How to Reliably Estimate the Tortuosity of an Animal's Path

ARTICLE IN PRESS

Journal of Theoretical Biology 229 (2004) 209–220

How to reliably estimate the tortuosity of an animal’s path: straightness, sinuosity, or fractal dimension? Simon Benhamou Behavioural Ecology Group, Centre d’Ecologie Fonctionelle et Evolutive, CNRS, 1919 Route de Mende, F-34293, Montpellier Cedex 5, France Received 14 November 2003; received in revised form 19 March 2004; accepted 26 March 2004

Abstract

The tortuosity of an animal’s path is a key parameter in orientation and searching behaviours. The tortuosity of an oriented path is inversely related to the efficiency of the orientation mechanism involved, the best mechanism being assumed to allow the animal to reach its goal along a straight line movement. The tortuosity of a random search path controls the local searching intensity, allowing the animal to adjust its search effort to the local profitability of the environment. This paper shows that (1) the efficiency of an oriented path can be reliably estimated by a straightness index computed as the ratio between the distance from the starting point to the goal and the path length travelled to reach the goal, but such a simple index, ranging between 0 and 1, cannot be applied to random search paths; (2) the tortuosity of a random search path, ranging between straight line movement and Brownian motion, can be reliably estimated by a sinuosity index which combines the mean cosine of changes of direction with the mean step length; and (3) in the current state of the art, the fractal analysis of animals’ paths, which may appear as an alternative and promising way to measure the tortuosity of a random search path as a fractal dimension ranging between 1 (straight line movement) and 2 (Brownian motion), is only liable to generate artifactual results. This paper also provides some help for distinguishing between oriented and random search paths, and depicts a general, comprehensive framework for analysing individual animals’ paths in a two-dimensional space. r 2004 Elsevier Ltd. All rights reserved.

Keywords: Fractal; Diffusion; Orientation; Random Search

1. Introduction more the animal is efficient, the more its path should be close to the straight line segment linking the starting Animals’ movements are involved in a large spectrum point to the goal. To measure this closeness, Batschelet of fundamental behavioural and ecological processes, (1981) promoted the use of a simple and intuitively such as navigation, migration, dispersal, space use and appealing straightness index ranging between 0 and 1, food searching. Although the animal’s speed can be a computed as the ratio between the initial beeline key factor, e.g. in prey detection (Gendron and Staddon, distance to the goal and the path length travelled to 1983; Knoppien and Reddingius, 1985) or in space use reach it. To my knowledge, however, no theoretical (Benhamou and Bovet, 1989), numerous questions arise study yet attempted to determine the reliability of this at the purely spatial level corresponding to the path index as a measure of the orientation efficiency. structure. The analysis of such a structure most often The tortuosity of random search paths is a key reduces to the main question of how to reliably measure parameter in space use, dispersal or food searching. The the ‘‘tortuosity’’ (i.e. the convoluted aspect) of the path index of sinuosity proposed by Bovet and Benhamou (e.g. Claussen et al., 1997). (1988) has proved valuable to tackle various theoretical The tortuosity of oriented paths is obviously linked to questions such as optimal search paths for a central the efficiency of the orientation mechanism involved:the place forager (Bovet and Benhamou, 1991), optimal food searching in a patchy environment (Benhamou, E-mail address: [email protected] 1992) and, maybe more surprisingly, gradient field- (S. Benhamou). based orientation (Benhamou, 1989; Benhamou and

Bovet, 1989). When applied to the analysis of actual distribution) and 1 (punctual distribution, all angles paths, however, it is restricted to a partial range of being equal). Note:arctan 2ðy; xÞ is defined as tortuosity and requires a regular path sampling (see arctanðy=xÞ for x > 0 and arctanðy=xÞþp radians for below). xo0: Since it was introduced by Dicke and Burrough (1988), the fractal analysis of animals’ paths appeared to be an alternative and promising method for measuring 2. Oriented paths the tortuosity of a random search path as a fractal dimension ranging between 1 (curvilinear path) and 2 At any location, the ratio between the magnitude of (fully jagged and wiggly path). The critique made by the movement component in the goal direction and the Turchin (1996) has thrown some doubt, however, on the magnitude of movement actually performed is equal to reliability of the fractal dimension for measuring an the cosine of the angular difference, hereafter referred to animal’s path tortuosity. as the directional error, between the movement direction This paper aims at providing a comprehensive (step orientation) y and the goal direction g: Conse- framework for analysing the structure of animals’ paths quently, the best measure of the orientation efficiency, through the answers to the following four questions:Is and thereby of the (inversely related) path tortuosity, the straightness index really a reliable estimator of the is provided by the expected value of the cosine of the animal’s orientation efficiency? Can this easily com- directional errors, E½cosðy À gÞ: It ranges between 0, for puted index be used as a reliable measure of the an animal which tends, on the average, to move tortuosity of a random search path as well? How can orthogonally to the goal direction, and 1, for an animal the sinuosity index be generalized to deal reliably with which moves in straight line towards the goal (negative any level of tortuosity of an irregularly sampled path values can be considered when an animal aims at ranging between the straight line movement and the moving away from a dangerous location rather than at Brownian motion? Is the fractal dimension a reliable approaching a goal). Two cases can be distinguished:(1) estimator of the tortuosity of an animal’s random search when the goal is located at infinity, so that the animal path? In the present paper, an estimator is considered to will never reach it, but only aims at maintaining its be reliable if it fulfils two conditions:(1) it is unbiased, course in the correct direction (e.g. a given compass preventing the occurrence of any over- or under- bearing for a migrating animal); (2) when the goal is estimation on the average, and (2) its coefficient of located at a finite distance from the starting point, and variation (standard deviation/mean) decreases with the the animal will reach it after having moved some sample size, making it possible to get a better accuracy variable path length. when considering longer paths. Except in the particular case of an animal that 2.1. Goal located at infinity (Never reached) naturally performs discrete steps (e.g. a bee moving from one flower to another; see Discussion), an animal is The situation occurring with a goal located at infinity assumed to move along a curvilinear path, which is or at a very large distance is quite simple because the discretized as a sequence of fixes at the recording level. It goal direction g (expressed with respect to the X-axis) is must be kept in mind that fixes are only sampling points: constant, whatever the animal’s current location ðXi; YiÞ: the lengths and orientations of the steps (i.e. movements The orientation efficiency E½cosðy À gÞ of a path made between successive fixes) have no biological meaning by of n steps with lengths li (i ¼ 1toi ¼ n) can be reliably themselves. This discrete step representation of curvi- estimated as the l-weightedP mean cosine of directional n linear paths proves very useful, however, because it errors,P cwðy À gÞ¼ i¼1 li cosðyi À gÞ=L; where L ¼ n provides a mathematically tractable way for both i¼1 li is the total path length travelled. The global analysing and modelling them. Many variables involved movement component in the goal direction, referred to in this paper being of angular nature, let us briefly recall as the drift G; corresponds to the distance between the a key notion of circular statistics (see Batschelet, 1981; starting point ðX0; Y0Þ and the projection of the animal’s Fisher, 1993 or Mardia, 1972) before entering the location after n steps ðXn; YnÞ on an axis running core of the subject. Any sample of n angular values through ðX0; Y0Þ with the orientation g : G ¼ x can be characterized by a mean vector, whose ðXn À X0ÞcosðgÞþðYn À Y0ÞsinðgÞ: It can be expressed PCartesian coordinates are the mean cosine cðxÞ¼ as the sum of the local (step by step)P movement n n Pi¼1 cosðxiÞ=n ¼ rðxÞ cos½fðxÞ and the mean sine sðxÞ¼ components in the goal direction: G ¼ i¼1 li cosðyi À n i¼1 sinðxiÞ=n ¼ rðxÞ sin½fðxÞ: The mean vector orien- gÞ¼Lcwðy À gÞ: The drift is therefore proportional to tation fðxÞ¼arctan2½sðxÞ; cðxÞ expresses the angular the path length, and the ratio G=L ¼ cwðy À gÞ consti- mean, and the mean vector length rðxÞ¼½cðxÞ2 þ tutes an unbiased estimator of the orientation efficiency sðxÞ20:5 expresses the concentration of the distribution ðEðG=LÞ¼E½cosðy À gÞÞ: The ratio G=L being equal to around its mean between 0 (uniform, fully dispersed a weighted mean, its variability obviously decreases ARTICLE IN PRESS S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 211 when the sample size (i.e. the step number) increases. In g is computed only step by step, whereas it varies the simplest case of a path with a constant step length p continuously during the animal’s movement. The longer and statistically independent step orientations yi; it can is the step the larger is the bias, and the unbiased be easily demonstrated that the variance of the drift estimation is therefore provided by the limit value to VðGÞ is proportional to the path length: VðGÞ¼ which the mean cosine of directional errors converges LpV½cosðy À gÞ; with L ¼ np: There usually exists some when the step length becomes infinitesimal. This limit correlation between the successive movement directions, value is equal to the ratio D=L; as illustrated by the however, so that the expression of VðGÞ is much more simple following example involving a constant step complex, but the main point (confirmed using computer length p: To reach a goal located 100 m eastward, an simulations) is that the variance remains proportional to animal moves 50 m northward, turns 90 clockwise, the path length. The variance of the ratio G=L is moves 100 m eastward, turns again 90 clockwise and therefore inversely proportional to the path length. This eventually moves 50 m southward (D=L ¼ 0:5). The ratio thus clearly constitutes a reliable estimator of the mean cosine of directional errors is equal to 0.65 for orientation efficiency of a path oriented towards an p ¼ 50 m (n ¼ 4), 0.533 for p ¼ 10 m (n ¼ 20), and 0.503 infinitely distant goal:this is an unbiased estimator for p ¼ 1m (n ¼ 200). More formally, let us call v the whose accuracy increases when considering longer path curvilinear abscissa, ranging from 0 to L.An paths. infinitesimal movement dv results in an infinitesimal variation in beeline distance from the animal’s current 2.2. Goal located at a finite distance (Reached after location to the goal dD ¼Àcosðy À gÞdv: The total n steps) beeline distance fromR the startingR point to the goal is 0 L then equal to D ¼À D dD ¼ 0 cosðy À gÞ dv: As dv=L When the goal is located at a finite distance D from represents the relative frequency of occurrence of the starting point, and the animal reaches it after having the various values of cosðy À gÞ obtained along the travelled some variable path length L; the orientation path, the straightness index D=L therefore gives the efficiency is often empirically measured as the ratio D=L; exact measure of the orientation efficiency of the path in referred to as the straightness index (Batschelet, 1981). question Z This is a popular estimate because it is very easy to L cosðy À gÞ compute and results in an intuitively appealing value D=L ¼ dv ¼ E ½cosðy À gÞ: ð1Þ L ðpathÞ ranging between 0 and 1. Batschelet (1981) stated that 0 the ratio D=L is practically equal to the mean vector The basic mean vector length of step orientations rðyÞ length of the step orientations rðyÞ: He concluded that hence usually results in an approximate value of the both can be used toP estimate the orientationP efficiency. orientation efficiency (it provides the exact value only n 2 n 2 0:5 In fact, withP D ¼½ð i¼1 li cosðyiÞÞ þð i¼1 li sinðyiÞÞ when the step length is constant). Obviously, the n and L ¼ i¼1 li; the ratio D=L is exactly equal to the ‘‘l- question of orientation efficiency concerns the orienta- weighted mean vector length’’ of the y distribution, and tion process at work rather than any given particular therefore to the basic (non-weighted) mean vector length path. Because of the random noise occurring in the rðyÞ if the step length is constant. This does not prove by environment and/or in the orientation process, the itself, however, that the straightness index reliably actual path (and thereby its length L) travelled by an estimates the orientation efficiency E½cosðy À gÞ: animal to reach a goal located at a given distance D is The theoretical approach of the situation occurring just a sample of the parent population of paths with a goal located at a finite distance is rather complex potentially generated by the orientation process used. because the goal direction g is no more constant but The particular value of D=L obtained for a given path depends on the animal’s current location: gi ¼ (Eq. (1)) is an unbiased estimator of the orientation arctan2½ðYn À YiÞ; ðXn À XiÞ; the goal location corre- efficiency ðEðD=LÞ¼E½cosðy À gÞÞ: Furthermore, using sponding to the last point of the path ðXn; YnÞ: As the computer simulations, it can be shown that the variance beeline distance from the startingP point to the goal can of this index is also inversely proportional to D (or L). n also be expressed as D ¼ i¼1 li cosðyi À g0Þ; the ratio The straightness index D=L thus appears to be a reliable D=L seems only to be an approximation of the l- (and easy to compute) estimator of the efficiency of the Pweighted mean cosine of directional errors cwðy À gÞ¼ orientation process used by an animal that orients itself n i¼1 li cosðyi À giÀ1Þ=L; where the goal direction is towards and eventually reaches a goal. taken to be constant and equal to the initial goal direction (g0) whatever the animal’s current location. 2.3. Correcting the straightness index Contrary to what occurs with a goal located at infinity, however, cwðy À gÞ appears itself to be a biased estimator The straightness index D=L is primarily a measure of of the orientation efficiency E½cosðy À gÞ when the goal the discrepancy between the path actually followed by is located at a finite distance, because the goal direction the animal and a perfectly oriented straight segment ARTICLE IN PRESS 212 S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 linking the goal to the starting point. The ratio G=L 3.1. Relationship between the diffusion distance and the can be considered as a straightness index as well:it path length plays exactly the same role for a path oriented towards a very distant, never reached goal. A general formula Insofar as the beeline distance D between the first and which applies to any situation (including the case of last points of a random search path, usually referred to an animal whose track was lost before it reaches a as the diffusion distance or the net displacement, is an goal located at a finite distance) is given by the ratio increasing function of the path length travelled and a DD=L; where DD is the difference between the initial and decreasing function of the path tortuosity, it is tempting final beeline distances to the goal, and L is the path to use the simple straightness index D=L to derive a length travelled between the initial and final locations tortuosity measure (e.g. 1 À D=L). Unfortunately, such recorded. an index has no meaning for random search paths Although the straightness index has been proven to be because it tends to 0, whatever the path tortuosity, when a reliable estimate of the orientation efficiency, the the path length L increases. As we saw before, the ratio physical structure of the environment is liable to weaken D=L is approximately (variable step length) or exactly the value obtained at the path level. For example, the (constant step length) equal to the mean vector length environment may be cluttered with many obstacles, rðyÞ: The successive step orientations y of a random forcing the animal to perform undesired detours before search path being correlated, there exists an initial bias reaching the goal. An animal may also choose to linked to the first step orientation. Without orientation perform some detours to go preferentially through some mechanism, this directional bias progressively vanishes: safe places or to avoid going through some other the step orientations tend to be uniformly distributed dangerous places. The actual efficiency of the orienta- and therefore D=LErðyÞ tends to 0 when the step tion mechanism involved should be then higher than the number increases. measure obtained at the path level. In contrast, the path The expected value of the squared diffusion distance length travelled may be minimized at the recording level. of an n–step random search path with a directional Similar to what happens in fractal analysis when the correlation c ðco1Þ and a randomly variable step length ruler length increases (see below), the path length l is equal to (Hall, 1977; Kareiva and Shigesada, 1983; measured as the sum of the step lengths tends to Marsh and Jones, 1988) decrease with the recording frequency, causing the c 1 À cn straightness index to take too high values if the EðD2Þ¼nEðl2Þþ2EðlÞ2 n À : ð2Þ 1 À c 1 À c recording frequency is low. As far as possible, these various factors should be taken into account to obtain a Eq. (2) can be rewritten as better estimate of the orientation mechanism efficiency 1 þ c 2cð1 À cnÞ than the crude value of the straightness index. EðD2Þ¼p2 n þ b2 À ; ð3Þ 1 À c ð1 À cÞ2 where p and b are the expectation and the coefficient 3. Random search paths of variation of the step length (EðlÞ¼p; Eðl2Þ¼p2ð1 þ b2Þ). This formula was previously pro- Wandering animals usually travel with both a vided by Tchen (1952) and Skellam (1973) in the persistence and a balance propensity:they tend to move particular case of a constant step length (b ¼ 0). When each new step in a direction (yiþ1) correlated to the n is large, Eq. (3) reduces to previous one (yi), and to turn to the right and to the left 1 þ c at random, with the same magnitude and frequency. A EðD2Þ¼Lp þ b2 ð4Þ random search path can therefore be modelled as a 1 À c correlated random walk. The distribution of turning with L ¼ np: It is worth noting that, when n is large but angles (ai ¼ yiþ1 À yi) is assumed to have a null mean the mean sine of turning angles sðaÞ is liable to take any (fðaÞ¼0) and to be symmetrical about it. The mean value (noted s), the general but intricate expression sine is then null (sðaÞ¼0), and the mean cosine is equal provided by Kareiva and Shigesada (1983) can be to the mean vector length (cðaÞ¼rðaÞ; hereafter referred reduced to a form just a little more complex than Eq. (4) simply to as c) and represents the correlation between 1 À c2 À s2 the orientations of successive steps. In contrast, the EðD2Þ¼Lp þ b2 : ð5Þ turning angles themselves are assumed to be statistically ð1 À cÞ2 þ s2 independent (this constraint is addressed in the Discus- Bovet and Benhamou (1988) demonstrated that, when sion). The extreme cases with c ¼ 0 (uniform distribu- n is large, the expected diffusion distance is equal to tion) and c ¼ 1 (punctual distribution) correspond to Brownian motion (non-correlated random walk) and p 0:5 EðDÞ¼ EðD2Þ : ð6Þ straight line movement, respectively. 4 ARTICLE IN PRESS S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 213

According to Wu et al. (2000) and Byers (2001), Eq. (6) been ‘‘linearized’’ by expressing them strictly between generates poor estimations of EðDÞ when n is small and c Àp and p rad. The standard deviation of a linear is close to 1. This holds true only if Eq. (6) is based on uniform distribution between Àp and p is equal to the approximate (Eq. (4)) rather than exact (Eq. (3)) p=O3 ¼ 1:81: Hence, the standard deviation of the value of EðD2Þ (Benhamou, 2004). Hence, when n is linearized distribution tends to 1.81 rad when the initial large enough (depending on c), EðDÞ increases propor- standard deviation s tends to infinity (s > 4 rad in tionally to OL: This confirms that the ratio D=L tends to practice). The two values diverge markedly for decrease when L increases (and reaches 0 when the path s > 1:2 rad. Eq. (7) can therefore be applied only to is infinitely long). In contrast, the ratio D=OL consti- paths characterized by a lower value of s or sq: The tutes an (inversely related) unbiased estimator of the second shortcoming comes from that Eq. (7) is based on tortuosity of a random search path. It cannot be the rediscretization of constant step correlated random considered as a reliable estimator, however, because of walks whereas actual paths, after recording, occurred as its intrinsic variability. Indeed, the variance VðDÞ¼ discrete sequences of fixes, with a randomly variable ð4=p À 1ÞEðDÞ2 involves a rather high coefficient of distance between successive fixes due to changes in the variation (0.52), whatever the path length travelled. animal’s speed and possibly in the recording frequency. Nevertheless, the form of this ratio shows that, contrary How to define a general sinuosity index able to deal to the straightness index of an oriented path, which is with any level of turning angle dispersion and with a dimensionless, the tortuosity of a random search path randomly variable step length? Let us recall that step has a dimension which is clearly related to the unit lengths and turning angles are useful tools for represent- length. ing a path but have no meaning by themselves:this is the way they are combined which makes it possible to define 3.2. Definition of the sinuosity a biologically meaningful sinuosity index. It appears retrospectively that the particular combination occur- Bovet and Benhamou (1988) modelled random search ring in Eq. (7) comes from the relation taking place paths as correlated random walks with a constant step between the expected squared diffusion distance and the length p ðb ¼ 0Þ: We assumed that the turning angles path length. Indeed, with a constant step length p were drawn, independently from each other, in a normal (b ¼ 0), Eq. (4) reduces to EðD2Þ¼Lpð1 þ cÞ=ð1 À cÞ: distribution wrapped on the circle with a null mean and Turning angles being drawn from a zero-centred a standard deviation s (o1.2 rad), and looked at the wrapped normal distribution with a standard deviation turning angle distribution obtained after rediscretization s; the value of the mean cosine is c ¼ expðÀs2=2Þ with a constant step length q (this consists in replacing (Mardia, 1972). In the restricted range, where the linear the original sequence of data points by a new sequence definition of the sinuosity is valid (so1:2 rad, i.e. such that the distance between any two successive new c > 0:5), the ratio ð1 þ cÞ=ð1 À cÞ is approximately equal points is equal to q; see appendix in Bovet and to 4=s2: The sinuosity index can therefore be written as Benhamou, 1988). Using computer simulations, we S ¼ 2½EðD2Þ=LÀ0:5: A general definition of the sinuosity showed that the resulting turning angles are normally follows: distributed with a null mean and a standard deviation 1 þ c À0:5 s ¼ 0:85sðq=pÞ0:5 (for s 1:2 rad). Segregating the S ¼ 2 p þ b2 : ð8Þ q qo 1 À c parameters initially introduced in the model (p and s) from the variables resulting from the rediscretization (q Other formulations are possible. For example, a and sq), we defined the sinuosity S of a random search ‘‘random search straightness index’’ may be defined as path as R ¼ 4=S2 ¼ p½ð1 þ cÞ=ð1 À cÞþb2; so that one gets R ¼ EðD2Þ=L (when n is large). S ¼ s=Op ¼ 1:18sq=Oq: ð7Þ To verify whether S constitutes a suitable combina- The first part of Eq. (7) can be used to simulate random tion of the three parameters involved (c; p; and b), I search paths with a given sinuosity by choosing looked at the evolution of the mean cosine of turning appropriate values of p and s; and the second part to angles obtained after rediscretization, cq; as a function compute the sinuosity of an actual path after redis- of the rediscretization step length q: For this purpose, I cretization with a step length q: However, Eq. (7) simulated two very long (3 Â 106 steps, to allow presents two shortcomings. The first comes from that statistical convergence) correlated random walks, one it rests on linear statistics. Drawing the turning angles with a constant simulation step length p (b ¼ 0), and the from a 0-centred wrapped normal distribution makes it other with a randomly variable simulation step length l possible to generate any path ranging from a straight with mean p (l ¼ pz2; where z2 is a random variable line (s ¼ 0) to Brownian motion (s>4 rad). At the following a w2 law with one degree of freedom: EðlÞ¼p; practical level of path analysis, however, the standard b2 ¼ 2), for each of the following 13 values of c:0, 0.1, deviation is computed on angular values which have 0.2,y, 0.8, 0.9, 0.95, 0.99, and 1. Turning angles were ARTICLE IN PRESS 214 S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 drawn, independently from each other, from a wrapped to ð1 þ b4=14Þ=ð2nÞ0:5; whatever the current values of p normal distribution with a null mean and standard and c (co1). S thus constitutes a reliable sinuosity deviation s ¼½À2LnðcÞ0:5: These paths were rediscre- index. It is inversely related to the ratio EðDÞ=OL tized with 33 values of q (0.6p, 0.7p,y, 1.9p,2p,3p,y, expected for a large step number. Being based on the 20p), and the mean cosine cq of the resultant turning whole path structure, its accuracy increases with the angle distributions was computed. The results (Fig. 1) path length or the sampling frequency, contrary to the are suitably approximated by the function cq ¼ 1 À 2=p simple D=OL ratio, which is based only on the relative arctan(qS2=2), with S as defined by Eq. (8). Whatever locations of the first and last points of the path. the arbitrary rediscretization step length q used, the Sometimes, turning angles may be noticeably not sinuosity can be computed as balanced. For example, thrushes searching for earth- worms in high-density patches tend to turn system- 0:5 2 p atically to the same side (Smith, 1974a, b), involving a S ¼ tan ð1 À cqÞ : ð9Þ q 2 mean sine of turning angles s markedly different from 0. An even more general sinuosity index can be defined Hence, Eq. (8) constitutes an appropriate combination theoretically based on Eq. (5) as of c, p, and b. Additional computer simulations À0:5 involving four values of b (0, 1, O2; and 2) showed that 1 À c2 À s2 S ¼ 2 p þ b2 : ð10Þ the coefficient of variation of S is approximately equal ð1 À cÞ2 þ s2 Eqs. (8) and (10) can be used to simulate correlated random walks with any given sinuosity by choosing appropriate values of c; p; and s (b may be set to 0; Fig. 2a). In the case where turning angles are drawn from a zero-centred wrapped normal distribution with c > 0:5(so1:2 rad), the simple linear expression of the sinuosity (first part of Eq. (7)) can be applied as well. The practical computation of the sinuosity of an actual random search path, based on Eqs. (8), (9) or (10) is detailed in the Discussion.

3.3. Fractal analysis

The fractal dimension F of an object measures its ability to fill the Euclidean space in which it is embedded (Mandelbrot, 1982). In particular, one gets F ¼ 1 for a straight line, and F ¼ 2 for Brownian motion, which covers the whole plane when the step number increases indefinitely. Between these two extremes, random search paths might be characterized by intermediates F values, reflecting various values of tortuosity. The dimension of a fractal line can be measured by the divider method, which consists in looking at how the line length varies with the ruler length (formally equivalent to a rediscretization step length) used to measure it (see Sugihara and May, 1990). The logarithm of a fractal line length is a linear function of the logarithm of the ruler length, with a slope equal to 1 À F: I simulated a correlated random walk made of 3 Â 106 Fig. 1. Influence of the redisretization step length q (expressed with steps for each of the following six values of mean cosine respect to the mean simulation step length p) on the mean cosine of of turning angles c:0 (Brownian motion), 0.4, 0.6, 0.8, turning angles cq obtained after rediscretization for various values 0.9 and 1 (straight line), with a constant step length of the simulation mean cosine of turning angle c: The dashed serving as unit length. The path lengths L were curves represent the equation cq ¼ 1 À 2=p arctan½2ðq=pÞ=ðf1 þ cg= measured with 109 ruler lengths:1.1, 1.2, y, 1.9, 2, f1 À cgþb2Þ: The value of c used is indicated on each curve. For clarity, the results obtained with some values of c (0.1, 0.2, 0.4 and 0.6) 2.5, 3, 4, ..., 99, and 100. The results (Fig. 3) confirm are omitted. The simulation step length is constant (b ¼ 0) in (a) and those got by Turchin (1996):animal’s random search variable (b ¼ O2) in (b). paths cannot be considered as fractal (i.e., indefinitely ARTICLE IN PRESS S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 215

fractal dimension is roughly constant corresponding to the size ranges of the various scales involved (e.g. Fritz et al., 2003). To test this hypothesis, I simulated a 3 Â 106 step random walk in a two-scale environment. The animal performed extensive searching (c ¼ 0:9) for 900 steps and very intensive searching (c ¼ 0) for 100 steps, and so on 3000 times, as if it encountered a highly proﬁtable patch every 1000 steps. The path length was then measured with the same ruler lengths as previously. The results (grey dots in Fig. 3) are similar to those obtained in a one-scale environment (black dots):the fractal analysis proved unable to detect that the simulated animal foraged in a two-scale environment.

4. Discussion

4.1. Distinguishing between oriented and random search paths

The tortuosity of an animal’s path can be reliably measured either as a straightness index or a sinuosity index, depending on whether the animal performed an oriented movement or a random search movement. Before using the appropriate index, straightness or sinuosity, it is therefore required to determine which type of movement the animal performed. In homing Fig. 2. (a) Example of a 600 step correlated random walk with a experiments, animals are assumed to orient, more or less constant step length p (b ¼ 0). Turning angles were drawn from a efficiently, towards their home (e.g. Papi, 1992). In some wrapped normal distribution with mean equal to fðaÞ and a standard other experiments, they are assumed to wander freely in deviation s ¼½À2LnðrðaÞÞ0:5; with fðaÞ¼0 and rðaÞ¼0:9(c ¼ 0:9; the environment, searching at random for items which s ¼ 0; S ¼ 0:46=Op) for the first and the last 200 steps (extensive cannot be detected at distance (e.g. Kareiva and mode), and fðaÞ¼p=6 rad and rðaÞ¼0:6(c ¼ 0:5; s ¼ 0:3; S ¼ 1:42=Op) for the middle 200 steps (intensive mode). (b) Values Shigesada, 1983). In many other experiments, however, of mean cosine c (black curve) and mean sine s (grey curve) based on this is not so clear. An animal reaching a given target the current, 25 previous, and 25 following turning angles (51 step width may have performed a random search path and sliding window). The changes in search mode appear as smooth encountered the target by chance, or it may have transitions involving a step number equal to the windows width. actively oriented itself towards the target. A very straight path suggests an oriented movement, but a broken) lines. Indeed, the slope of the log–log relation slightly more tortuous path may result from a less (and thereby the fractal dimension F) obtained is not efficient orientation mechanism as well as from a low constant, but varies to a large extent as a function of the sinuosity random search. ruler length:it tends to 0 ( F ¼ 1) for small ruler lengths A complex statistical method based on angular (F reaches 1 with an infinitesimal ruler length) and to À1 distribution analyses was developed by Benhamou and (F ¼ 2) for the largest ruler lengths. Like the ratio D=L; Bovet (1992) to determine whether or not an orientation the fractal dimension F appears to have no real meaning mechanism is involved, and the type (scalar of vectorial) for characterizing random search paths. Further com- of orientation information used. It requires a large puter simulations showed that this is also the case for amount of precise data, however, to be effective (e.g., oriented paths:the fractal dimension remains close to 1 Dejean and Benhamou, 1993). A simpler method whatever the orientation efficiency. consists in computing the difference between the square With (1994) suggested that the fractal dimension of of the mean vector length of step orientations, r2ðyÞ; and movement patterns should depend on the scales at the square of the mean vector length of turning angles, which the organisms are interacting with the patch r2ðaÞEc2. This quantity is, on the average, positive for a structure of the landscape. Hence, despite the fractal movement oriented in a given constant direction (goal dimension of a path appears to have no value by itself, a located at infinity) and negative for a random search fractal analysis might highlight the hierarchical scales movement (Marsh and Jones, 1988). The variances encountered, the ranges of ruler lengths for which the involved are however so high that distinguishing ARTICLE IN PRESS 216 S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220

Fig. 3. Influence of the ‘‘ruler length’’ (corresponding to a redisretization step length) on the path length for various values of the simulation mean cosine of turning angles c: Both the ruler and path lengths are expressed in simulation step lengths p; and plotted on a logarithm scale. The results were obtained with an initial (simulated) path length equal to 3 Â 106 p: The black dots correspond to paths simulated with a constant value of c (0, 0.4, 0.6, 0.8, 0.9 and 1), whereas the grey dots correspond to a path simulated with a value of c alternating every 1000 steps (i.e. 3000 times) between 0.9 (900 steps) and 0 (100 steps). between the two types of movement is not statistically eE1À1.45 arctan½ð1 À cÞðn À 1Þ=ð3cÞ=p for a constant possible when the step number n is low. In addition, the step length (a randomly variable step length leads to a case of a goal reached after n steps is not considered. slightly higher risk). The minimum risk, obtained with An alternative method rests on the way the beeline large values of n; is therefore e ¼ 0:275: Hence, a reliable distance D (or the drift G for a goal located at infinity) is statistical decision can be taken only at the population related to the path length L: Indeed, on the average, L is level, considering several paths reaching a target (with N proportional to D for an oriented path, whereas for a paths characterized by similar values of n and c; the random search path, D is proportional to OL (when n is statistical test rests on a binomial law BðN; eÞ). large or c is not too close to 1; otherwise Eq. (3) cannot be suitably approximated by Eq. (4), and D is then 4.2. Practical computation of the sinuosity of actual almost proportional to L). Using simply the final values random search paths of D and L is rather ineffective because of the very high variability involved (Marsh and Jones, 1988). Looking One of the basic assumptions of the correlated at the whole relation between D and L in a backward random walk model, on which rests the definition of procedure (starting at the penultimate point of the path the sinuosity index S; is that turning angles are and moving backward along the path down to the first statistically independent. Depending on the recording point) is more informative. Considering an animal that frequency used (from 25 fixes per second in videotrack- reaches a target after n steps, this procedure consists in: ing to a few fixes per day in Argos satellite tracking), the (1) computing the current backward beeline distance sequence of fixes obtained constitutes a more or less n 2 2 0:5 i ¼½ðXn À XnÀiÞ þðYn À YnÀPiÞ and the current accurate representation of the original path. Recording i backward path length Łi = j¼1 pnþ1Àj; for i ¼ 1to a path with a high frequency results in a ‘‘quasi- i ¼ n (2) plotting ni as a function of Łi and as a function continuous’’ path representation preserving a maximum of OŁi and (3) for each plot, computing the sum of the of details (provided the accuracy of the fixes is good linear regression residual squares with a null intercept enough; otherwise the data need filtering before (n0 ¼ Ł0 ¼ 0; whatever the type of path). The type of analysis). Contrary to what occurs for a path recorded path involved is likely to correspond to the minimum with a low frequency, successive turning angles mea- sum. Additional computer simulations involving ran- sured at the fix level are then positively correlated, dom search movements showed however that the because of motion constraints:the long-term average of probability of obtaining a relationship closer to the turning angles remains null, but the probability of linear line than the square root curve, corresponding to occurrence of a (anti)clockwise turning angle is greater the probability of erroneously rejecting the null hypoth- when the previous turning angle was (anti)clockwise. To esis, is rather high at the individual path level: get rid of this auto-correlation, it is necessary to ARTICLE IN PRESS S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 217 resample the path at a larger scale, either by filtering out case occurs when the path was recorded with an a fraction of the fixes (subsampling), or by rediscretizing intermediate frequency, and was then rediscretized with the path with a constant step length. As the accuracy of a constant step longer than the mean distance between the sinuosity measure increases with the step number, it fixes to get statistically independent turning angles. In is preferable to use the minimum resampling required to this case, the sinuosity value should be computed based respect the non-correlation constraint. The recorded on Eq. (9) which counterbalances the grazing effect, path should therefore be resampled several times with an provided the mean of turning angles does not differ increasing (filtration or rediscretization) step length, significantly from 0. Otherwise, the sinuosity cannot be starting with a length close to the average distance computed because the way the mean cosine c and the between fixes, until the turning auto-correlation be- mean sine s are altered by the rediscretization is comes statistically non-significant. unknown (and beyond the scope of this paper). In any Insofar as the measure of the sinuosity of an actual case, the sinuosity cannot be computed if there is a random search path depends on the way the path was significant cross-correlation between the magnitude of discretized at the recording level, and possibly redis- turning angles and the step lengths. The path rediscretized afterwards, it is important to decipher the cretization with a constant step length is a useful mean effects due to these processes. Three effects appear to be to get rid of such a correlation. Note that, to simply involved:‘‘scattering’’, ‘‘smoothing’’, and ‘‘grazing’’. compare different random search paths which were The first two effects correspond to an increasing rediscretized with the same constant step length q; a dispersion of the turning angles, and to an increasing simplified sinuosity index may be computed as Sc ¼ loss of details, respectively, when the step length 1 À c (provided the mean of turning angles remains close increases. As long as the path does not degenerate into to 0). Brownian motion (involving a complete loss of essential A worth noting point is that the sinuosity index S; information due to a too low recording frequency and/ contrary to straightness index D=L; is not dimensionless or a too long rediscretization step), the way the mean but related to the unit length. Random search paths are cosine of turning angles varies with the step length is often used to find prey items whose density is also predictable, making it possible to compute the sinuosity related to a unit length. Expressing the sinuosity with index. In addition, the rediscretization process tends to the same unit length as the one used to express the prey plane the sharpest asperities of the path, resulting in a item density ensures the results to be consistent turning angle distribution more concentrated than (Benhamou 1992, 1994). The choice of the unit length expected. This grazing effect is best exemplified when a may be critical when comparing sinuosities of paths constant step length non-correlated random walk is made by animals of very different sizes, e.g. an ant and rediscretized with a rediscretization step length close to an albatross (if such a comparison proved relevant). The the simulation step length (q=pE1inFig. 1a):the mean two movement patterns may look similar, because of cosine of turning angles increases from c ¼ 0tocq ¼ close mean cosine values c; whereas the actual mean step 0:28: Similarly, the 1.18 factor in Eq. (7) counter- lengths p are very different, say, 1 cm for the ant and balances this effect. As its magnitude appears to be 1 hm for the albatross. If the sinuosity is expressed with inversely related to c, this effect should not occur with respect to a common metric unit, e.g. 1 m, the values the rediscretization of the quasi-continuous tracks obtained will be extremely different. However, the obtained with a high recording frequency, which are biological meaning of a 1-m distance for the two species usually characterized by small and positively correlated is also very different in terms of both prey density and turning angles. Similarly, the initial discretization (at the perceptual range detection. Hence, for interspecific recoding level) of an actual curvilinear path should not comparative purpose, the sinuosity values have to be involve any grazing effect. ‘‘rescaled’’ to biologically relevant unit lengths (e.g. in The practical computation of the sinuosity of an rad/cm0.5 for the ant and rad/hm0.5 for the albatross). actual random search path hence depends on whether a grazing effect is involved or not. The second case occurs 4.3. Analysing composite paths when the path was recorded either with a low frequency, so that the turning angles measured at the fix level were Paths recorded for a long time are likely to encompass statistically independent, or with a high frequency both oriented and random search sections. A global (quasi-continuous track) and was then resampled to analysis, in terms of correlated random walk (Bergman get statistically independent turning angles. In this case, et al., 2000) or fractal dimension (Fritz et al., 2003), is the sinuosity value can be computed based on Eq. (8) or therefore liable to provide dubious results. The correct (10), depending on whether the mean of turning angles procedure requires first to identify the various parts does not or does differ significantly from 0, respectively corresponding to oriented stages and to random search (with p set to q and b set to 0 for quasi-continuous tracks stages (Dejean and Benhamou, 1993; Girard et al., rediscretized with a constant step length q). The first 2004), and possibly to the simultaneous use of the two ARTICLE IN PRESS 218 S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 kinds of process (see below). In addition, during a phase may correspond to a random search for favour- random search stage, an animal may alternate between able stopping places as well as to an oriented move in a extensive and intensive search modes, involving a direction liable to change at each stop. If the animal separate computation of the sinuosity for each mode. tends to perform the successive moves in a given The best situation occurs when one can a priori predict preferred direction, the global pattern is directionally where the animal should perform intensive searching biased and can no more be accounted for by a simple and extensive searching (and then test the hypothesis correlated random walk model (Samu et al., 2003). that the animal used two modes), based on additional An animal may search for prey at random, and at the information about the habitat structure or behavioural same time, tends globally to move in a given direction. events such as prey captures (Smith 1974a, b; Mellgren This results in second type of composite path where the and Roper, 1996). When such an information is missing, two types of processes are used simultaneously rather the areas concerned may be deduced from the movement than alternately. In such biased random walks, the data only (involving a less powerful analysis). For this diffusion distance tends to increase faster than predicted purpose, one can look at the mean cosine and mean sine by the correlated random walk model (Firle et al., 1998; of the turning angle distribution measured on a sliding Marell( et al., 2002; Samu et al., 2003). A useful means to window. The width of the window should be large model a biased random walk involves differential enough to ensure statistical stability (the narrower is the klinokinesis (Benhamou and Bovet, 1989), i.e. a random window, the noisier are the results). On the average, the search path whose local sinuosity Si depends on the mean cosine is lower and/or the mean sine is closer to 0 previous movement orientation yiÀ1 with respect to a when the animal performs extensive searching than given preferred direction g; e.g. Si ¼ S0½1 À k cosðyiÀ1 À when it performs intensive searching (see Fig. 2b). If the gÞ; where S0 is the basic sinuosity, which controls the shift from intensive to extensive searching is progressive, overall searching intensity, and k (0oko1) is the the observed movement pattern has to be analysed in the klinokinetic factor, which controls the variations in framework of a modified model where the sinuosity sinuosity, and thereby the movement bias (drift G) in the decreases with the path length (Morales and Ellner, preferred direction (kEG=L). The analysis of this kind 2002), because the mere correlated random walk model of movement pattern then involves to specify the values is unable to account for the data. of both S0 and k (Benhamou and Bovet, 1992). Some animals perform naturally discrete steps start- Alternatively, the diffusion distance may increase slower ing and ending at specific places, e.g. a bee moving from than predicted (e.g. Fortin, 2003). This occurs in one flower to another. This results in a composite path particular when an animal tends to perform search which usually requires a two-level analysis (Benhamou, loops, e.g. a mammal which simultaneously searches for 1990). The first, global level of analysis focuses on the prey items and restricts its movement within the given general movement pattern from place to place, the area corresponding to its home range, or an insect moves between two successively visited places being searching for its nest hole after a homing path. Search schematically represented as straight segments. The loops involve simultaneously a centripetal orientation global movement pattern may correspond to an oriented component and a random search component. Analysis path, the places visited serving as beacons (piloting of such patterns obviously require specific models behaviour). It can then be quantified using a global (Benhamou, 1989, 1994), because neither an analysis in straightness index Dg=Lg; where Dg is the beeline pure terms of orientation efficiency nor an analysis in distance between the first and last places visited, and pure terms of sinuosity can provide reliable results:as Lg is the sum of the move lengths between places. If the the beeline distance D is repeatedly reset to a value close global movement pattern involves a random search to 0, the straightness index D=L has no meaning, and the path, Eq. (8) or (10) should result in a reliable sinuosity global movement pattern clearly does not fit a simple value, provided there are no significant auto-correla- correlated random walk model. tions between successive turning angles and cross- correlation with move lengths. The second, local level 4.4. Weakness of the fractal analysis of animals’ paths of analysis focuses on the oriented moves performed between successively visited places:the orientation Rediscretizing or measuring a path with an increasing efficiency of each move can be quantifiedP as a localP step or ruler length involves some loss of information, straightness index Di=Li; and the ratio Di= Li because the longer is the step or ruler, the larger is the provides a mean local efficiency value for the whole number of smaller scale details ignored. The main path. Many animals also perform ‘‘saltatory searching’’ information about the structure of the path appears to (O’Brien et al., 1990), which consists in naturally be roughly conserved, however, because of some discretized movements at the temporal level. The stop intrinsic self-similarity, suggesting that the tortuosity phases of these stop-and-go movements do not occur at of random search paths may be characterized in the obvious places (such as flowers for a bee), so that any go form of a fractal dimension (Dicke and Burrough, ARTICLE IN PRESS S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220 219

1988). There is in fact no self-similarity:looking at a tortuosity of a random search path whereas the random search path at different scales does not result in apparent fractal dimension F; which is just an analogue the same pattern because the directional correlation (i.e. of the mean cosine of turning angles and fully ignores the mean cosine of turning angles) varies with the scale. the mean step length, is not. In some way, the variation of the former counter- The related concepts of chaos, strange attractor and balances the variation of the latter, as long as the path fractal have been largely used in biology (Briggs and has not degenerated into Brownian Motion (the Peat, 1989; Hastings et al., 1993; Sugihara and May, distribution can then no more flatten to counterbalance 1990; Halley et al., 2004). Applying the fractal approach the increase in step length). When recorded with a high to the study of random search paths might be fruitful, frequency, actual animals’ paths look like curvilinear but first requires building up a fractal movement model lines, which are theoretically differentiable at any as an alternative to the correlated random walk model. location. It is therefore hard to think about them as A fractal movement model was proposed by Dicke and fractal lines (i.e., continuous but non-differentiable lines; Burrough (1988; see also Sugihara and May, 1990), see Sugihara and May, 1990). Turchin (1996) and the based on a fractal Brownian function. To my knowl- present paper provide clear evidence that animals’ edge, however, it has never been proved yet that this random search paths, at least when they are modelled type of model may constitute a more suitable represen- as correlated random walks, are not fractal. Random tation of actual animals’ random search paths than the walks are able to cover the whole plane, whatever the correlated random walk model. The way to further correlation value co1: This means that the true fractal theoretical research on this topic is wide open. dimension of these walks is equal to 2, whatever the apparent F value computed from the local slope of the log(path length)-log(ruler length) relation. Acknowledgements Halley et al. (2004) warned against measuring fractal dimensions of non-fractal objects which may seem to be I thank Pierre Bovet, Laurent Dagorn, Charlotte fractal at some spatial scales (apparent fractality). Girard and an anonymous referee for stimulating Beyond the logical argument that measuring the fractal discussions and/or helpful comments and suggestions. dimension of a non-fractal object has no meaning and is The Monte-Carlo simulations and rediscretization therefore only liable to generate artifactual results, a programs run thanks to FreePascal compiler (freely strong mathematical argument against the use of the available at www.freepascal.org). apparent fractal dimension F (as computed from the local slope of the log–log relation) to measure the path tortuosity was provided paradoxally by Nams (1996) in References a paper advocating the opposite point of view. Indeed, he showed that F is approximately equal to 2=½1 þ Batschelet, E., 1981. Circular Statistics in Biology. Academic Press, log2ð1 þ cÞ: In other terms, F is no more than a London. monotonously decreasing function of the mean cosine of Benhamou, S., 1989. An olfactory orientation model for mammals’ turning angles c (with 2 > F > 1 for 0oco1). The movements in their home ranges. J. Theor. Biol. 139, 379–388. decrease of the local slope (equal to 1 À F; from 0 to À1; Benhamou, S., 1990. An analysis of movements of the wood mouse Apodemus sylvaticus in its home range. Behav. Proc. 22, 235–250. see Fig. 3) is eventually the simple reflect of the decrease Benhamou, S., 1992,. Efficiency of area-concentrated searching (from 1 to 0) of the mean cosine of turning angles (see behaviour in a continuous patchy environment. J. Theor. Biol. Fig. 1). As we have seen, there are two key parameters in 159, 67–81. random search path analysis:the mean cosine of turning Benhamou, S., 1994. Spatial memory and searching efficiency. Anim. angles and the mean step length. In a seminal paper, Behav. 47, 1423–1433. Benhamou, S., 2004. On the expected net displacement of animals’ Siniff and Jessen (1969) considered the distributions of random movements. Ecol. Model. 171, 207–208. turning angles and of step lengths, but only separately. Benhamou, S., Bovet, P., 1989. How animals use their environment:a In-most cases, turning angles and step lengths are just new look at kinesis. Anim. Behav. 38, 375–383. useful tools for representing a curvilinear path, but have Benhamou, S., Bovet, P., 1992. Distinguishing between elementary no biological meaning by themselves:both distributions orientation mechanisms by means of path analysis. Anim. Behav. 43, 371–377. depend on the arbitrary (re)discretization used at the Bergman, C.M., Schaefer, J.A., Luttich, S.N., 2000. Caribou move- level of movement recording and analysis. The sinuosity ment as a correlated random walk. Oecologia 123, 364–374. index S; since its introduction by Bovet and Benhamou Bovet, P., Benhamou, S., 1988. Spatial analysis of animals’ movements (1988) to the generalized definition proposed in the using a correlated random walk model. J. Theor. Biol. 131, present paper, combines the two key parameters 419–433. Bovet, P., Benhamou, S., 1991. Optimal sinuosity in central place explicitly to obtain a measure of tortuosity which is foraging movements. Anim. Behav. 42, 57–62. independent of the arbitrary (re)discretization used. This Briggs, J., Peat, F.D., 1989. Turbulent Mirror. Harper & Row, is why the sinuosity S is a reliable measure of the path New York. ARTICLE IN PRESS 220 S. Benhamou / Journal of Theoretical Biology 229 (2004) 209–220

Byers, J.A., 2001. Correlated random walk equations of animal Marell,( A., Ball, J.P., Hofgaard, A., 2002. Foraging and movement dispersal resolved by simulation. Ecology 82, 1680–1690. paths of female reindeer:insights from fractal analysis, correlated Claussen, D.L., Finkler, M.S., Smith, M.M., 1997. Thread trailing of random walks, and Levy! flights. Can. J. Zool. 80, 854–865. turtles:methods for evaluating spatial movements and pathway Marsh, L.M., Jones, R.E., 1988. The form and consequences of structure. Can. J. Zool. 75, 2120–2128. random walk movement models. J. Theor. Biol. 133, 113–131. Dejean, A., Benhamou, S., 1993. Orientation and foraging movements Mellgren, R.L., Roper, T.J., 1986. Spatial learning and discrimination in a patchy environment by the ant Serrastruma lujae. Behav. Proc. of food patches in the European badger (Meles meles). Anim. 30, 233–243. Behav. 34, 1129–1134. Dicke, M., Burrough, P.A., 1988. Using fractal dimensions for Morales, J.M., Ellner, S.P., 2002. Scaling up animal movements in characterizing tortuosity of animal trails. Physiol. Entomol. 13, heterogeneous landscapes:the importance of behavior. Ecology 83, 393–398. 2240–2247. Firle, S., Bommarco, R., Ekbom, B., Natiello, M., 1998. The influence Nams, V.O., 1996. The Vfractal:a new estimator for fractal dimension of movement and resting behavior on the range of three carabid in animal movement paths. Landscape Ecol. 11, 289–297. beetles. Ecology 79, 2113–2122. O’Brien, W.J., Browman, H.I., Evans, B.I., 1990. Search strategies of Fisher, N.I., 1993. Statistical Analysis of Circular Data. Cambridge foraging animals. Am. Sci. 78, 152–160. University Press, Cambridge, USA. Papi, F. (Ed.), 1992. Animal Homing. Chapman & Hall, London. Fortin, D., 2003. Searching behavior and use of sampling information Samu, F., Sziranyi,! A., Kiss, B., 2003. Foraging in agricultural fields: by free-ranging bison (Bos bison). Behav. Ecol. Sociobiol. 54, local ‘sit-and-move’ strategy scales up to risk-averse habitat use in a 194–203. wolf spider. Anim. Behav. 66, 939–947. Fritz, H., Said, S., Weimerskirch, H., 2003. Scale-dependent hierarch- Siniff, D.B., Jessen, C.R., 1969. A simulation model of animal ical adjustments of movement patterns in a long-range foraging movement patterns. Adv. Ecol. Res. 6, 185–219. seabird. Proc. R. Soc. London B 270, 1143–1148. Skellam, J.G., 1973. The formulation and interpretation of mathema- Gendron, R.P., Staddon, J.E., 1983. Searching for cryptic prey:the tical models of diffusionary processes in population biology. In: effect of search rate. Am. Nat. 121, 172–186. Bartlett, M.S., Hiorns, R.W. (Eds.), The Mathematical Theory of Girard, C., Benhamou, S., Dagorn, L., 2004. FAD:fish aggregating or the Dynamics of Biological Populations. Academic Press, London, fish attracting device? a new analysis of yellowfin tuna (Thunnus pp. 63–85. albacares) movements around floating objects. Anim. Behav. 67, Smith, J.N., 1974a. The food searching behaviour of two European 319–326. thrushes. I. Description and analysis of search paths. Behaviour 48, Hall, R.L., 1977. Amoeboid movements as a correlated walk. J. Math. 276–302. Biol. 4, 327–335. Smith, J.N., 1974b. The food searching behaviour of two European Halley, J.M., Hartley, S., Kallimanis, A.S., Kunin, W.E., Lennon, J.J., thrushes. II. The adaptiveness of search patterns. Behaviour 49, Sgardelis, S.P., 2004. Uses and abuses of fractal methodology in 1–61. ecology. Ecol. Lett. 7, 254–271. Sugihara, G., May, R.M., 1990. Applications of fractals in ecology. Hastings, A., Hom, C.L., Ellner, S., Turchin, P., Godfray, H.C., 1993. TREE 5, 79–86. Chaos in ecology:is mother nature a strange attractor? Annu. Rev. Tchen, C.M., 1952. Random flight with multiple partial correlations. Ecol. Syst. 24, 1–33. J. Chem. Phys. 20, 214–217. Kareiva, P.M., Shigesada, N., 1983. Analyzing insect movement as a Turchin, P., 1996. Fractal analyses of animal movement:a critique. correlated random walk. Oecologia 56, 234–238. Ecology 77, 2086–2090. Knoppien, P., Reddingius, J., 1985. Predators with two modes of With, K.A., 1994. Using fractal analysis to assess how species perceive searching:a mathematical model. J. Theor. Biol. 114, 273–301. landscape structure. Landscape Ecol. 9, 25–36. Mandelbrot, B., 1982. The Fractal Geometry of Nature. Freeman, Wu, H.-I., Li, B.-L., Springer, T.A., Neill, W.H., 2000. Modelling New York. animal movement as a persistent random walk in two dimensions: Mardia, K.V., 1972. Statistics of Directional Data. Academic Press, expected magnitude of net displacement. Ecol. Model. 132, London. 115–124.