Characteristic Spatial and Temporal Scales Unify Models of Animal Movement.Author(s): Eliezer Gurarie and Otso OvaskainenSource: The American Naturalist, Vol. 178, No. 1 (July 2011), pp. 113-123Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/660285 .Accessed: 10/07/2011 14:22

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vol. 178, no. 1 the american naturalist july 2011

Characteristic Spatial and Temporal Scales Unify

Models of Animal Movement

Eliezer Gurarie1,2,* and Otso Ovaskainen1

1. Department of Biosciences, University of Helsinki, Viikinkaari 1, FI-00014 Helsinki, Finland; 2. National Marine MammalLaboratory, Alaska Fisheries Science Center, National Oceanic and Atmospheric Administration, 7600 Sand Point Way NE, Seattle,Washington 98115

Submitted November 28, 2010; Accepted March 7, 2011; Electronically published June 1, 2011

abstract: Animal movements have been modeled with diffusionat large scales and with more detailed movement models at smallerscales. We argue that the biologically relevant behavior of a wideclass of movement models can be efficiently summarized with twoparameters: the characteristic temporal and spatial scales of move-ment. We define these scales so that they describe movement behaviorboth at short scales (through the velocity autocorrelation function)and at long scales (through the diffusion coefficient). We derive thesescales for two types of commonly used movement models: the dis-crete-step correlated random walk, with either constant or randomstep intervals, and the continuous-time correlated velocity model.For a given set of characteristic scales, the models produce verysimilar trajectories and encounter rates between moving searchersand stationary targets. Thus, we argue that characteristic scales pro-vide a unifying currency that can be used to parameterize a widerange of ecological phenomena related to movement.

Keywords: autocorrelation, correlated random walk, diffusion, en-counter rates, correlated velocity movement model.

Introduction

Animal movement is fundamental to many ecological phe-nomena occurring on a wide range of temporal and spatialscales. At the smallest scales, movements reflect immediatetactical responses to stimuli. At the largest scales, move-ments are related to dispersal, migration, and colonization.Processes related to foraging, predator avoidance, or mateencounter often occur at intermediate scales. At all scales,movements are constrained by bioenergetic limitations,physical constraints, and behavioral imperatives.

Because animal movements are the result of complexbehavioral responses to internal states, environmentalcues, and biophysical constraints, it is generally impossibleto measure or model all of these interactions explicitly(Codling et al. 2008; Nathan et al. 2008). Consequently,

* Corresponding author; e-mail: eli.gurarie@noaa.gov.

Am. Nat. 2011. Vol. 178, pp. 113–123. � 2011 by The University of Chicago.

0003-0147/2011/17801-52566$15.00. All rights reserved.

DOI: 10.1086/660285

movements are often modeled as a stochastic process withintrinsically random velocities and orientations summa-rized by probability densities. The earliest models of ran-dom movement were based on simple random-walk as-sumptions of Brownian motion, leading to diffusion(Skellam 1951; Okubo 1980; Turchin 1998). Diffusionmodels, which are well understood mathematically (Ovas-kainen and Cornell 2003; Ovaskainen 2008), have provenuseful for modeling a variety of complex systems, includ-ing identifying and predicting patterns of dispersal in het-erogeneous habitats (Turchin 1998; Ovaskainen et al.2008b) and explaining variability in heterogeneous pop-ulations of migrating and dispersing organisms (Gurarieet al. 2009b). Brownian motion can be a useful model ofmovement at temporal scales substantially greater than thescale of autocorrelation (Bartumeus et al. 2005; Visser andKiørboe 2006) and have consequently been most fruitfullyapplied to explain relatively coarse-resolution data such asmark-recapture data (Ovaskainen et al. 2008a).

A natural extension of the simple random-walk modelis the correlated random-walk (CRW) model, which in-cludes directional persistence. While an extensive math-ematical analysis of CRWs dates over half a century (Patlak1953a, 1953b), CRWs gained prominence only after a sem-inal study of butterfly movements by Kareiva and Shige-sada (1983). Since then, parameters of the CRW have beenestimated for a variety of actual animal movement datasets (e.g., Byers 2001; Morales et al. 2004; Fortin et al.2005; Coltelli et al. 2008; Patterson et al. 2008). In parallel,the statistical properties of the CRW have been extensivelyexplored (see Bovet and Benhamou 1988; Viswanathan etal. 2005; Bartumeus et al. 2008).

While the CRW is most directly applicable to discretelysampled movements with independent steps, an alternativeapproach is to model movement as the integral of a sto-chastic velocity process. A straightforward example is theOrnstein-Uhlenbeck (OU) mean-reversion process (Uh-lenbeck and Ornstein 1930), which was developed as avelocity model to describe the movements of video-tracked

114 The American Naturalist

microscopic organisms (Dunn and Brown 1987; Alt 1988,1990). Alternative formulations of this model have beenstudied and applied to a variety of organisms, ranging frommotile algae to marine mammals (Blackwell 1997; Johnsonet al. 2008; Gurarie et al. 2009b, 2011; Nouvellet et al.2009; Reynolds 2010). These correlated velocity modelsare most relevant to very high-resolution data with non-independent steps or irregularly sampled data where a con-tinuous description of movement is a natural startingpoint (Johnson et al. 2008; Gurarie et al. 2009b; Nouvelletet al. 2009).

The approaches outlined above differ with respect totheir structural assumptions and the scales at which theyoperate. On the one hand, correlation in movement isnecessarily observed at high sampling rates; on the other,diffusion models are valid at scales that are longer thanthe length and time scales of autocorrelation in the move-ment. Several studies have approached the question ofidentifying the scales that characterize the transition be-tween ballistic and diffusive limits, leading to alternativedefinitions of characteristic scales derived for various CRW(Jeanson et al. 2003; Viswanathan et al. 2005; Visser andKiørboe 2006) and continuous-movement (Nouvellet etal. 2009; Gurarie et al. 2011) models. All of these resultsshare the same intuitive interpretation in terms of quan-tifying the time scale at which correlated movements be-come uncorrelated.

In this study, we propose a definition for the temporaland spatial scales of stochastic movements that can beapplied to general discrete- and continuous-movementmodels. These scales rigorously quantify the transition be-tween the ballistic and diffusive limits, provide a commoncurrency by which to compare diverse types of movementmodels, and thus generalize and unify the special casespresented in the earlier literature. We illustrate via sim-ulation the utility of the characteristic scales by showingthat they are sufficient for predicting mesoscale phenom-ena such as encounter rates between searchers and targets.We thus argue that the characteristic scales of movementare sufficient for parameterizing a wide range of ecologicalphenomena.

Methods

Movement Models

Though actual movements contain many different behav-ioral modes and may be influenced by a variety of externalstimuli, the discussion here is constrained to a single be-havioral unit of movement that is homogeneous and sta-tionary, that is, described by the same set of parametersindependent of absolute spatial location and time and withno external biases. Homogeneous and stationary move-

ments will tend toward diffusion behavior at long timescales while having a varying degree of autocorrelation atthe short scale. We consider here for simplicity only two-dimensional movement processes and denote the positionof the individual at time t by , and we analyze2z(t) � Rmodels in which finite steps are taken at regular or randomintervals and models in which the organism moves con-tinuously in time.

Correlated Random Walks. Correlated random walks(CRWs) are movement models in which the individualtakes discrete steps at finite time intervals. Step lengths

and time intervals can be drawn from arbitraryL 1 0 T 1 0joint or independent distributions. An important sum-mary statistic is the velocity variability parameter,

2AFvF Sl p ,

2AFvFS

where the speed (all symbols used to describeFvF p L /Ti i i

the movement models and encounter rate simulations aresummarized in table 1). Note that if the organisml p 1moves with constant speed, while for any randoml 1 1distribution of . We restrict the discussion here to twoFvFspecial cases. If the locations of the organism are measuredat regular intervals, it is natural to use the fixed-intervalCRW with a constant interval T. In the exponential-interval CRW, the intervals are drawn from an exponentialdistribution with mean . The fixed-interval CRW is aAT SMarkov chain in discrete time, while the exponential-interval CRW is a Markov process in continuous time(Grimmett and Stirzaker 2001).

The turning angles v between steps are derived from adistribution in , typically assumed to be unimodal[�p, p)and symmetric around 0. The degree of correlation canbe measured by the mean cosine of the turning angles,

: corresponds to an uncorrelated ran-k p Acos (v)S k p 0dom walk, and corresponds to perfectly lineark p 1movement.

Correlated Velocity Models. The second family of move-ment models is based on formulating a stochastic modelfor velocity and obtaining the position by integratingv(t)the velocity of the organism over time via

t

′ ′z(t) p z(0) � v(t )dt .�0

As one example, a two-dimensional version of the OUprocess (Uhlenbeck and Ornstein 1930; Gillespie 1996),formulated as the stochastic differential equation

Characteristic Scales of Movement 115

Table 1: Key parameters and quantities

Parameter Quantity

Correlated random walk:k Angle clustering coefficient: Acos (v)Sl Velocity variability index: 2 2AFvF S/AFvFS

2ALS, AL S Moments of step length distribution2ATS, AT S Moments of time interval distribution

Correlated velocity movement:t Time scale of autocorrelationb Standard deviation of Wiener stochasticity

All movements:t Characteristic time scale of autocorrelationj Characteristic length scale of autocorrelation

Encounter rates:a Encounter radiusr Mean density of targetsh Mean number of targets per clusterg Mean distance of targets from center of respective cluster

1dv p (v � v)dt � bdw , (1)m t

t

results in a continuously varying velocity vector. The ve-locity is perturbed by the two-dimensional Wiener processwt with variance b2 and returns from fluctuations towardthe mean velocity vm with a relaxation time t. As with theCRW, we do not consider biased movements and thus set

. We refer to this model as the correlated velocityv p 0m

movement (CVM) model. Though not nearly as wide-spread as the CRW model, this model has been appliedto biological movements of cells (Dunn and Brown 1987),diverse unicellular organisms (Alt 1988, 1990; Gurarie etal. 2011), wood mice (Blackwell 1997), and marine mam-mals (Johnson et al. 2008; Gurarie et al. 2009a).

Characteristic Scales of Movement

The fundamental descriptions of the CVM and CRW mod-els suggest that they cannot be mapped to each other one-to-one. For example, the CVM is defined regardless of thefrequency at which it is sampled, whereas the assumptionof independent move lengths and turning angles betweenconsecutive steps makes the properties of the CRW dependprofoundly on the sampling intervals. In order to intro-duce a common currency for both kinds of movementmodels, we interpret the CRW as a continuous-time pro-cess. Thus, rather than considering the intervals Ti as wait-ing times between instantaneous movements, we considermovement between positions zi and as occurring atz i�1

constant speed .L /Ti i

The short-term behavior of a movement process can becharacterized by the velocity autocorrelation function(VAF; Alt 1990; Takagi et al. 2008),

Av(t � Dt) 7 v(t)SC (Dt) p , (2)v 2AFv(t)F S

where the expectation is taken over initial times t. For ahomogeneous movement process in continuous time withno bias or persistent rotation, such as the CVM, the VAFhas the simple exponential form (Alt 1988)

�Dt/tC (Dt) p e . (3)v

The autocorrelation function of the CRW is not exactlyexponential, so the CVM and the CRW cannot be matchedexactly for their detailed short-scale behavior. However,we can define a characteristic time scale by determininga specific target value for the VAF. Because it is natural toconsider t as the characteristic time scale of the CVM, wedefine the characteristic time scale t for any movementprocess as the time lag at which the velocity autocorre-lation function attains the value of .C (t) p 1/ev

At long times, most unbiased and unconstrained sto-chastic movements are asymptotically diffusive; that is, theexpected squared displacement increases linearly with timeas

2AF(z(t) � z(0))F S p 2dDt � o(t), (4)

where the constant D is commonly referred to as the dif-fusion coefficient; d is the dimensionality of the process,restricted to in our discussion; and is a termd p 2 o(t)that becomes negligible at large t (Codling et al. 2008).The diffusion constant is sufficient to describe the as-ymptotic behavior of a movement process. However, Dhas a nonintuitive unit of distance squared over time, sowe replace it with the spatial distance scale, which (in twodimensions) can be expressed as

�j p 2 Dt, (5)

116 The American Naturalist

such that the coefficient of proportionality of the expectedsquared displacement at large time is just .2j /t

The characteristic length and time scales of movementcan be computed for any movement model following(j, t)

the definitions above. While these two parameters do notcapture all details of a movement process, we hypothesizethat many ecologically relevant key features of the move-ment process at all scales (short, long, and intermediate)are summarized by these parameters.

Can Characteristic Scales Predict Encounter Rates?

To test the extent to which the characteristic scales (j, t)provide a common currency to compare different modeltypes, we first derive them for the three movement modelsintroduced above, reparameterize the movement modelsin terms of the scales, and examine the properties of theensuing trajectories. We explore whether the characteristicscales are sufficient to predict a nontrivial ecological pro-cess, using encounter rates as a test statistic. Encounterrates are fundamental to many ecological processes, suchas foraging, predator avoidance, and mate encounter, andthey depend nontrivially on short- and long-term char-acteristics of movements (Hutchinson and Waser 2007).While there are many ways to define encounter rates, weconsider here the mean first hitting time, that is, the ex-pectation of the time that a searcher released in a randomlocation in a space populated with point targets first comeswithin a certain distance of any target. We denote thedensity of targets (number per unit area) by r and theencounter radius by . We consider three types ofa p 1landscape configurations: a regular hexagonal lattice, com-plete spatial randomness (spatial Poisson process), and aclustered random landscape. We generated clustered land-scapes by distributing cluster locations using a spatial Pois-son process, and distributing a Poisson-distributed (meanh) number of targets with exponentially distributed (meang) distances from the cluster location. For all geometries,the simulated landscapes were continuously generatedwhen the searcher moved to an area where targets werenot yet generated, thereby simulating infinite boundaryconditions. For each combination of parameter values,target distributions, and movement models, we performed200 simulations, over which we computed the mean andstandard error of the first hitting time.

Results

Characteristic Scales of the CRW

The expected squared displacement after n steps for a two-dimensional CRW was derived by Kareiva and Shigesada(1983). With some reorganization, their result can be ex-

pressed (for both the fixed-interval CRW and the random-interval CRW) as , where2 2AFz(t) � z(0)F S p j tt � o(t)

2 2 2j AL S � ALS [2k/(1 � k)]p (6)

t AT S

and represents a term that grows slower than linearlyo(t)with time. In general (see app. A), the VAF of a CRW isgiven by

�1

iC (Dt) p p (Dt) � p (Dt)k , (7)�v 0 il ip1

where is the probability that i steps have been takenp (Dt)i

in time interval Dt, starting from any random time.

Fixed-Interval CRW. Because of our continuous-time in-terpretation of CRW, decreases linearly from 1 top (Dt)0

0 in the interval while increases linearly0 ≤ Dt ≤ T p (Dt)1

from 0 to 1 over the same interval. A general formula foris derived in appendix A. Substituting these resultsp (Dt)i

into equation (7) yields

Dt k1 � 1 � for Dt ≤ T( )T lC (Dt) p . (8)v 1

Dt/T{ k for Dt 1 Tl

The second case in equation (8) is, in fact, exact only forinteger multiples of T, that is, . At fractional mul-Dt p nTtiples, the exact solution is a linear interpolation betweenthe integer multiple values (fig. 1A). From the definitionof t, we obtain

1 l l1 � T for k ≤( )( )e l � k e

t p . (9)log (l) � 1 l{ T for k 1

log (k) e

Note that for the simple fixed-interval random walk,and ; that is, the characteristic timek p 0 t p (1 � 1/e)T

scale is, somewhat counterintuitively, shorter than the in-terval between steps. As expected, t increases with k; thatis, the characteristic time scale is long if the random walkhas a high level of persistence. Corresponding formulaefor j can be obtained directly from equation (6).

Exponential-Interval CRW. In this model, the number ofsteps taken within time Dt is Poisson distributed with pa-rameter , yieldingDt/AT S

�Dt 1 kDtC (Dt) p exp 1 � �1 � exp , (10)v ( ){ [ ( )]}AT S l AT S

from which t can be solved by requiring that C (t) pv

Figure 1: Simulations of movement models (left panels) and empirical and theoretical velocity autocorrelation functions (VAFs; right panels).A, Correlated random walk (CRW) with a fixed-interval time step ; B, exponential-interval CRW with . Velocities haveT p 1/2 AT S p 1/2a Weibull distribution, with shape parameter 2, such that , and values of k have been selected so that the characteristic time scalel p 4/pt is 1, 2, or 4, matching the values of t in the correlated velocity movement (CVM) trajectories (C). For all models, values of j are setequal to t by selecting appropriate mean step lengths for the CRW and the b parameter for the CVM. The right panels illustrateALScorresponding empirical VAFs (circles) obtained by averaging over 100 trajectories and theoretical VAFs (lines) from equations (3), (8), and(10). The dotted horizontal lines show , illustrating the definition of the characteristic time scale.1/e

118 The American Naturalist

Figure 2: Plots of the characteristic scale t as a function of the velocity variability index l (right) and the clustering coefficient k (left) forfixed-interval (top) and exponential-interval (bottom) correlated random walks. The fixed-interval results are based on equation (9) andthe exponential-interval results on equation (10). The fixed and mean step durations T and are set to 1 in both cases.AT S

and j can be obtained using equation (5). The de-1/ependency of t on model parameters is illustrated in figure2. While the general solution can be obtained only nu-merically, two limiting cases are worth noting. First, forconstant speed ( ), andl p 1 t p AT S/(1 � k) j p

. Second, for uncorrelated random2 1/2ALS[(1 � k)/(1 � k) ]walk ( ), and .1/2k p 0 t p AT S j p ALS(1/l)

Characteristic Scales of CVM

The CVM model is parameterized in terms of its char-acteristic scales in a more natural way than the CRWmodel. In particular, the VAF of the CVM is given exactlyby equation (2), and thus, its characteristic time scale isdirectly the parameter t (fig. 1C). Furthermore, it can beshown (app. B) that

t2 2 3 �t/tAFz(t) � z(0)F S p 2b t � e � 1 . (11)( )t

At time scales greater than t, the second and third termsin parentheses are negligible, giving . We note3 1/2j p (2t ) b

additionally that the expected stationary speed of move-ment of a CVM, defined as , can ben p lim AFv(t)FSs tr�

expressed in terms of the characteristic scales (app. B)

�p j jn p ≈ 0.627 . (12)s � t t2 2

For a comprehensive treatment of the statistical propertiesof the OU process, from which all the properties of theCVM are derived, the reader is encouraged to consultGillespie (1996).

Simulation Results

Simulated trajectories of the fixed-interval CRW, the ex-ponential-interval CRW, and the CVM models look vi-sually very similar when their characteristic scales (j, t)are equal (fig. 1). Higher values of k for the CRW modelslead to less tortuous movements than do lower values ofk, as do higher values of t for the CVM (figs. 1, 2). Figure1 further confirms that the velocity autocorrelation func-tions derived above correspond to simulation results. In

Characteristic Scales of Movement 119

Figure 3: Hitting-time simulations. Upper panels are schematics of three different target geometries: hexagonal lattice (A), Poisson pointprocess (B), and clustered landscape (C; , ). The lower panels are corresponding simulation results. The points and the errorh p 10 g p 5bars show the means � 2 SEs based on 200 replications of the first hitting-time process. The symbols refer to the three movement models:fixed-interval correlated random walk (CRW; circles), exponential-interval CRW (triangles), and correlated velocity movement (CVM; squares).Simulations were performed for a range of density (r) values between 0.001 and 0.008, coded by color, and for a range of characteristicscale values from 0.5 to 8. Other parameter values are as in figure 1.j p t

the CVM, the velocity autocorrelation function decays ex-ponentially, while for the CRWs, the decay is only ap-proximately exponential.

Simulation results of the hitting-time experiment arepresented in figure 3. Differences in hitting times spannedmany orders of magnitude, depending on the density anddistribution of targets and on the parameters of the move-ment model. As expected, hitting times decrease with in-creasing target density, and more directed movement (highvalues of ) leads to shorter hitting times than doesj p t

tortuous movement. The regular lattice had lower meanhitting times than the Poisson target geometry, which hadlower hitting times than the clustered target geometry. In

line with our hypothesis, all movement models led to al-most identical hitting times when their characteristic scaleswere matched. This is a nontrivial result, given the wideoverall variation in hitting times and the fact that quali-tatively different models were matched using the two pa-rameters only.

Comparisons with Earlier Results

The concept of characteristic scales has been addressedseveral times in the literature on movement analysis. Webriefly review these studies here, with the aim of illustratingthe equivalence to our more general results. In all cases,

120 The American Naturalist

we restate the earlier results in terms of the notation in-troduced here.

Viswanathan et al. (2005) analyzed a CRW with constantstep length, citing the exponential decay typical of Markovprocesses to identify a dimensionless correlation functionequivalent to a discrete version of the VAF (eq. [2]). Theapproximate exponential decay term is then expressed as

. This corresponds to a special case of equa-t p �T/ log (k)tion (9) for constant step lengths ( ) and constrainedl p 1to . It bears noting that this study was later�1k 1 e ≈ 0.368revisited in some detail by Bartumeus et al. (2008) in thecontext of extensive simulations of encounter rates.

In a study exploring the transition between ballistic anddiffusive movement and encounter rates, Visser andKiørboe (2006) consider the characteristic temporal andspatial scales of “run-tumble” movements, which are sim-ilar to the random-interval CRWs discussed here. Citinga general result applicable to one-dimensional autocor-related movements (Berg 1993), they report a character-istic time scale . This is equal to the char-t p AT S/(1 � k)acteristic time scale of the exponential-interval CRWobtained here (eq. [10] and accompanying text) in thespecial case of constant speed ( ). Our results thusl p 1generalize the cited results by showing how variability instep length and speed alters the characteristic spatial andtemporal scales (fig. 2).

More recently, Nouvellet et al. (2009) presented a gen-eral model of continuous stochastic movements, also de-fining position as an integral of velocity. Rather than definean explicit model for velocity, the authors simply assumedthat velocity follows some autocorrelation function

. They considered (among other options) the ex-C (Dt)v

ponential form , corresponding to theC (Dt) p exp (�t/t)v

CVM presented here, noting that t “concisely characterizesthe decay of correlations,” matching our concept of thecharacteristic time scale (Nouvellet et al. 2009, p. 509).The authors propose that an estimate for t can be obtainedfrom the behavior of the mean square displacement

, which can be estimated from data directly2AFz(t) � z(0)F Swithout assuming any specific movement model. Theyproposed (though did not formally prove) that t corre-sponds to the time lag that leads to AFz(2t) �

. From equation (11), we obtain2 2z(0)F S/AFz(t) � z(0)F S p 3

2AFz(2t) � z(0)F S 1p � e p 3.086161 … ,

2AFz(t) � z(0)F S e

thereby showing that the authors’ assertion is indeed avery good approximation, at least in the case of the CVM.

Discussion

The idea of scales of movement that characterize the tran-sition between short-term correlated movements and

long-term diffusive behavior has cropped up with increas-ing frequency in the literature on movement modeling(e.g., Taylor 1921; Berg 1993; Jeanson et al. 2003; Vis-wanathan et al. 2005; Visser and Kiørboe 2006; Nouvelletet al. 2009). Most often, the question of characteristicscales has been addressed out of necessity when dealingwith intricacies of characterizing movements, complexsimulations of ecological processes, or interpretation ofdetailed movement data. In this work, our aim has beento define the characteristic scales in such a way that theyapply to any kind of homogeneous stochastic movementmodel and thus to unify the earlier results.

Because the characteristic scales are an intrinsic propertyof any unbiased homogeneous stochastic movement, theyalso allow for a unification of diverse types of movementmodels. One immediate application is the unification ofthe very movement models considered here. The CVM is,perhaps, a theoretically more appealing model than theCRW because many of its asymptotic and short-scale prop-erties are analytically tractable and it describes a contin-uous process independent of the sampling interval. On theother hand, the CRW parameters have the advantage ofbeing very straightforward to estimate from empirical data.The characteristic scales lead to an immediate parameter-ization of the CVM with respect to the measured CRWparameters.

Our primary conclusion is that these two parametersprovide a compact summary of a movement process.Short-term behavior, specifically the nature of the auto-correlation function and the tangential velocity of move-ment, are governed by the ratio of the length scale andthe time scale. Long-term behavior is related via the dif-fusion coefficient to the ratio of the square of the lengthscale and the time scale. The strength and the limitationof our approach are the use of only two parameters. Ob-viously, more degrees of freedom are needed to capturethe full behavior of the movement model. For example,Nouvellet et al. (2009) proposed characterizing move-ments through the behavior of mean-squared displace-ment over time, that is, a function rather than a scalar.Because our characteristic temporal scale is definedthrough the velocity autocorrelation function, it dependsboth on the geometry of the track and on variation inmovement speed and thus cannot disentangle these twoaspects of movement. Nonetheless, our simulations in-cluded a range of movements from much smaller to muchgreater scales than the typical distances between targets,yet the encounter rates were well predicted by the scales.The agreements between the simulation results suggest thatthe characteristic scales also captured essential processesoccurring at intermediate scales, regardless of the specificmovement model.

Another important limitation of our approach is that

Characteristic Scales of Movement 121

we have considered only homogeneous and unbiased pro-cesses, both generally unrealistic assumptions. Often, thelong-term behavior of a movement process cannot simplybe extrapolated from the short-term behavior, as it maybe significantly influenced by behavioral switching, rarelong-distance dispersal events, or environmental con-straints. For example, many empirical data on organismsranging from unicellular algae to whales indicate shiftsbetween foraging movements and more linear displace-ment movements (e.g., Laidre et al. 2004; Polin et al. 2009)or behavioral transitions (Gurarie et al. 2009a), often with-out a significant change in actual movement speeds. Eachmovement mode, however, fulfills a different ecologicalpurpose and is associated with its own characteristic scales.Thus, identifying the scales of each movement mode mayaid in compactly summarizing the heterogeneous processand may shed light on the ecological role of thatmovement.

It has been noted before that tortuous movements (i.e.,with small magnitudes of the characteristic scales) are in-efficient for encountering targets, regardless of the absolutevelocity of the movement (Visser and Kiørboe 2006; Bart-umeus et al. 2008), a prediction that is supported by theresults of the encounter rate simulations presented here.This observation has led to the hypothesis that movementscales should be greater than the encounter scales of tar-geted prey yet smaller than the encounter scales of re-spective predators (Visser and Kiørboe 2006). We hopethat the identification of the characteristic scales as a fun-damental, tractable property of movement will facilitatethe eventual development of a consistent theory of en-counter rates, in which encounter rates can be predictedas analytic functions of the movement scales, densities,encounter radii, and target geometry. We hope further thatthe identification of these scales will serve as a first stepfor developing a theory for animal movement that is flex-ible enough to capture ecologically relevant characteristicsand facilitate comparisons between study systems whileavoiding the level of detail often present in individual-based simulation models. A theory of this type is neededto understand the interactions between inherent propertiesof animal movement, inter- and intraspecific interactions,and environmental variability, as all of these interactionsdepend on their spatial and temporal scales.

Acknowledgments

The study was supported by the Academy of Finland (grant124242 to O.O.) and the European Research Council (ERCStarting Grant 205905 to O.O.). We are indebted to J. J.Anderson and F. P. N. Laidre for encouraging the pursuitof many of the key ideas and to two anonymous reviewers

for constructive comments and in particular for pointingout related results in the literature. The findings and con-clusions in the article are those of the authors and do notnecessarily represent the views of the National MarineFisheries Service, National Oceanic and AtmosphericAdministration.

APPENDIX A

Velocity Autocorrelation Function of CorrelatedRandom-Walk Model

We first consider any type of a correlated random walk(CRW; fixed or exponential time intervals) with

. Letting vi denote the velocity for step i andAcos (v)S p k

vi the turning angle after the ith step, it holds that

v 7 v p FvFFv F cos (v � v � … � v ).i i�k i i�k i i�1 i�k�1

Because the step lengths and turning angles are assumedto be independent, we obtain

2

LAv 7 v S p G H,i i ( )T

2LAv 7 v S p kG H .i i�1 T

To proceed to higher-order steps, we note that

cos (a � b) p cos (a) cos (b) � sin (a) sin (b)

and that consecutive turning angles are assumed to beindependent. Assuming that the turning-angle distributionis symmetric around 0, , ,2Asin vS p 0 Acos (v � v )S p ki i�1

and, more generally, kAcos (v � v � … � v )S p ki i�1 i�k�1

(see also Benhamou 2006). This yields

�1

iC (Dt) p p (Dt) � p (Dt)k , (A1)�v 0 il ip1

where pi is the probability that, starting from a randomtime, I steps are taken before time Dt.

In the case of fixed-time CRW, if ,Dt ≤ T p p 1 �0

, , and . In this case, equation (A1)Dt/T p p Dt/T p p 011 i 1

reduces to

Dt k DtC (Dt) p 1 � � . (A2)v ( )T l T

For , we note that at the discrete times ,Dt 1 T Dt p jTwhere , and . Thus,j p (1, 2, 3, …) p p 1 p p 0ipj i(j

, , , and so on,2 3C (T) p k/l C (2T) p k /l C (3T) p k /lv v v

and equation (A2) is approximately

122 The American Naturalist

1Dt/TC (Dt) ≈ k . (A3)v

l

This approximation is exact for integer multiples of T,while fractional values are linear interpolations betweenthe integer multiples. Using the expressions in equations(A2) and (A3) and setting leads to the CRWC (t) p 1/ev

characteristic time scale in equation (9).

APPENDIX B

Properties of the Correlated Velocity Movement Model

We aim first to derive the expected stationary velocity ofthe correlated velocity movement (CVM) model, definedas

n p lim AFv(t)FS,str�

assuming a CVM with parameter values b and t, initialposition , and arbitrary initial velocity ( ).Z(0) p 0 v , vx y

Each of the individual components of velocity are one-dimensional Ornstein-Uhlenbeck (OU) processes (Uh-lenbeck and Ornstein 1930) and have a Gaussian distri-bution (Gillespie 1996), with

�t/tAV (t)S p v e ,x x

�t/tAV(t)S p v e , (B1)y y

2b t�2t/tVar (V (t)) p (1 � e ),k 2

where is the index of dimension. In the long term,k � x, ythe exponential terms in the mean and variance die out,leading to stationary-velocity variables with mean 0 andvariance . Thus, we obtain2b t/2

2 2�n p lim A v (t) � v (t)Ss x ytr�

�b t2 2�p A X � X S, (B2)1 2�2

where X1 and X2 represent independent standard normalvariables (mean 0, variance 1). The root-squared sum oftwo standard independent normal variables is a x distri-bution, with 2 df, the mean of which is , giving�p/2

2� �pb t p jn p p . (B3)s �2 t2 2

To obtain the expected squared displacement of a CVMwith initial position , we note thatz(0) p 0

2 2 2Az (t)S p AZ (t)S � AZ (t)S. (B4)x y

Recalling that is a Gaussian variable (Gillespie 1996),Z (t)k

we can rewrite in terms of the standard normal var-Z (t)k

iable X via , where the mean andZ (t) p s(t)X � m(t) m(t)k

variance are given by (Gillespie 1996)2s (t)

�t/tm(t) p AZ (t)S p v t(1 � e ), (B5)k k

2s (t) p Var (Z (t))k

�2t/tt 1 � e2 3 �t/tp b t � 2(1 � e ) � . (B6)[ ]t 2

The mean of the expanded squared displacement is

2 2AZ (t)S p A(s(t)X � m(t)) Sk

2 2 2p s (t)AX S � m (t) � 2s(t)m(t)AX S (B7)

2 2p s (t) � m (t).

Equations (B5)–(B7) give the expected squared displace-ment for a one-dimensional OU process (Uhlenbeck andOrnstein 1930) with arbitrary initial velocity . To obtainVk

an overall expectation, we integrate over the stationaryVk

velocity . By equation (B1), 2lim V (t) lim AV (t)S ptr� k tr� k

, leading to . Substi-2 2 2 3 2b t/2 m (t) p b t [1 � exp (�t/t)] /2tuting these results into equation (B7) and summing thetwo components in equation (B4) gives the final expres-sion

2 2 3 �t/tAz (t)S p 2b t (tt � e � 1).

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Associate Editor: Volker GrimmEditor: Mark A. McPeek