You have free access to this content

Minimizing errors in identifying Lévy flight behaviour of organisms

  1. DAVID W. SIMS1,
  2. DAVID RIGHTON1 and
  3. JONATHAN W. PITCHFORD2

Version of Record online: 23 JAN 2007

DOI: 10.1111/j.1365-2656.2006.01208.x

Issue

Journal of Animal Ecology

Journal of Animal Ecology

Volume 76, Issue 2, pages 222–229, March 2007


Additional Information

How to Cite

SIMS, D. W., RIGHTON, D. and PITCHFORD, J. W. (2007), Minimizing errors in identifying Lévy flight behaviour of organisms. Journal of Animal Ecology, 76: 222–229. doi: 10.1111/j.1365-2656.2006.01208.x

Author Information

  1. 1

    Marine Biological Association of the United Kingdom, The Laboratory, Citadel Hill, Plymouth PL1 2PB, UK; Centre for Environment, Fisheries and Aquaculture Sciences, Pakefield Road, Lowestoft NR33 0HT, UK;

  2. 2

    Department of Biology, University of York, Heslington, York YO10 5YW, UK

* David Sims, Marine Biological Association of the UK, The Laboratory, Citadel Hill, Plymouth PL1 2PB, UK. Tel.: 01752 633227. Fax: 01752 633102. E-mail: dws@mba.ac.uk

Publication History

  1. Issue online: 29 JAN 2007
  2. Version of Record online: 23 JAN 2007
  3. Received 18 September 2006; accepted 28 November 2006

SEARCH

SEARCH BY CITATION

ARTICLE TOOLS

Enhanced Article (HTML) Get PDF (254K)Get PDF (254K)

Keywords:

  • animal movement;
  • optimal foraging theory;
  • search strategy;
  • random walk

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References
  • 1
    Lévy flights are specialized random walks with fundamental properties such as superdiffusivity and scale invariance that have recently been applied in optimal foraging theory. Lévy flights have movement lengths chosen from a probability distribution with a power-law tail, which theoretically increases the chances of a forager encountering new prey patches and may represent an optimal solution for foraging across complex, natural habitats.
  • 2
    An increasing number of studies are detecting Lévy behaviour in diverse organisms such as microbes, insects, birds, and mammals including humans. A principal method for detecting Lévy flight is whether the exponent (µ) of the power-law distribution of movement lengths falls within the range 1 < µ ≤ 3. The exponent can be determined from the histogram of frequency vs. movement (step) lengths, but different plotting methods have been used to derive the Lévy exponent across different studies.
  • 3
    Here we investigate using simulations how different plotting methods influence the µ-value and show that the power-law plotting method based on 2k (logarithmic) binning with normalization prior to log transformation of both axes yields low error (1·4%) in identifying Lévy flights. Furthermore, increasing sample size reduced variation about the recovered values of µ, for example by 83% as sample number increased from n = 50 up to 5000.
  • 4
    Simple log transformation of the axes of the histogram of frequency vs. step length underestimated µ by c.40%, whereas two other methods, 2k (logarithmic) binning without normalization and calculation of a cumulative distribution function for the data, both estimate the regression slope as 1 − µ. Correction of the slope therefore yields an accurate Lévy exponent with estimation errors of 1·4 and 4·5%, respectively.
  • 5
    Empirical reanalysis of data in published studies indicates that simple log transformation results in significant errors in estimating µ, which in turn affects reliability of the biological interpretation. The potential for detecting Lévy flight motion when it is not present is minimized by the approach described. We also show that using a large number of steps in movement analysis such as this will also increase the accuracy with which optimal Lévy flight behaviour can be detected.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References

A central issue in behavioural ecology is understanding how organisms search for resources within heterogeneous natural environments (MacArthur & Pianka 1966; Stephens & Krebs 1986). Organisms are often assumed to move through an environment in a manner that optimizes their chances of encountering resource targets, such as food, potential mates or preferred refuging locations. For a forager searching for prey in a stable, unchanging environment, prior expectation of when and where to find items will inform a deterministic search pattern (Stephens & Krebs 1986; Houston & McNamara 1999). However, foragers in environments that couple complex prey distributions with stochastic dynamics will not be able to attain a universal knowledge of prey availability. This raises the question of how should a forager best search across complex landscapes to optimize the probability of encountering suitable prey densities?

Recent progress in optimal foraging theory has focused on probabilistic searches described by a category of random-walk models known as Lévy flights (Viswanathan et al. 2000; Bartumeus et al. 2005). Lévy flights are specialized random walks that comprise ‘walk clusters’ of relatively short step lengths, or flight intervals (distances between turns), connected by longer movements between them, with this pattern repeated at all scales resulting in scale invariant or fractal patterns (Bartumeus et al. 2005). In a Lévy flight the step lengths are chosen from a probability distribution with a power-law tail, resulting in step lengths with no characteristic scale: inline image with 1 < µ ≤ 3 where lj is the flight length. Theoretical studies indicate Lévy flights represent an optimal solution to the biological search problem in complex landscapes where prey are sparsely and randomly distributed outside an organism's sensory detection range (Viswanathan et al. 1999, 2000; Bartumeus et al. 2005). An advantage to predators of selecting step lengths with a Lévy distribution compared with simple Brownian motion is that Lévy flight increases the probability of encountering new patches (Viswanathan et al. 2000, 2002; Bartumeus et al. 2002). Lévy search strategies are also robust to changes in environmental parameters such as the availability of patchy resources (Raposo et al. 2003; Santos et al. 2004).

Lévy behaviour has been detected among diverse organisms, including amoeba (Schuster & Levandowsky 1996), zooplankton (Bartumeus et al. 2003), bumblebees (Viswanathan et al. 1999), wandering albatrosses (Viswanathan et al. 1996), deer (Viswanathan et al. 1999), jackals (Atkinson et al. 2002), spider monkeys (Ramos-Fernández et al. 2004) and humans (Brockmann, Hufnagel & Geisel 2006). This illustrates that the number of studies detecting Lévy-type behaviour in organism movement patterns is increasing: our literature search revealed that over 30 biological and ecological studies citing Lévy flight behaviour have been published since 2000. However, a key requirement of investigations to identify the presence of Lévy flight behaviour in organisms is an accurate determination of the Lévy exponent (µ) of the power-law distribution of step-length frequency against step length. To reveal the power-law form of the distribution a histogram of step-length frequency against step-length distance is plotted on logarithmic scales (Newman 2005). The exponent is calculated for a movement path consisting of a series of consecutive steps from the gradient of the linear regression of the log-log plot. Movement patterns with superdiffusive Lévy characteristics have exponents in the range between 1 and 3 (Viswanathan et al. 1996) with Brownian motion emerging at µ-values ≥ 3 (normal diffusion) (Bartumeus et al. 2005). Furthermore, modelling studies indicate that optimal Lévy flight search patterns occur with µ = 2 (Viswanathan et al. 1999, 2000, 2001; da Luz et al. 2001), an assertion that finds empirical support from insect, seabird and terrestrial mammal data (Viswanathan et al. 1996, 1999; Atkinson et al. 2002). Therefore, any inaccurate estimation of µ will have important implications for biological interpretation; it will influence whether or not Lévy flights are detected, at least initially, and in addition whether such movements are interpreted to converge on the theoretically optimal search pattern.

Survey of the literature identifying Lévy behaviour in organisms reveals differences in the methods used to estimate µ. Some studies, mostly by biologists, use a simple log-log plot of the histogram of step-length frequency against step-length bin width (e.g. 5, 10, 15 …n) to derive the power-law exponent (e.g. Mårell, Ball & Hofgaard 2002; Austin, Bowen & McMillan 2004; Bertrand et al. 2005; Weimerskirch, Gault & Cherel 2005). In contrast, other studies, usually by physicists or mathematical biologists, vary the width of the bins in the histogram with each bin being a fixed multiple wider than the previous bin. Unequal bin widths are used to obtain a more homogeneous number of data per bin than is possible with equal-sized bins, which reduces statistical errors in the power-law tail (Newman 2005; Pueyo 2006). Usually bins are increased logarithmically with each bin k (e.g. 1, 2, 3 … ) increased by 2k (e.g. 2, 4, 8 … ). This can be further refined with the frequency per logarithmic bin normalized by dividing by bin width and n (total number of steps) to obtain the probability density of each bin (Viswanathan et al. 1996; Bartumeus et al. 2005; Newman 2005; Pueyo 2006). Another way of plotting power-law data is to calculate a cumulative distribution function (Newman 2005). However, there has not been a detailed analysis of how these principal methods differ in the accuracy of determining Lévy power-law exponents even though it may have important implications for identifying such behaviour in organisms.

The purpose of the current study was to examine differences in plotting methods by quantifying the effects of the different binning methodologies on Lévy flights with different power-law exponents within the range 1–3. Our approach was to model power-law distributions of Lévy flights with known µ-values and then to calculate the effects on µ of four different binning methods. The results of simulations were then described mathematically. Finally, biological data from published results were reanalysed using an appropriate binning method to illustrate how plotting errors can result in spurious interpretation of behaviour.

Methods and results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References

model simulations

Ten thousand sets of random numbers (n = 1000) were generated from a probability density distribution P(l) = αl−µ (see Mathematical description) with an exponent µ = 2·00 to represent consecutive step lengths in a movement path describing a theoretically optimal Lévy flight. Simulations were conducted in R programming language and step lengths were constrained to the range 0·5–2048 for computational ease. For each set of random numbers, the step length data were plotted as a histogram of frequency against bin k (Fig. 1a) and re-plotted after log-10 transformation of both axes (LT) (Fig. 1b). The second binning method was to place data into bins of width 2k, where k is an integer before log-10 transformation (LB), which yields a straight line relationship with equally spaced data points due to use of the geometric bin width of each 2k bin, calculated from 2k−1 (Fig. 1c). The third method was to place data into 2k bin widths then divide the frequency of observations in the untransformed variable in each bin by the bin width and total frequency before log-10 transformation (LBN) (Fig. 1d). This normalization procedure determines the (logarithmically) central value for each bin by estimating its probability density f(nk) = (1/2k)(sk/S) where sk is the frequency of steps in bin k, S is the total number of steps and 2k is the width of the bin (Pueyo 2006). Finally, we calculated a cumulative distribution (CD) for the observed data by plotting, for any given value l on the horizontal axis, the proportion of observations that were equal to or larger than l, again on log-10 transformed axes (Fig. 1e; see Mathematical description). We repeated these procedures for six similarly derived power-law distributions with µ exponents of 1·50, 1·75, 2·25, 2·50, 2·75 and 3·00 and compared the fitted linear relationship of each across the four binning methods. The effect of sample size on the estimated value of µ was examined by generating 10 000 sets of random numbers for each of the following sample sizes: 50, 102, 5 × 102, 5 × 103, 104, 105 and 106 random numbers.

image

Figure 1. (a) Histogram plot of 1000 random numbers generated from a power-law distribution with Lévy exponent µ = −2·00 (see eqn 1 in text for explanation). (b) Histogram of the same data after log-10 transformation of both axes (LT), as for eqn 2 in text. (c) Histogram of the data in (a) after placing frequency data into bins of width 2k (termed logarithmic binning) (LB), where k is an integer, then log-10 transforming the results (eqn 3). (d) Histogram of the data in (a) after dividing the frequency of observations in the untransformed variable in each logarithmic bin by the bin width and N (total number of steps) before log-10 transformation (LBN). (e) A cumulative distribution (CD) of the same data given in (a).

Download figure to PowerPoint

Although each method suffered slight positive biasing in the recovered µ exponents due to the imposition in our simulations of a maximum step size, we found that the third binning method (2k binning with normalization, LBN) most closely recovered the expected µ-values direct from the histogram regression slope with low error (mean error, 1·4% ± 6·4 SD) (Figs 2 and 3a). By contrast, linear regression slopes from simple LT of both axes did not accurately recover µ-values (mean error, 39% ± 9·6 SD) (Figs 2 and 3a). The two other methods (2k binning without normalization, LB; cumulative distribution, CD) recovered µ-values accurately from regression slopes (1 −µ) after correction (mean errors: LB, 1·4% ± 6·4 SD; CD, 4·5% ± 12·3 SD) (Figs 2 and 3a) (see Mathematical description for interpretation). Without correction, error in estimating µ directly from the regression slopes was c. 50%. Increasing sample sizes had the effect of increasing the kurtosis of the step length distributions, reducing the variation about the recovered values of µ (Fig. 3b,c). For example, variation about recovered µ was reduced by 82·8% as sample size increased from 50 to 5000 steps, and by 99·9% as n steps increased from 100 to 1 000 000 (Fig. 3b).

image

Figure 2. Estimated values of the Lévy exponent µ recovered from known Lévy distributions using different plotting methods. The actual µ-values of power laws subjected to the different plotting methods are joined by the solid grey line. Open square symbols denote values after log-10 transformation (LT), circles represent values from log-10 transformation after dividing through the 2k binned data by each bin width and total N (LBN). Triangle symbols illustrate µ-values derived from a cumulative distribution function (CD). Note that CD values for µ < 2 are not shown because biasing due to the imposition of a maximum step length make these values unreliable. Estimates for each µ-value were derived from 10 000 sets of 1000 random numbers. Estimates for the method log-10 transformation after 2k binning (LB) is not shown because the corrected slope was the same as the results for LBN. LB and CD methods yield slopes with 1 −µ, so correction is required to estimate µ.

Download figure to PowerPoint

image

Figure 3. (a) Summary of the mean percentage error (± 1 SD; n = 7) of each plotting method across the seven Lévy distribution datasets (µ: 1·5, 1·75, 2·0, 2·25, 2·5, 2·75, 3·0). Plotting method: LT, log-10 transformation of frequency vs. step (move) length; LB, log-10 transformation after 2k (logarithmic) binning using the geometric midpoint of each bin; LBN, log-10 transformation after dividing through the 2k binned data by each bin width and total N (normalization); CD, cumulative distribution function. The effect of sample size on (b) the magnitude of error variation about recovered µ-values following 2k binning and normalization, and on (c) kurtosis of the frequency distribution of recovered µ-values.

Download figure to PowerPoint

mathematical description

Different methods are available to estimate the slope of power-law distributions. Here we briefly explain the mathematics underpinning the plotting methods used to interpret the simulation output. Suppose that the distribution of step lengths genuinely follows some power-law distribution, so that

  • P(l) =αl − µ( eqn 1)

In the above, P(l) is the probability density of having a step length l, and the constant α simply normalizes this density to ensure that its integral from the minimum step length to infinity is one. Then clearly ln(P(l)) = ln α − µ ln l so that plotting a histogram of ln(frequency per unit step length) vs. ln(step length) should result in a straight line relationship with slope –µ, as described by Newman (2005; shown here from simulated data in Fig. 1b). However, suppose we take logarithms of each observed step length:

  • x= ln(l)( eqn 2)

Then the logarithmically transformed step lengths x have a probability density function given by

  • image(eqn 3)

so that

  • ln(f(x)) = ln(α) +x(1 −µ)( eqn 4)

Equation 4 therefore states that if we were to take logarithms of the step length data, place them into equally spaced bins on an x-axis, and then plot the number of data points in each bin on a vertical axis (here termed the LT method), then we would observe a straight line relationship with a slope of (1 − µ). This is exactly what is observed in Figs 1(c) and 2: the slope when geometric bin widths are used (i.e. bin widths that are equally spaced on a logarithmic scale) is always (1 − µ). Note that the plot resulting from this procedure is not a histogram representing a frequency distribution, as to produce a histogram requires either equally spaced bins in the untransformed variable, or to divide the frequency of observations in a bin by the bin width in the untransformed variable. Only when frequencies are divided by bin width and total frequency (in the untransformed variable), here termed the LBN method, do we recover a straight line graph with a slope of –µ (Fig. 1d).

An alternative method is to use a form of cumulative distribution (CD) function (sensu Newman (2005), also known as the probability of excedance), CD(l), defined as the probability that an observation is larger than or equal to l. Integrating the probability density in eqn 1 gives

  • image( eqn 5)

so that plotting CD(l) cumulative distribution function (Newman 2005) also leads to a straight line graph of slope 1 − µ when plotted on logarithmic axes, as shown in Fig. 2. Note that this method does not require the data to be binned; the fact that no information is lost via a binning process is therefore a strength of the CD method.

To summarize, if the data obey eqn 1 then plotting a genuine histogram of ln(frequency per unit step length) vs. ln(step length), that is, correctly taking into account the changing bin widths by normalization, results in a straight line graph with slope –µ. Simply placing data into bins of geometrically increasing size and plotting the resulting logged frequency, or alternatively plotting the empirical cumulative distribution function on logged axes, results in a straight line graph with slope 1 − µ. This explains why the slopes of linear regressions derived using 2k binning without normalization and the cumulative distribution function both underestimate µ by 1 (see Fig. 3a). The same arguments, and the same relationships between graph slopes and µ, hold regardless of the chosen logarithm base (10, e, 2, … ).

empirical analysis

We analysed published data from two studies where Lévy flight distributions were apparently identified in the satellite-tracked movements of grey seals Halichoerus grypus (Austin et al. 2004) and in the feeding locations of wandering albatrosses Diomedea exulans (Weimerskirch et al. 2005). The purpose of our reanalysis of these data was to test whether differences in binning methodology (1) falsely detected Lévy flights when they were not present, or (2) resulted in inaccurate estimation of µ when Lévy flights were present. From frequency distributions given in each study we calculated the new µ-value using the method of 2k binning with normalization prior to log-10 transformation (LBN) (see Fig. 2).

Austin et al. (2004) used log transformation of both axes (frequency vs. movement (step) length) (LT, as in Fig. 1b) to determine µ and found 30% of tracked grey seal movements fitted a Lévy distribution (µ-value range, 1·12–1·30). One of the seal movement-length histograms from Austin et al. (2004) (seal no. 6118) yielded an exponent of 1·26 (Fig. 4a,b). However, the relationship between log frequency and log step length is not linear (Fig. 4b) indicating the distribution of steps is unlikely to follow a power law. Therefore, the binning method they use is inappropriate for the data. We reanalysed Austin et al.'s (2004) data using the 2k binning with normalization method (LBN) and find no evidence for the existence of Lévy flights (Fig. 4c).

image

Figure 4. Reanalysis of empirical data from published studies. (a) Frequency histogram of movement lengths of a grey seal from Austin et al. (2004), together with (b) the log-transformed data indicate the absence of a power-law distribution of movement lengths (cf. Figure 1). Austin et al. calculated a Lévy flight exponent of 1·26; however, (c) reanalysis of the seal data using 2k logarithmic binning and normalization prior to log-10 transformation provides no evidence for the presence of Lévy flights in the data (e.g. µ < 1) and confirms the absence of a power-law distribution. (d) Frequency histogram of distances between feeding events of wandering albatrosses (Weimerskirch et al. 2005), and (e) the calculated µ-value of 1·26 using the log-10 transformation method. (f) Reanalysis confirms presence of a Lévy distribution in prey encounter by albatrosses, but the more accurate plotting method shows the exponent is higher and closer to the theorized optimal search pattern of µ≈ 2.

Download figure to PowerPoint

Weimerskirch et al. (2005) determined the frequency of different distances between prey captures by satellite-tracked albatrosses fitted with stomach temperature sensors. The histogram of frequency vs. distance between prey is given in Fig. 4(d) and simple LT of both axes provided a µ-value of 1·26 (Fig. 4e). This was interpreted by Weimerskirch et al. (2005) to indicate prey encounter by wandering albatrosses conformed to a Lévy flight. Although we did not have access to the original data that were separated into 10-km bins, we re-binned the frequency data into 2k logarithmic bins (where k is equal to n + 0·5) as conservatively as possible to avoid bias towards low distance bins, which would increase µ anomalously. Our reanalysis of the frequency distribution confirms that prey encounter in albatrosses does conform to a Lévy flight motion; however, our approximate µ-value was 1·68, which is substantially higher than determined using the less accurate method (Fig. 4f). The difference between these two estimations by different methods is < 1 unit, which in broad terms is contrary to that predicted by our simulations and mathematical description. It is likely this difference can be accounted for by the bias associated with our re-binning the histogram data (see above) (Fig. 4d). Figure 2 shows how the numerical difference in µ-values between plotting methods varies by < 1 unit in some instances.

Overall, our results show that using the simple LT method for deriving Lévy exponents, commonly used among biologists in published studies, results in significant errors. The LBN estimates µ accurately direct from the slope of the histogram plot, whereas although the LB and CD methods also estimate µ accurately, they yield histogram plot slopes of 1 − µ so correction is necessary.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References

There is an increasing number of studies showing the existence of Lévy flight searches among diverse organisms (Bartumeus et al. 2005; also see Introduction). Consequently there is a need to locate any sources of errors associated with the methodology used to identify such movement patterns. It appears that the method used to detect Lévy flights in movement data varies between studies, variation that may introduce significant error into the assessment of whether Lévy behaviour is present. However, until now, there has not been a formal investigation of how different methods influence the reliability of estimating the Lévy exponent, or how sample size influences reliability of estimation.

The results of this study indicate that the use of 2k (logarithmic) binning with normalization prior to log-10 transformation of both axes (frequency vs. step length) provides an accurate method for identifying Lévy flight in organism movement data directly from the slope of the linear regression. We have shown explicitly that simple LT of both axes, a method used widely (Fig. 3a), does not provide an accurate estimate of the Lévy exponent µ, which in our study resulted in its underestimation by c. 40%. The methods of 2k (logarithmic) binning without normalization (LB) and the CD function provide an accurate estimate of µ if taken from a corrected linear regression according to 1 − µ. These results are nontrivial because the exponent of the power law controls the range of correlations in the movement pattern, and hence, the macroscopic properties of the movement (Bartumeus et al. 2005). Therefore error in µ estimation has important implications for how movements are interpreted, and may influence directly therefore the strength of a particular study's findings.

On the basis of our model simulations and mathematical description we were able to hypothesize that simple log-log plots would introduce statistical errors in the power-law tail of the frequency distribution of step lengths (Newman 2005) resulting in Lévy exponents with low accuracy when directly compared with exponents derived from logarithmic binning followed by normalization and log transformation. Reanalysis of published data supported this hypothesis and demonstrated two main problems with using less accurate methods. First, the potential for wrongly ascribing Lévy flights to non-Lévy distributions. In support of this we found movement patterns of seals (Austin et al. 2004) were identified as Lévy flights when they were not power-law distributions. This may account for why the detection rate of Lévy flights among seal trackings was low in the latter study on account of Lévy motion being detected arbitrarily. Second, there is potential for greater error in estimating µ using less accurate methods that affect biological interpretation in a more subtle, but no less important, way. For example, the study of foraging in wandering albatrosses strongly indicated that prey encounter by the seabirds conformed to a Lévy flight motion (Weimerskirch et al. 2005). However, because the µ-value of 1·26 was not close to µ = 2, the optimal search pattern predicted by theory (Viswanathan et al. 1999), it was concluded that prey encounter may not be optimal for albatrosses. As we described, reanalysis using a more accurate method yielded µ = 1·68, which is substantially closer to 2 than the original estimate. Clearly, this indicates prey encounter by wandering albatrosses may not be particularly suboptimal. These examples illustrate that biological interpretation is sensitive to errors associated with plotting methods used to identify Lévy flights.

Our simulations also illustrate how a low number of step lengths measured for tracked animals can influence significantly the accuracy with which µ can be estimated. In our simulations, the standard deviation of the estimated Lévy exponent dropped from 0·3 to 0·09 when the number of steps used to recover the exponent was increased from 50 to 1000. This indicates that animal movement data sets need to be appropriately large to detect accurately a behavioural signal such as an optimal Lévy flight.

Several studies in the literature have used simple log-log plots to identify Lévy flights in animal behaviour (e.g. Mårell et al. 2002; Austin et al. 2004; Bertrand et al. 2005; Weimerskirch et al. 2005). We limited our reanalysis of data to two studies to illustrate specific points; however, our demonstration that the simple log-transformation power-law plotting method has low accuracy raises the question of how many similar errors have been introduced into behavioural studies by either falsely detecting Lévy flight motion in organisms, or by failing to identify its genuine presence. The majority of investigations we found have low error probabilities because they used the accurate method of 2k logarithmic binning with normalization to identify the power-law exponent, coupled with other techniques to detect Lévy flight phenomena (Viswanathan et al. 1996, 1999; Bartumeus et al. 2003, 2005). For example, studies have used net squared displacement of movement coordinates (Bartumeus et al. 2005), the root mean square fluctuation of the displacements (Viswanathan et al. 1996; Atkinson et al. 2002), or spectral analysis (Bartumeus et al. 2003) to reveal the existence within data series of long-range correlations that characterize Lévy statistics. Maximum likelihood estimates for Lévy parameters are also available (Johnson, Kotz & Balakrishnan 1994; Newman 2005). The number of ecological studies citing Lévy flight behaviour is increasing as more authors become aware of their existence, and the need for accurate and unambiguous methods is clear. We advocate not only the use of accurate plotting methods shown here to identify the presence of Lévy flights, but these other techniques also.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References

We are indebted to Frederic Bartumeus for valuable discussions and for making available to us preprint manuscripts on power laws, and to Paul Baxter for statistical insight. Gillian Frost assisted with the development of the programming code for the Lévy simulations. DWS is supported by a NERC-funded MBA Research Fellowship and DR by Defra-funded contract MF0154. This research is part of the European Tracking of Predators in the Atlantic (EUTOPIA) programme in the European Census of Marine Life.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and results
  5. Discussion
  6. Acknowledgements
  7. References
  • Atkinson, R.P.D., Rhodes, C.J., Macdonald, D.W. & Anderson, R.M. (2002) Scale-free dynamics in the movement pattern of jackals. Oikos, 98, 134–140.
    Direct Link:
  • Austin, D., Bowen, W.D. & McMillan, J.I. (2004) Intraspecific variation in movement patterns: modelling individual behaviour in a large marine predator. Oikos, 105, 15–30.
    Direct Link:
  • Bartumeus, F., Catalan, J., Fulco, U.L., Lyra, M.L. & Viswanathan, G.M. (2002) Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. Physical Review Letters, 88, 097901.
  • Bartumeus, F., Peters, F.S., Pueyo, S., Marrasé, C. & Catalan, J. (2003) Helical Lévy walks: adjusting searching statistics to resource availability in microzooplankton. Proceedings of the National Academy of Sciences USA, 100, 12771–12775.
  • Bartumeus, F., Da Luz, M.G.E., Viswanathan, G.M. & Catalan, J. (2005) Animal search strategies: a quantitative random-walk analysis. Ecology, 86, 3078–3087.
    Direct Link:
  • Bertrand, S., Burgos, J.M., Gerlotto, F. & Atiquipa, J. (2005) Lévy trajectories of Peruvian purse-seiners as an indicator of the spatial distribution of anchovy (Engraulis ringens). ICES Journal of Marine Science, 62, 477–482.
  • Brockmann, D., Hufnagel, L. & Geisel, T. (2006) The scaling laws of human travel. Nature, 439, 462–465.
  • Houston, A.I. & McNamara, J.M. (1999) Models of Adaptive Behaviour: an Approach Based on State. Cambridge University Press, Cambridge, UK.
  • Johnson, N.L., Kotz, S. & Balakrishnan, N. (1994) Continuous Univariate Distributions. Wiley, New York.
  • Da Luz, M.G.E., Buldyrev, S.V., Havlin, S., Raposo, E.P., Stanley, H.E. & Viswanathan, G.M. (2001) Improvements in the statistical approach to random Lévy flight searches. Physica A, 295, 89–92.
  • MacArthur, R.H. & Pianka, E.R. (1966) On optimal use of a patchy environment. American Naturalist, 100, 603–609.
  • Mårell, A., Ball, J.P. & Hofgaard, A. (2002) Foraging and movement paths of female reindeer: insights from fractal analysis, correlated random walks, and Lévy flights. Canadian Journal of Zoology, 80, 854–865.
  • Newman, M.E.J. (2005) Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46, 323–351.
  • Pueyo, S. (2006) Diversity: between neutrality and structure. Oikos, 112, 392–405.
    Direct Link:
  • Ramos-Fernández, G., Mateos, J.L., Miramontes, O., Cocho, G., Larralde, H. & Ayala-Orozco, B. (2004) Lévy walk patterns in foraging movements of spider monkeys (Ateles geoffroyi). Behavioural Ecology and Sociobiology, 55, 223–230.
  • Raposo, E.P., Buldyrev, S.V., Da Luz, M.G.E., Santos, M.C., Stanley, H.E. & Viswanathan, G.M. (2003) Dynamical robustness of Lévy search strategies. Physical Review Letters, 91, 240601.
  • Santos, M.C., Raposo, E.P., Viswanathan, G.M. & Da Luz, M.G.E. (2004) Optimal random searches of revisitable targets: Crossover from superdiffusive to ballistic random walks. Europhysics Letters, 67, 734–740.
  • Schuster, F.L. & Levandowsky, M. (1996) Chemosensory responses of Acanthamoeba castellani: Visual analysis of random movement and responses to chemical signals. Journal of Eukaryotic Microbiology, 43, 150–158.
    Direct Link:
  • Stephens, D.W. & Krebs, J.R. (1986) Foraging Theory. Princeton University Press, Princeton, NJ.
  • Viswanathan, G.M., Afanasyev, V., Buldyrev, S.V., Murphy, E.J., Prince, P.A. & Stanley, H.E. (1996) Lévy flight search patterns of wandering albatrosses. Nature, 381, 413–415.
  • Viswanathan, G.M., Buldyrev, S.V., Havlin, S., Da Luz, M.G.E., Raposo, E.P. & Stanley, H.E. (1999) Optimizing the success of random searches. Nature, 401, 911–914.
  • Viswanathan, G.M., Afanasyev, V., Buldyrev, S.V., Havlin, S., Da Luz, M.G.E., Raposo, E.P. & Stanley, H.E. (2000) Lévy flights in random searches. Physica A, 282, 1–12.
  • Viswanathan, G.M., Afanasyev, V., Buldyrev, S.V., Havlin, S., Da Luz, M.G.E., Raposo, E.P. & Stanley, H.E. (2001) Lévy flight search patterns of biological organisms. Physica A, 295, 85–88.
  • Viswanathan, G.M., Bartumeus, F., Buldyrev, S.V., Catalan, J., Fulco, U.L., Havlin, S., Da Luz, M.G.E., Lyra, M.L., Raposo, E.P. & Stanley, H.E. (2002) Lévy flight random searches in biological phenomena. Physica A, 314, 208–213.
  • Weimerskirch, H., Gault, A. & Cherel, Y. (2005) Prey distribution and patchiness: factors in foraging success and efficiency of wandering albatrosses. Ecology, 86, 2611–2622.
    Direct Link:
Enhanced Article (HTML) Get PDF (254K)Get PDF (254K)