- Top of page
- Materials and methods
- Supporting Information
A recent advance within the field of movement ecology that aims to progress our understanding of the mechanisms underpinning search behaviour of diverse organisms has focused on the identification of specialised random walks, such as Lévy flights, principally through the analysis of move step-length distributions arising from recorded movement paths (Viswanathan et al. 1996). A Lévy flight is a special category of super-diffusive random walk where the distribution of move step-lengths fits an inverse power law such that P(l) ≈ l−μ where 1 < μ ≤ 3 where l is the move step-length and μ the power-law exponent. These movement patterns are characterised by clusters of short move steps connected by rare long relocations, with the pattern being repeated at all scales. Lévy flights (or walks) have generated interest because they have been shown theoretically to optimise searches for sparse resources such as prey, when located beyond an organism's sensory range (Viswanathan et al. 1999, 2011). Empirical studies have now identified movement patterns consistent with Lévy flights (walks) in individuals from diverse species including insects (Maye et al. 2007; Reynolds et al. 2009; Bazazi et al. 2012; Reynolds 2012), jellyfish (Hays et al. 2012), sharks, bony fish, turtles and penguins (Sims et al. 2008, 2012; Humphries et al. 2010) and seabirds (Humphries et al. 2012), as well as from single cells such as E. coli and T-cells (Korobkova et al. 2004; Harris et al. 2012).
Arguably, the most robust evidence for Lévy flight movement patterns in animals has come from studies of pelagic marine predators where move step-lengths have been derived from the depth recordings of electronic tags (Sims et al. 2008, 2012; Humphries et al. 2010; Hays et al. 2012). In one-dimensional (1D) data such as this, vertical displacement step-lengths are straightforward to compute because putative turning points are simple to identify and are unambiguous, being the points where there is a change of direction (i.e. diving or ascending) between consecutive steps. While the 1D turning points identified in this way do not correspond exactly to the actual turning points in the original 3D movement of the animal, the overall scaling properties of Lévy flights are preserved. The step-lengths are analysed using maximum likelihood estimation (MLE) to estimate exponents and goodness-of-fit (GOF) for power-law or exponential distributions (Clauset, Shalizi & Newman 2009; Humphries et al. 2010). Testing of the Lévy flight foraging (LFF) hypothesis (Viswanathan, Raposo & da Luz 2008; Viswanathan et al. 2011) is mainly concerned with power-law and exponential distributions because the LFF hypothesis predicts that Lévy flight searching is optimal when prey is sparse, whereas simple Brownian (exponential) movements are expected when prey is abundant.
The analysis of horizontal movement paths – in terms of a discrete random walk – for the presence of Lévy or Brownian patterns using MLE requires the identification of turning points in order for the step-lengths to be computed. While Lévy walk characteristics can be identified using functions such as root mean square fluctuation, or mean square displacement, these methods do not provide estimates of exponents and cannot be used to test fully the LFF hypothesis. Turning points are relatively straightforward to identify in the low spatial resolution datasets of animals tracked using Argos satellite transmitters, but the large and variable error fields make such data unsuitable for rigorous testing for Lévy flight behaviour (Bradshaw, Sims & Hays 2007). In some movement data, such as from bacteria, T-cells or desert locusts, the recorded movements are essentially discrete, comprising, for example in the case of E. coli, runs and tumbles; in these cases, turn identification is also straightforward. This is also true of some lower resolution data, which is already closer to a discrete approximation of the original movement path.
Tags equipped with Global Positioning System (GPS) sensors now provide data with high spatial accuracy and temporal resolution (Weimerskirch et al. 2002; Sims et al. 2009); however, testing high-resolution GPS data for Lévy flight patterns has proved problematic because of the difficulties in objectively identifying turning points in a tortuous path (Codling & Plank 2011; Humphries et al. 2012). Various methods have been proposed for the identification of turning points, for example, the location of acute turning angles (e.g. Reynolds et al. 2007), or the deviation of the movement path from an arbitrary corridor encompassing the trajectory (Turchin 1998; de Knegt et al. 2007). However, while the results are dependent on the parameters chosen (Plank & Codling 2009), it can be difficult to set a threshold turning angle, or corridor width, that have a sound basis in the biology of the animal and which are not to some degree contentious. Consequently, the discretisation of the path into steps is in some cases somewhat arbitrary (Reynolds 2010) and parameter choices and the results of the analysis are sometimes difficult to justify. To illustrate the problems, the consequences of differing turn-angle thresholds in GPS data are explored in a sensitivity analysis of wandering albatross data presented in Appendix S1 (section 2). In summary, we found that the number of truncated Pareto-Lévy (TP) distribution fits to the 27 datasets was 4, 17 and 25, for the turn angles 45, 90 and 135°, respectively. While both the number of fits, and the closeness of the fit in many cases, make the results of the analysis using 135° compelling, there is no clear biological justification for that choice of angle. Therefore, it appears that there are potentially significant amounts of 2D data, from diverse species, that at present cannot be used reliably in either Lévy flight or any other random walk movement analysis (e.g. correlated random walks), severely limiting widespread testing of these ideas in ecology.
A study by Sims et al. (2008) showed mathematically that a Lévy flight can be projected from 3D to 2D and to 1D with preservation of the power-law exponent. From this, it was suggested that the power-law-distributed 1D vertical displacements of marine predators were indicative of an overarching 3D Lévy flight movement pattern. However, no empirical study has yet shown this to be the case, nor has the dimensional symmetry of a Lévy flight been demonstrated for the TP-Lévy distribution, which was found to be the most common power-law distribution to best fit animal movement data (Humphries et al. 2010).
Here, we verify the purported dimensional symmetry of Lévy movement patterns using simulated 3D Lévy distributed move step-length datasets, with a range of exponents, from which any one of the dimensions can be used to form a 1D dataset. The conditions under which the symmetry holds are explored and compared with the results from exponential and composite Brownian (CB) datasets. We examine the effects that tag measurement errors, such as low spatial or temporal resolution, would have on the reliability of identifying move step-lengths, and detecting Lévy flight behaviour, in naturally complex datasets. By extending the symmetry paradigm, we then present a methodology for the identification of step-lengths that can be applied to 2D or 3D data and demonstrate the utility of this methodology in a re-analysis of wandering albatross (Diomedea exulans) GPS location data previously analysed for the presence of Lévy flight patterns in the distribution of landing sites (Humphries et al. 2012). Software developed to test these ideas is freely available to download with this study.
- Top of page
- Materials and methods
- Supporting Information
We have demonstrated that a range of move step-length distributions typically found in empirical datasets project into 1D with sufficient fidelity, even with common empirical measurement errors, to be clearly identified in a subsequent MLE analysis. Using this finding, we propose a method for the identification of turns and step-lengths in high-resolution 2D or 3D datasets where previous methods have been found to be unsatisfactory.
The new method was illustrated using high-resolution GPS data from wandering albatross (Diomedea exulans). We found Lévy movement patterns were prevalent, occurring in both dimensions for 20 of the 27 birds analysed. The majority of fits were found to be very good, with significant P-values computed for 23 individual dimension datasets and with five birds having significant P-values for both dimensions. In a previous study, where the distribution of landing locations was analysed (Humphries et al. 2012), only four of the D. exulans datasets were best fit by a TP distribution. Because D. exulans landed relatively infrequently during foraging trips, the analysis of the previous study resulted in very few data points which was shown to reduce the likelihood of a TP fit. Consequently, the study by Humphries et al. (2012) detected fewer examples of Lévy flight behaviour, compared with the black browed albatross (Thalassarche melanophrys) datasets for which far more data points were available. In the current study however, with significantly more data to analyse, we found very good support for the presence of Lévy flight movement patterns for the majority of wandering albatrosses, improving the rate of detection of Lévy flight behaviour from 15% to 74% of individual birds. Given the significant increase in the detection of Lévy patterns with the new method, it seems likely that previous studies purporting to demonstrate biological Lévy flight, or questioning its existence, might have been strengthened (e.g. Viswanathan et al. 1996) or have drawn quite different conclusions (e.g. Edwards et al. 2007; Edwards 2011), had this new methodology been employed. Furthermore, the movements analysed here represent characteristics of the actual flight path of the birds, rather than the distribution of the landings, which might correspond to the fractal nature of the prey distribution (Miramontes, Boyer & Bartumeus 2012). The movements analysed here therefore better reflect the complex paths taken during animal foraging/searching behaviour.
In the previous study by Humphries et al. (2012), the Lévy exponents were found to be lower than the theoretical optimum value of 2·0 (mean 1·19). Here, however, the mean value of 1·75 (SD 0·31) is closer to the optimum. Exponents <2·0 have been shown theoretically to be optimum values where targets (i.e. prey) are nonrevisitable (i.e. single prey items as opposed to patches; Viswanathan et al. 1999), and it was proposed by Humphries et al. (2012) that this was the most likely scenario for wandering albatrosses foraging in the open ocean. Eight datasets analysed here have a value slightly larger than 2·0, (mean 2·26, SD 0·16) suggesting more abundant prey. The results presented here therefore suggest a broader range of behaviours and of foraging environments than previously concluded. That not all datasets were found to fit a Lévy distribution is unsurprising because albatrosses are known to use olfactory clues (Nevitt, Losekoot & Weimerskirch 2008) and will on those occasions be performing directed, rather than random searches, and will not be expected to fit a Lévy pattern.
Given that Lévy movement patterns have been shown to optimise random searches for sparse targets (Viswanathan et al. 1999, 2002, 2011), our finding that Lévy movement patterns are prevalent in wandering albatrosses engaged in foraging trips in the open ocean, where prey patches are sparsely and unpredictably distributed, provides further evidence that adaptations for stochastic search movements described by Lévy flights may have naturally evolved (Humphries et al. 2012; Sims et al. 2012).
Applicability to empirical datasets
In principle, the method we have outlined here can be applied to almost any 2D or 3D movement dataset. It should be emphasised that the method is intended to identify turns and step-lengths in movement data suitable for subsequent analysis (such as MLE), but does not in itself provide direct evidence of the process that might have generated the observed step-length distribution, be it a correlated random walk, a Lévy walk or some other process, such as a continuous time random walk (Reynolds 2010).
Furthermore, very low-resolution datasets, such as those produced via Argos satellites, where there are few locations per day, will not capture sufficient fine-scale behaviour for robust conclusions to be drawn about the precise spatial form of the movement pattern (Bradshaw, Sims & Hays 2007). As was found with the original analysis of the D. exulans landings data (Humphries et al. 2012), there are sometimes too few data points to sample the distribution sufficiently for a power law to be discernible. Turchin (1998) comments that there is no generally acceptable solution to deal with over-sampled data, however, with 1D data, over-sampling is preferable to under-sampling. With over-sampled 1D data, it is straightforward to coalesce steps that have been artificially divided by the sampling interval and therefore to recover accurately the correct turning points. With under-sampled data, certainly at the further extremes, too much of the original movement is lost and cannot be recovered. Therefore, high-resolution datasets, such as those obtained using GPS, are expected to give much better results.
The method presented here would be well suited to other GPS datasets, for example, golden eagles (Aquila chrysaetos; Lanzone et al. 2012) or griffon vultures (Gyps fulvus; Nathan et al. 2012), both of which perform extensive foraging flights and might be expected to exhibit Lévy flight patterns. Terrestrial GPS studies of predators such as wolves (Canis lupus; Gurarie et al. 2011), lynx (Lynx Canadensis; Olson et al. 2011) or cougar (Puma concolor; Knopff et al. 2010) could also make use of this methodology for testing the possibility of scale independence in animal movement or for identifying the times/locations where optimal search strategies may be employed. The method may also benefit studies of the movements of nematodes (Ohkubo et al. 2010) or even individual immune cells (Harris et al. 2012).
It would also be interesting to apply this method to studies that have led to debate, such as that by de Jager et al. (2011), where the movements of mussels (Mytilus edulis) were found to follow a Lévy-like walk pattern but which was challenged by Jansen, Mashanova & Petrovskii (2012). The method presented here is simpler to implement than the methods employed in that study and requires no sensitivity testing to determine the best parameters.
The simplicity of the method, together with the now very abundant, high-resolution, 2D data available for analysis from a very broad range of studies, should allow a more thorough test of the LFF hypothesis, as well as the provision of a useful tool for the analysis of movement in diverse organisms. To encourage research in this area, the software we developed to perform the MLE analysis is freely available for download with this study from the Sims Lab web site (www.mba.ac.uk/simslab/).