Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.
During the past 25 years, the collection of animal movement data via automated electronic tags has become a key method of observing the behaviour of free-living animals. These data allow behavioural observation over long periods of time where, for a wide variety of logistical reasons, direct observation is not possible [1,2]. From such data we can infer behavioural processes, basic aspects of a species' ecology such as mechanisms of dispersal and migration, and how species interact with conspecifics, other ecosystem members and landscapes [3–6].
The miniaturization, decreased cost and increasing reliability of animal-borne tracking devices has allowed the tracking of hundreds of species and the production of mountainous volumes of animal movement data. Improved electronics are also increasing precision and temporal resolution. The rapidly improving technical capacity to remotely observe animal movement is sparking the birth of the field of movement ecology . However, the situation is not ideal because statistical methods for analysing movement have generally not kept pace with the rapid development of tracking technology. New technologies such as accelerometry are being deployed on animals faster than methods are developed to analyse these novel, often complex data streams (e.g. ).
Movement data are often information-rich, allowing robust inference of behavioural processes. However, these data are typically collected with no independent behavioural observations. The lack of such observations can make it difficult to properly interpret movement patterns or put an animal's movement into a broader behavioural context. In the absence of behavioural or environmental observations that would otherwise constrain possible explanations of an observed movement pattern, movement processes may be misunderstood and/or fitted with biologically inappropriate models. As we will show, these observations can change the statistical results and biological interpretation of movement patterns.
Because there are often no direct observations that might otherwise help develop a set of biologically informed hypotheses, many researchers instead rely on classic model selection methods to select from a set of candidate models that may or may not be appropriate. In such analyses, the most parsimonious or best-fitting model is selected using criterion such as lowest AIC or BIC (Akaike and Bayesian information criteria, respectively ). This is a well-supported and justified approach so long as the set of candidate models is biologically justified . In many analyses of movement data, however, the candidate models are limited to a small number of simple probability distributions and do not consider autocorrelation or other patterns in step-length order. This has led to considerable debate about what the appropriate set of candidate distributions should include or are biologically defensible [10–12].
During the development of modern movement models, diffusion, random walk and various correlated random walk (CRW) models were often employed to analyse movement, many of which were developed and borrowed from physicists (see [10,13,14] for some historical context). These early random walk models, many of which are still used today, rely heavily on Gaussian and exponential probability distributions, both of which have finite variance. A number of authors suggested that the finite variances of these distributions do not capture animal movement processes well, which frequently have longer steps than they predict. Instead, these authors proposed that the longest step-lengths be fit by a power-law (PL, Pareto) distribution, with infinite variance [15,16]. Authors eventually refined this by bounding the PL to capture the long steps but remove the infinite variance [17–19]. Further, many have argued that the presence of PL distributions in animal movement are indicative of a behavioural mechanism and optimal search [15,16,18–20]. Many question if such optimality exists in nature, because the conditions under which it has been shown are restricted. When more realistic assumptions are imposed a wide variety of alternative models perform better [21–23]. The use of memory is one of the conditions outside the assumptions of Lévy flight optimality, which most animals are known to use in order to refine search and movement strategies . Such strategies are usually superior to Lévy flights [22–25]. In any event, random walk processes where step-lengths are drawn from any heavy-tailed distributions such as a Pareto are known as ‘Lévy flights' or ‘Lévy walks’.
In most cases, there are no ancillary observations to corroborate or test if any particular behavioural mechanism produces the apparent PL distributed step-lengths. Lack of behavioural observations connecting apparent Lévy flight patterns to clear behavioural processes combined with concerns that statistical methods for detecting Lévy flights may not be appropriate, has led to a debate in the emerging field of movement ecology [10–12,17,26–28]. Although step-lengths are often distributed with much ‘fatter’ tails than would be predicted from exponential or Gaussian distributions, it has been contended that mixing multiple movement behaviours, behavioural intermittence, the action of memory on movement processes, or indeed many other complex behavioural interactions, can all lead to movement step-lengths with apparent PL tails [5,12,24–26,29,30]. For example, a significant number of studies have recently pointed out that variation in the movement patterns of individuals composing a population, whose individual movement kernels have Gaussian distributions, can result in fat-tailed step-length distributions at the population level [31–36]. In many of the cases where long tails exist, it may or may not be appropriate to call such movement patterns ‘Lévy flights’, because they were formed by mixing two or more distinct kinds of movement, each with a different underlying behavioural motivation. It is also unclear that the label ‘Lévy flight’ is appropriate in any situation where memory acts on the movement process. Lévy flights are random walks that assume draws from the distribution are independent and identically distributed, meaning that step-lengths should not be correlated, are unrelated to memorized information and are unrelated to information animals perceive in their environment. Without ancillary independent behavioural observations, it can be very difficult to determine whether these conditions exist or discriminate step-lengths produced from a mixture of behaviours or mixtures of individuals with different behavioural repertoires (i.e. steps drawn from mixtures of exponential distributions) from those drawn from a single PL distribution [21,22,36].
Here we use a mix of direct behavioural observations and GPS tracking to measure the movement of two closely related butterfly species, congeneric checkerspots belonging to the genus Euphydryas. Unexpectedly, the step-lengths of one of the species appeared PL distributed, while the other appeared exponentially distributed. The apparent difference was surprising for several reasons. First, these butterflies are extremely closely related; second, we only tracked gravid females who were searching for either oviposition sites or nectar resources and third, females were searching for the same host plant (Plantago lanceolata) in similar habitats. Thus, we expected the step-lengths and movement patterns to be similar. As our movement pathways were accompanied by direct behavioural observation, the situation afforded an excellent opportunity to understand the mechanistic behavioural processes that lead to PL or exponentially distributed step-lengths.
To understand how these different movement patterns might have arisen, we fit the step-lengths to a series of probability distributions often considered as the whole set of candidate models in studies investigating Lévy processes in animal movement [17–20], as well as a mixed-exponential probability distribution as suggested by others investigating the ‘Lévy flight paradigm’ [12,37]. For comparison, we also fit a Weibull distribution, which has become a probability distribution of choice for modelling step-lengths in movement models (e.g. [38,39]) and in many respects can be considered a more general version of the exponential. We then investigated the influence of an important directly observed behaviour on step-length in these two butterflies: intraspecific interaction, particularly the effect of male harassment on female movement in the two study species. In our analysis, we clearly show that apparent PL step-lengths arise as the result of mixing two otherwise exponentially distributed movement modes—harassed evasion of males, which causes longer steps, and unharassed movement, a search process with much shorter step-lengths. We discuss the implications of our findings in the larger context of Lévy flight paradigm of animal movement.
2. Material and methods
2.1. Study species and populations
We tracked the movements and directly observed the behaviour of two congeneric butterfly species—Euphydryas editha taylori, an endangered subspecies endemic to temperate regions of western North America, and Euphydryas phaeton, a more widely distributed species found throughout temperate mesic regions of eastern North America. These species were selected because they were historically highly specialized to a single or small number of host plants, but both independently adopted the exotic weed (Plantago lanceolata) as a new host. Our primary research goal was to understand how these species interact behaviourally and demographically with the exotic host plant. We report those findings in Severns & Breed . However, secondary to this goal, we collected fine-scale movement data and behavioural observations on intraspecific interactions, which are highly informative on the current debate over the nature and behavioural mechanics of PL distributed step-lengths in animal movement data.
Both E. phaeton and E. e. taylori form small local colonies and their landscape-level population dynamic are understood to be governed by a classic metapopulation dynamic. In 2011, we observed the movement of gravid females from two populations of E. phaeton in Massachusetts, USA. The population at Stevens-Coolidge Place (hereafter SCP; 42.682° N, −71.116° W) uses the native host-plant Chelone glabra, while a second population at the Bullitt Meadow (hereafter Bullitt; 42.501° N, −72.754° W) uses the recently adopted exotic host Plantago lanceolata . In 2012, we observed the movement of E. e. taylori from a single population occupying two fields separated by approximately 200 m. This population is near Corvallis, Oregon (44.58° N, −123.37° W), and also uses the exotic Plantago lanceolata as its primary host plant. At all study sites, the host plant was widespread and resources were not patchy from the perspective of the observers. However, female butterflies are known to search not just for the host plant, but also assess host plant quality. How they make such assessments is unclear, but they can have significant fitness consequences .
2.2. Butterfly tracking
We opportunistically followed encountered individuals, recording GPS locations (Garmin eTrex Venture HC) every 15 s for 15 min for E. phaeton and every 20 s for 20 min for E. e. taylori, to quantify short-term movements in all three study populations (details reported in ). The longer observation steps for E. e. taylori were used because only one observer was available, and observations on shorter time-steps proved too difficult. Individual females were selected for collection of movement pathways after they were observed stereotypically sampling host plants for oviposition sites. We chose to follow gravid females for two reasons. First, selecting host plants for oviposition is the key behavioural control adult butterflies have over host-plant selection. Second, and more importantly for the results reported here, gravid females searching for hosts were unequivocally directly observed engaging in a search for resources, a key aspect of the ‘Lévy foraging hypothesis' . Moreover, these animals move around the habitat steadily while searching, producing relatively long, high-quality movement pathways. We should note that, although we used a consumer grade GPS, due to the slow-varying nature of GPS error, step-lengths collected on such short time intervals are actually extremely precise, as we report in Severns & Breed .
At each time step, we also made focal behavioural observations (behaviours recorded included flying, perching, basking, courtship, inspecting host plants for oviposition, nectaring, oviposition and mating) to construct individual activity budgets. Perching was recorded when individuals were at rest with wings closed over their thorax, minimizing the exposure of wings to the sun. Basking occurred when individuals were at rest with their open wings exposed to the sun (dorsal basking). Flying, mating, nectaring and oviposition are self-explanatory.
For this analysis, courtship events were key. These events are intraspecific interactions and occur when males attempt to mate with females, typically interrupting other behaviours such as nectaring or searching for host plant. We recorded whether females were receptive to male courtship by noting whether females rapidly fluttered their wings and curled their abdomen away from courting males, the stereotypical rejection behaviour exhibited by female checkerspot butterflies . Courting events very often escalated to harassment by males, especially in E. e. taylori, who would attempt to forcibly mate unreceptive females. This elicited a variety of responses, including hiding and fleeing. Fleeing females typically made unusually long, fast flights. Hiding females crawled beneath vegetation and remained still in a perched or lateral position immediately after or during courtship attempts.
The literature has explored life-history strategies that lead to male harassment, and, in a range of species, conflict has evolved between males and females. For females, a small number of matings maximizes fitness, while male fitness is often maximized by mating as many times as possible. This conflict can lead to male harassment and the elicitation of avoidance behaviours by females [43,44]. It also leads to strong agonistic interactions between males which we frequently observed. Female–female intraspecific interactions were essentially not observed in our populations; females ignored other females when they encountered each other in both E. phaeton and E. e. taylori.
Most of the individuals we followed were previously uniquely marked with metallic ink as part of a parallel mark–recapture study. In many cases, we were able to follow individuals for the entire prescribed observation interval of 15 or 20 min, but occasionally individuals could not be followed or were otherwise lost, so some tracks were shorter than others. Daily flight periods of both species are short, usually only several hours each day, and inclement weather can abruptly end an observation period. We implemented rules for aborting behavioural observations in the case of inclement weather or if individuals entered a protracted bout of resting. In these instances, after 5 min of inactivity we terminated observations and a new individual was selected and followed. Observations were made when weather conditions were ideal for butterfly flight, always under full or near full sun, between 10.00 h and 17.00 h local time.
Our sample sizes included 40 E. e. taylori, 25 E. phaeton from the Bullitt population using P. lanceolata host plant and 20 E. phaeton from the SCP population using the native C. glabra host plant. In total, we collected 573 move steps from E. phaeton at SCP, 483 steps from E. phaeton at Bullitt and 674 steps from the Oregon E. e. taylori population. For the step-length distribution fits, step-lengths were pooled for each of the three populations.
2.3. Fitting step-length distributions
Our overall approach to fitting step-length distributions is based on the methods detailed in Edwards et al. [17,28]. Adapting the R code provided in Edwards et al. , we used a maximum-likelihood optimization method to fit the standard set of four candidate probability distributions usually tested in Lévy flight or PL animal movement analyses. Distributions were fit to our step-length data using the nlm function in R , following Edwards et al. [17,28]. The four models and probability density functions f(x) for step-lengths x are taken from Edwards et al.  and Edwards . First is the classic Lévy flight model of an unbounded PL 2.1 with exponent μ, minimum movement length a and normalization constant . The second model is the simplest possible alternative of an unbounded exponential (Exp) 2.2 with parameter λ. The third model is a bounded power law (PLB) 2.3 where b is the upper bound and normalization constant for μ ≠ 1 and for μ = 1 . The forth candidate model is a bounded-exponential distribution (ExpB) 2.4 with normalization constant . For both ExpB and PLB, b is set to the maximum value in the data. Likelihood functions are derived in Edwards et al.  and Edwards  and provided in electronic supplementary material, appendix A.
The same numerical likelihood fitting procedure was adapted to fit an additional two candidate distributions. First is a mixture of two unbounded-exponential distributions (2Exp): 2.5 where p represents the mixture probability of the two exponential distributions . Second is a Weibull distribution (Weib) 2.6 which has two parameters (k, λ), and is in the exponential family but can take a much wider range of shapes including a hump shape or an exaggerated exponential shape with higher probability of short and long steps compared with an exponential. For the special case of k = 1, the Weib and Exp distributions are identical. Weibull distributions have become a common choice in modelling and analysis of animal step-lengths and movement processes (e.g. [38,39]).
In all cases a is the minimum step-length in the dataset. Note that, because we directly observed movements, we know short steps were not attributable to observation error and they are numerically important. Thus, we could not justify ignoring some or all of these short steps—often fitting procedures have excluded the shorter steps and only fit the distribution tails .
The numerical likelihood maximizations yielded parameter estimates and a minimum log-likelihood for each of the seven candidate models. From the log-likelihood, an AIC score was calculated for each of the candidate models to compare fit: 2.7 where K is the number of parameters being estimated in the model. After fitting and model selection, the selected model was tested for goodness of fit (GOF) using a Williams corrected G statistic (a likelihood ratio test) .
Mixtures of exponentials here represent the marginal distributions of more complex models that include behavioural intermittence and state-switching CRW models that have been developed to understand intermittence in movement processes. These more complicated models usually include an autocorrelation function or dynamic, an explicit state transition equation, and are fit as Hidden Markov Models, usually using a Bayesian approach. This complexity and the Bayesian approach do not allow them to be easily compared with simpler models such as probability density functions (though it is possible). However, because a mixture of exponential distributions is the marginal distribution of these more complicated state-switching models, they can represent the step-lengths these complex processes produce well [12,37].
After fitting step-lengths from the entire set of relocation data, we used the direct behavioural observations made during tracking to group the steps made by females into two categories: harassed by conspecific males or unharassed. We then repeated the step-length distribution fitting described above on the two groups of step-lengths. In addition, we selected 32 (16 E. e. taylori, 16 E. phaeton) of our best observed individuals (tracks in which at least 15 and as many as 40 step-lengths were observed). Using the same numerical methods, we fit the candidate models to these individual step-length datasets to understand individual variation (results are presented in electronic supplementary material, appendix D). Finally, we explored the effect of courting/harassment on step-lengths of E. e. taylori using a set of simple mixed-effects models. These models included a random effect of individual, which allowed us to assess the effect of courting on movement while accounting for individual random effects in our step-length data. These models were fit using the lmer function from the lme4 package in R. They include step-length as the response, were called with a log-link, included site and courtship status as fixed-effects, and individual as a random effect.
3.1. Distributions fit to whole step-length datasets
In our first analysis, we ignored the directly observed behavioural information because in nearly all similar analyses of animal movement step-lengths, such data are not available. The results indicate that, when considering the four step-length distributions usually included in analyses of PL governed movement patterns, E. e. taylori step-lengths are best fit by a bounded PL (figure 1 and table 1a). Step-lengths produced by both populations of E. phaeton are best fit by exponential distributions (figure 1 and table 1a). The difference in the shapes of the step-length distributions is clearly visible in the histograms shown in figure 1a,d,g. The histograms in figure 1 and the fact that these two closely related butterflies fit very different step-length distributions indicates that the movement processes strongly differ.
When considering the additional two candidate step-length distributions (2Exp, Weib), the new distributions fit better than the original four distributions for all three datasets (table 1a,b). For E. e. taylori, the 2Exp model fitted best. For both populations of E. phaeton, the Weib model fitted best. When these six candidate distributions were fit to individuals, we found that for the 16 E. e. taylori tracks tested, eight were best fit by either the 2Exp model or a mixture of Weibulls (see the electronic supplementary material, appendix C for details on fitting a mixed-Weibull, and electronic supplementary material, appendix D for model fits to individual butterfly pathways), while the remaining eight were best fit by either a Weibull, exponential or bounded-exponential. It is tempting to attribute this to individual variation, but the more prudent interpretation is that most individuals express a biphasic movement pattern, but in half of the observed individuals, only one of the phases was observed because of the short sampling window. For the 16 individual E. phaeton, 12 of the individuals were best fit by either a bounded exponential, exponential or Weibull distribution, while four were best fit by mixed distributions, suggesting a movement pattern that mostly consists of a single movement phase, but where a second phase with longer steps occurs much more rarely than in E. e. taylori (electronic supplementary material, appendices C and D).
3.2. Goodness of fit
In the first analysis including the four candidate models, the GOF test suggested the data are not consistent with a bounded PL model for E. e. taylori in spite of being selected (p < 0.001). Although this model is favoured in this first-round analysis, the fit is poor lending further motivation for testing additional models. For E. phaeton, the GOF tests suggest that the data are not consistent with coming from the selected Exp model for the Bullitt population (GOF p-value = 0.001) but are for the SCP population (GOF p-value = 0.08). When the additional alternative models were included (table 1b), the GOF tests imply that the data are consistent with coming from 2Exp for E. e. taylori (GOF p-value = 0.26) and Weib for both E. phaeton populations (GOF p-values = 0.64 and 0.27 at Bullitt and SCP, respectively). The differing results for the two species are intriguing, because these animals are very closely related and otherwise performing the same search for oviposition sites.
3.3. Step-length distributions of harassed versus non-harassed steps
Having noticed a strong difference in courting behaviour by males in our direct observations, we investigated further a hypothesis that male harassment induces long, evasive moves in females, and that mixing harassed and unharassed moves creates the apparent PL distributed set of overall step-lengths. To test this, we used the direct behavioural observations to investigate whether step-lengths made by females were affected by harassing or courting males.
Despite being closely related, E. e. taylori males were far more aggressive towards females compared with male E. phaeton. Butterfly population densities were similar at both sites, but female E. e. taylori were courted or harassed during 43% of moves, implying they are spending close to half of their day interacting with males. These interactions were often very aggressive, even violent, with multiple males forcing females to the ground and engaging in a scrum of attempted copulation. Females typically responded by hiding under a leaf or fleeing, often fleeing out of the suitable habitat patch (figure 2).
By contrast, females at the SCP and Bullitt populations of E. phaeton were courted 6% and 8% of the time, respectively. Moreover, these interactions seemed less aggressive than E. e. taylori, and when courting males were rejected by an unreceptive female, they usually quickly gave up on a courting attempt. Females rarely resorted to hiding or fleeing from courting males (figure 3).
When the E. e. taylori step-lengths were sorted into steps that occurred while being harassed and steps that occurred while not harassed and fit to the four usual candidate distributions of PL animal movement analyses, we find that both the harassed step-lengths and unharassed step-lengths are best fit by simple exponential distributions (table 2). Recall that when these distributions are fit to the unsorted steps, they are best fit by a bounded PL (table 1a). Thus, dividing up the original dataset based on behavioural information changes the best-fitting model and thus the biological conclusion. However, the GOF test indicates that neither harassed nor unharassed moves are consistent with coming from an Exp (GOF p-value = 0.015, <0.001, for harassed and unharassed, respectively).
When the Weib model is introduced as a candidate (2Exp was not tested because step-lengths were sorted into single states), it is selected over the Exp for both harassed and unharassed moves, and the GOF shows good fit in both cases (p = 0.10 and 0.07 for harassed and unharrassed moves, respectively). This suggests that in fact the process as a whole may be best fit by a mixture of Weibull distributions (see the electronic supplementary material, appendix C). Not shown here are the E. phaeton results, as very few moves were harassed. However, these were similarly always best fit by the Weib when split into harassed versus unharassed categories. This was not unexpected given the Weib was the best-fitting model for the entire E. phaeton step-length datasets from both the Bullitt and SCP populations.
Finally, exploring E. e. taylori movement with a simple set of mixed-effects models reveals that harassment has a strong positive influence on step-length, with courtship causing much longer step-lengths (p ≤ 10−16) for E. e. taylori. This is also not surprising given the step-length distribution histograms of harassed versus unharassed steps in figure 4. In addition, there was weak support for a model (p = 0.06 compared to the harassment only model) where courtship interacted with site. This suggests the way females react to male harassment is patch specific and influenced by habitat or other conditions.
Our results have three clear implications. First, several previous analyses showed that it is possible to refute findings of PL step-length distributions by testing the exponential distribution, the simplest alternative null model [17,27,28]. We have shown that the alternative Weibull distribution and mixtures of exponentials, both related to the simple exponential, can even better describe movement data. Future analyses should therefore test a wider suite of biologically sensible alternatives against PL and bounded PL distributions.
Second, we have shown that if only the four distributions usually considered in PL analyses are tested (PL, exponential, PLB and bounded exponential), E. e. taylori step-lengths best fit a bounded PL. In many previous studies, this comparison and the selection of a bounded PL (using other methods that include fitting only the tail of the distribution and not the entire set of step-lengths) has been considered evidence for a Lévy flight and support for the ‘Lévy foraging hypothesis' [47,48] (but see ). However, when we introduced an equally complex (Weibull) or only slightly more complex (mixture of two exponentials) alternative distributions, both fit better than a bounded PL, and thus there is no support for a Lévy flight.
Third, we could test if a mixed-exponential was biologically appropriate by using our direct behavioural observations to sort steps into behavioural classes. When we did this, both distributions were best fit by a simple exponential, when comparing the four usual unimodal candidates. The Weibull, which is a more flexible unimodal model from the exponential family, fit both harassed and unharassed steps best when it was introduced for comparison. Like the exponential, Weibull distributions can also be mixed, and we demonstrate such mixing in the electronic supplementary material, appendix C. Unsurprisingly, mixtures of two Weibull distributions, which have six parameters, fit two of our three step-length datasets better than the candidate models considered in the main text (electronic supplementary material, table C1).
Thus, when steps were appropriately sorted with relevant behaviour information, the evidence for PL step-lengths, indicative of Lévy flight movement, disappeared. The longer steps of E. e. taylori movement were due to harassment by males and were therefore not part of a search strategy. The unharassed steps were part of searches for suitable host-plant and/or nectar flowers in both E. e. taylori and E. e. phaeton. These unharassed step-lengths were distributed either exponentially or as a Weibull distribution, depending upon the candidate models tested, across all the populations we studied.
Some have suggested [33,34,36] that fat-tailed step-lengths at the population level can be caused by variation in individual movement patterns. However, our results fitting step-length distributions to individuals of the two species suggest that is not the case here. Where the population-level fits indicate a single step-length distribution (i.e. E. phaeton), most of the individuals are also best fit by an unmixed distribution, while where population-level fits indicate a mixed step-length distribution (i.e. E. e. taylori), most of the individuals also express a mixture distribution. The interpretation is thus that the long-tailed distribution is due to a state-switching process, and, with the behavioural observations, we can understand that the process is driven by interactions with conspecific males. This is an interesting analogous finding to that of Hills et al. , who showed that recent interactions with the environment, such as locating a food item, affect subsequent step-lengths through memory. In this case, the environmental interaction is not the location of a food item but harassment by conspecific males.
Our results offer insight in relating statistically selected PL movement patterns to PL or Lévy flight movement or search processes when the set of models are limited to overly simple candidates. When movement steps are considered without environmental, behavioural or physiological contexts, or without considering temporal correlation, PL or bounded PL distributed step-lengths will occasionally, and perhaps often, be favoured over a simple exponential distribution.
However, our results indicate that this is an apparent emergent pattern, and thus not a behavioural paradigm or optimal process. In fact, we might expect apparent PL movement patterns in any particle, animal or otherwise, that engages in intermittent powered movement of any variety [12,26]. It remains possible that some animals genuinely engage in Lévy search strategies, but future studies must carefully investigate environmental and behavioural factors that might cause PL patterns in step-lengths to emerge before concluding that they arise from an internal optimal behavioural process.
While complex state-switching CRW models have now been used to analyse and fit animal movement for over a decade [49–52], these models can be difficult to compare with the simple step-length distributions used in PL movement analyses. Mixed-exponential distributions have been suggested as a simple and appropriate proxy for these more complex composite CRWs that can be easily compared to other probability distributions . These simple mixtures have been shown to fit much better than PL distributions . Our results strongly support the validity of this suggestion both statistically and biologically. Moreover, our results also suggest that a Weibull distribution, a more general form of an exponential which will also produce Brownian motion when used as the basis of a random walk, might be a better choice for a null step-length model against which PL models should be compared. The Weibull distribution has no more parameters than a bounded PL. Thus, convincing evidence of Lévy flight movement patterns or processes should either include some of these more complex step-length distributions as candidate models, or clearly justify biologically why these alternative mixture models or Weibull distributions are not appropriate candidates for the system being analysed.
Finally, for researchers interested in understanding the relationship between an animal's movement and its environment, the suite of new and powerful models available to predict and estimate these relationships is growing every day (e.g. [5,39,53,54]). These are better choices for practical biological inference than any models which test for PL distributions in movement step-lengths.
We recommend that both Weibull and mixed-exponential distributions be considered in future investigations of PL movement processes. This is especially true where no direct behavioural observations are available to objectively classify step-lengths. The absence of direct observation of behavioural state does not imply they do not exist nor does it justify that more complex step-length models can be excluded from any statistical analysis of a movement pattern or process.
The work was funded by a grant from SERDP awarded to Elizabeth Crone (G.A.B. and P.M.S.), and an NSERC Banting Fellowship to G.A.B.
We thank Ed Easterling for access to butterfly populations in Oregon, the Trustees of Reservations for population access in Massachusetts, and Vivian Kimball for assistance in the field. The analysis and early drafts were improved by comments and discussions with Marie Auger-Méthé, Mark Lewis, Jonathan Potts and Ulrike Schlägel.
- Received August 18, 2014.
- Accepted November 26, 2014.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.