of animal movement at larger spatial
scales or at longer temporal scales has given rise to a
new category of random-walk models known as Le´vy
walks (Levandowsky et al. 1988
a,
Viswanathan et al.
1996). Animal paths involving large spatial or temporal
scales (i.e., large-scale animal movement), turn out to
be a combination of ‘‘walk clusters’’ with long travels
between them. The heterogeneous multiscale-like sam-
pling pattern generated by such paths are closely re-
lated to fractal geometries (Mandelbrot 1977) and bet-
ter modeled by random walks with Le´vy statistics. LWs
have their origin in the field of statistical mechanics
and find wide application in physics (Shlesinger et al.
November 2005 3079
ANIMAL SEARCH STRATEGIES
1995, Klafter et al. 1996, Weeks and Swinney 1998)
and natural sciences such as geology and biology (Met-
zler and Klafter 2004). Although they have only re-
cently gained attention in optimal foraging theory (Vis-
wanathan et al. 1996, 1999), they appeared in an eco-
logical context around the same decade as CRWs. The
first mention of Le´vy walks as animal search strategies
can be found in Shlesinger and Klafter (1986:283).
After that, Le´vy walks were formally considered by
plankton ecologists (Levandowsky et al. 1988
a
,
b,
Klafter et al. 1989).
CRW and LW models have been adjusted success-
fully to a wide range of empirical data (CRWs [Kareiva
and Shigesada 1983, Bovet and Benhamou 1988, Tur-
chin 1991, Crist et al. 1992, Johnson et al. 1992, Berg-
man et al. 2000], LWs [Viswanathan et al. 1996, Lev-
andowsky et al. 1997, Atkinson et al. 2002, Bartumeus
et al. 2003, Ramos-Ferna´ndez et al. 2004]). Recent
works have introduced the idea of hierarchical scale
adjustments on animal displacements (Fritz et al.
2003), and have fitted field data of specific species
(Marell et al. 2002, Austin et al. 2004) by using both
models. All these studies have shown that CRWs and
LWs can be used as fitting procedures to analyze animal
movement. Nevertheless, there is a lack of an expli-
cative framework for such an approach, which severely
limits the biological interpretation of the obtained re-
sults. A better understanding of random searching pro-
cesses may help to develop random-walk models with
sound explicative power, sensu Ginzberg and Jensen
(2004). This knowledge could clarify how animals face
environmental uncertainty and reduced perceptual ca-
pabilities in large-scale displacements (Lima and Zoll-
ner 1996). Further, a solid relationship between animal
behavior and the statistical properties of movement
could be established, thus uncovering useful links be-
tween the behavioral (Bell 1991) and the pattern-based
approaches common in spatial ecology (Okubo 1980,
Tilman and Kareiva 1997).
Although some theoretical studies have already
shown the potential role of CRWs and LWs in the un-
derstanding of animal random search strategies (Zoll-
ner and Lima 1999, Viswanathan et al. 1999), two fun-
damental questions about CRW and LW models still
need to be addressed. First, quantifying their efficiency
as random search strategies based on their respective
statistical and scaling properties. Second, developing
adequate biological interpretations of such properties
in a random search context. The present contribution
is a first effort to clarify the above points. For doing
so, we have structured our analysis as follows. First,
we demonstrate quantitatively relevant differences in
the statistical properties of CRWs and LWs. Then, we
discuss how such properties explain the different ef-
ficiencies obtained when the models are used as random
search strategies in the ecological context. Finally, we
suggest how the present results may lead to a better
theoretical understanding of some fundamental aspects
of large-scale animal displacements in real ecosystems.
We would like to emphasize that our goal is not to
provide recipes to analyze specific empirical data and
determine which models would lead to a better fitting
in a particular case. Instead, our purpose is to provide
general criteria to evaluate why we should expect one
of the models to fit better. Providing explicative power
to random-walk models is especially necessary if such
models are going to be used as null models, as well as
if deviations from such null models are going to be
interpreted biologically.
M
ETHODS
Random walks constitute probabilistic discrete step
models that involve strong simplifications of real an-
imal movement behavior. In relation to more complex
behavioralist models including many parameters, ran-
dom-walk models ultimately express behavioral min-
imalism (Lima and Zollner 1996, Turchin 1998). Their
main basic assumption holds that real animal move-
ments consist of a discrete series of displacement
events (i.e., move lengths) separated by successive re-
orientation events (i.e., turning angles). Discretization
of complex movement behaviors will determine (after
a large enough number of successive moves) the sta-
tistical distribution of displacement lengths on the one
hand, and the statistical distribution of changes of di-
rection (i.e., turning angles) on the other hand. From
successive random draws of such distributions, we can
obtain different movement path realizations. All the
paths obtained by this method have statistical equiv-
alence.
The models
We have used three random-walk models in our
quantitative analysis. Correlated random walks
(CRWs), Le´vy walks (LWs), and a new model based
on the previous ones which we have named Le´vy-mod-
ulated correlated random walks (LMCRWs). Each
model controls the directional persistence of the move-
ment (i.e., the degree of correlation in the random walk)
in a different way. Below we briefly discuss each model
and the simulation procedures, leaving to the Appendix
all the technical details.
CRW models combine a Gaussian (or other expo-
nentially decaying) distribution of move lengths (i.e.,
displacement events) with a nonuniform angular dis-
tribution of turning angles (i.e., reorientation events).
These models control directional persistence (i.e., the
degree of correlation in the random walk) via the prob-
ability distribution of turning angles. In our study, we
have used a wrapped Cauchy distribution (WCD [Bat-
schelet 1981, Haefner and Crist 1994]) for the turning
angles. Directional persistence is controlled by chang-
ing the shape parameter of the WCD (
r
). For
r5
0,
we obtain a uniform distribution with no correlation
between successive steps, thus Brownian motion
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FREDERIC BARTUMEUS ET AL.
Ecology, Vol. 86, No. 11
F
IG
. 1. (a) Shape of the wrapped Cauchy distribution used in the correlated random walk, for different values of the
shape parameter
r
. (b) Examples of correlated random walks, generated by wrapped Cauchy distributions with different shape
parameters. (c) Power-law distributions used in the Le´vy walk, for different values of the Le´vy exponent
m
. (d) Examples
of Le´vy walks, generated by power-law distributions with different Le´vy exponents.
emerges. For
r5
1, we get a delta distribution at 0
8
(Fig. 1a), leading to straight-line searches (Fig. 1b).
LW models involve a uniform distribution for the
turning angles, but a power-law distribution for the
move lengths (i.e., the so-called flights). The exponent
of the power-law is named the Le´vy index (1
,m#
3, see Fig. 1c) and controls the range of correlations
in the movement. LW models thus comprise a rich
variety of paths ranging from Brownian motion (
m $
3) to straight-line paths (
m →
1; Fig. 1d).
Finally, the LMCRW model introduced here gen-
erates a random walk with (i) a WCD for the turning
angles within a flight, (ii) a Gaussian distribution of
move steps within a flight, (iii) a uniform distribution
for the turning angles between flights, and (iv) a power-
law distribution of flight lengths. As in the LW model,
the directional persistence of LMCRW is also intro-
duced through a power-law distribution of move
lengths (i.e., flights) but we can also modulate or con-
trol the degree of directional persistence during flight
lengths through a WCD of turning angles (i.e., by
changing the value of
r
). This new model can reveal
which type of directional persistence controls the op-
timization of random searches, whether the power-law
distribution of move lengths or the WCD of turning
angles.
The simulations
The statistical properties of random-walk models
should be evaluated at the long-term limit (i.e., large
spatial scales and long temporal scales). When running
simulations, this means that both the turning angle and
the move length probability distributions should be
thoroughly sampled (i.e., this is especially important
with long-tailed probability distributions). The long-
term statistical properties of random searches only
emerge once a minimum amount of time and space are
included in the search. The spatiotemporal scales re-
quired for that are not fixed, but are organism specific.
A first group of simulations studied the behavior of
a relevant macroscopic property of random walks: the
mean square displacement (msd), defined as the
squared distance that an organism moves from its start-
ing location to another point during a given time, av-
eraged over many different random walkers. Msd is
related to the CRW metric of net squared displacement
but is not exactly the same (see the Appendix for more
details). In this set of simulations, we computed the
November 2005 3081
ANIMAL SEARCH STRATEGIES
msd for a set of random walkers moving in a two-
dimensional arena at different times considering dif-
ferent parameter values for
r
and
m
in CRWs and LWs,
respectively.
We devised a second group of simulations in order
to determine the search efficiencies (
lh
) of the three
types of random walks (i.e., CRW, LW, and LMCRW
models). The objects that are looked for are called tar-
gets. In general, a target may represent any important
resource for a searcher (i.e., food, mates, breeding hab-
itats, nesting sites, etc.). In our simulations, targets are
nonmobile, thus we prefer the term target sites (e.g.,
static resources, suitable habitats, etc.). We defined the
search efficiency function
h
as the ratio of the number
of target sites visited to the total distance traversed by
the searcher. Note that in LWs,
h5h
(
m
), in CRWs,
h
5h
(
r
), and in the LMCRWs,
h5h
(
m
,
r
). Specifically,
the simulations quantified the average search efficiency
of a set of random walkers provided with a radial de-
tection distance
r
d
, that looked for nonmobile circular
items with radius
r
t
(i.e., target sites) in a two-dimen-
sional space with periodic boundary conditions. Target
sites were uniformly distributed in an otherwise ho-
mogeneous arena. The scaling of the search scenarios
is based on a unique key parameter: the mean free path
(
l
), which is defined as the average distance between
two target sites. The mean free path is inversely related
to the density of target sites and the searcher’s detection
radius and gives us the idea of how far the searcher
moves before ‘‘detecting’’ a target (see Appendix). We
defined three different search scenarios with increasing
values of
l
representing a decreasing gradient of target
site densities (we kept the same searcher’s detection
radius for the three search scenarios). To represent dif-
ferent search strategies, we ran the simulations using
different parameter values for each random-walk model
(i.e., LW and CRW). The product
lh
allows us to obtain
a metric for the search efficiency that is independent
of the target site density.
We considered two kinds of encounter dynamics in
the efficiency simulations: destructive and nondestruc-
tive. In the case of nondestructive searches, the search-
er can visit the same target site many times. This ac-
counts for those cases in which target sites become
only temporarily depleted or searchers become satiated
and leave the area. In the case of destructive searches,
the target site found by the searcher becomes unde-
tectable in subsequent displacements—the target site
‘‘disappears.’’ In this case, just to make averages al-
ways with the same target density, we generated a new
target site at random in the searching space. Both types
of encounter dynamics may represent real ecological
situations and should demand different random-search
strategies in order to optimize the rate of encounters
(Viswanathan et al. 1999). The nondestructive and de-
structive searching scenarios represent the limit cases
of a continuum of possible target regeneration dynam-
ics (Raposo et al. 2003). Moreover, the nondestructive
case with uniformly distributed targets bears a simi-
larity to a destructive case with patchy or fractal target-
site distributions (Viswanathan et al. 1999). Thus, these
simulations cover a wide range of natural searching
situations.
R
ESULTS
On the macroscopic properties of CRWs and LWs
Random-walk theory assumes that a particularly rel-
evant macroscopic property of random walks involves
the scaling in relation to time of the mean square dis-
placement (msd) of the diffusing organisms:
^
R
(
t
)
2
&;
t
a
, where
a
characterizes the behavior of diffusive pro-
cesses. In normal (i.e., Fickian) diffusive processes, the
msd increases linearly with time (
a5
1). The simplest
example of this is particles (or organisms) moving in-
dependently and executing uncorrelated random walks;
i.e., pure Brownian motion. On the other hand, pro-
cesses that lead to a nonlinear dependence of msd over
time, known as anomalous diffusion, typically occur
in complex or long-range correlated phenomena (Gefen
et al. 1983). Anomalous diffusion arises due to long-
range statistical dependence between steps in a random
walk and can involve a subdiffusive (
a,
1) or a
superdiffusive (
a.
1) process. The fastest possible
superdiffusion occurs when particles (or organisms)
execute unbroken straight-line paths corresponding to
ballistic motion or dispersal with
a5
2.
As stated above, CRW models control persistence
(i.e., the degree of correlation in the random walk) via
the probability distribution of turning angles. However,
from the macroscopic point of view, CRWs represent
simple Markovian processes that, by their very nature,
cannot generate long-range correlations in the move-
ment (Johnson et al. 1992). Thus, for CRWs, the msd
can depart from the linear increase with time only over
a particular range of temporal and spatial scales, but,
at the long-term limit, the relation always becomes lin-
ear. Therefore, at the long-term limit, CRWs models
appear like uncorrelated random walks: they can only
give rise to Brownian motion. Fig. 2a shows the be-
havior of the msd in a CRW as we vary the shape
parameter
r
of the WCD (used to correlate the steps).
For any value
r,
1 (even for values close to 1, e.g.,
r5
0.95) the macroscopic behavior of movement con-
verges rapidly (
t
ø
100) to the Brownian motion do-
main. Only for the limit case of
r5
1 do we obtain a
ballistic motion. Thus, there is no a smooth way to go
from Brownian to ballistic motion by changing the
turning angle distribution parameter of CRWs (
r
). In-
stead, only two macroscopic motions emerge in the
long term limit: pure Brownian (
r,
1) or ballistic
dispersal behavior (
r5
1).
When persistence arises through a power-law distri-
bution of move lengths instead of Markovian short-
range angle correlations, a new property emerges be-
cause of long-range move length correlations. A grad-
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FREDERIC BARTUMEUS ET AL.
Ecology, Vol. 86, No. 11
F
IG
. 2. Mean square displacement (msd), obtained by averaging 500 individuals, in relation to time for (a) correlated
random walks and (b) Le´vy walks. The scaling behavior indicates short (
a5
1) and long-range (
a.
1) correlations in the
random walks;
a5
2 indicates ballistic (i.e., straight-line) motion.
ual change in the Le´vy exponent (i.e.,
m
) corresponds
to a gradual change in the diffusivity (i.e.,
a
) that does
not vanish at the long-term limit. A gradual transition
from normal diffusion (
a5
1 for
m $
3) to ballistic
motion (
a →
2 for
m →
1) becomes possible for LWs
(Fig. 2b). Therefore, different Le´vy exponents of the
power-law distribution of move lengths provide a
whole variety of super-diffusive behaviors (1
,a,
2 for 1
,m,
3). Thus, changing the Le´vy exponent
implies a qualitative change in the macroscopic and
long-term properties of the movement as a whole.
On the search efficiency of the random-walk models
Fig. 3 shows the changes in the searching efficiency
measured as
lh
of both a CRW (
h
(
r
)) and a LW (
h
(
m
))
when varying the parameters controlling the degree of
persistence in the walk. We have considered three
search scenarios (
l5
100, 1000, and 5000 [see
Meth-
ods
and Appendix]) representing a decreasing gradient
from high to low target densities, and two encounter
dynamics: destructive and nondestructive.
In all cases, LWs are more efficient than CRWs. As
density diminishes (i.e.,
l
increases), LWs become
even more efficient (than CRWs) in both dynamical
types of searches, but with different optimal Le´vy ex-
ponents (
m
opt
). In the destructive case,
m
opt
→
1 and, in
the nondestructive case,
m
opt
ø
2. These results agree
with previous works on Le´vy random-walk searches
(Viswanathan et al. 1996, 1999). Note the convergence
of CRWs (
r5
0) with LWs (
m5
3) and CRW (
r5
1) with LW (
m →
1). In the former, both models cor-
respond essentially to Brownian motion, whereas, in
the latter, they give rise to straight-line motion (i.e.,
ballistic dispersal behavior). The relevant differences
appear precisely in the transition from Brownian to
ballistic motion. Within the whole range of possible
random walks from the Brownian (pure random walk)
to the ballistic (straight-line walk), searchers perform-
ing LWs exhibit higher efficiency than searchers per-
forming CRWs in the long-term encounter statistics
(i.e.,
h
(
m
opt
)
$ h
(
r
opt
) in Fig. 3).
In destructive searches (Fig. 3), revisiting target sites
penalizes the search efficiency because targets are con-
sumed. Therefore, the larger the persistence in the
movement, the larger the search efficiency. Persistence
increases with increasing
r
in CRWs and decreasing
m
in LWs. However, changes in Le´vy exponent not only
modify short-term persistence of the walk but also in-
volve concomitant changes in the macroscopic prop-
erties of the movement that the CRWs do not have. As
m
decreases, superdiffusivity of movement is enhanced
(see Fig. 2b). Superdiffusion increases the efficiency
beyond short-ranged persistence; that is why LWs are
more efficient than CRWs in destructive searches.
In nondestructive searches (Fig. 3), revisiting sites
is not penalized because targets are not consumed.
Therefore, persistence and superdiffusivity do not in-
fluence search efficiency significantly. Indeed, they are
useful to avoid empty areas created by destructive en-
counter dynamics. This fact explains why the efficiency
of CRWs in the nondestructive case is
r
independent.
However, the higher values for the LWs efficiency
(which furthermore remains dependent on
m
) may be
due to another particular LW property not shared with
CRWs, namely, scale invariance. Thus, our results
clearly show that scale invariance plays a crucial role
in optimizing encounter rates in the nondestructive
cases.
Fig. 4 shows the searching efficiency measured as
lh
for the LMCRW model
h
(
m
,
r
) and
l5
5000 in
destructive and nondestructive searches. Changes in the
r
and
m
parameters account for different searching
