## Introduction

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.