^{1}

^{*}

^{2}

^{3}

^{4}

^{5}

^{4}

^{6}

The authors have declared that no competing interests exist.

Wrote the paper: AME MPF GAB IDJ. Conceived the analysis: AME MPF GAB IDJ. Contributed to understanding: AME MPF GAB IDJ. Obtained and analysed data: AME. Derived mathematics: AME MPF.

Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions.

We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (

We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.

Technological advances are revealing new insights regarding animal movements in marine ecosystems

One approach to analysing movement data is in the context of Lévy flights and Lévy walks. Lévy flights are random walks for which each movement step is drawn from a probability distribution that has a heavy power-law tail

The first step to identify Lévy movement patterns involves correctly testing whether the movement data are consistent with coming from a distribution with a heavy power-law tail (here, ‘heavy’ means that the distribution has infinite variance). This testing has long been done using regression-based techniques, though these have been shown to be inaccurate and problematic

Recent work

Given the aforementioned results, it is prudent to verify that the techniques applied in related works are valid. Here we investigate the methods used in recent studies concerning movements of marine predators

In

Both studies used likelihood methods to analyse data and reach conclusions. However, we demonstrate three issues with the likelihood calculations; each applies to one or both studies. For clarity, we focus on each study in turn.

Using correct likelihood methods we first re-analyse an example data set from

Issue one is that likelihood was calculated in

The results of our re-analysis of the bigeye tuna data contradict the original conclusions for those data. The problems identified here with the original methods of

We then describe some methodological issues of

We also discuss some aspects of the methods in another study

The issues we demonstrate reinforce that likelihood, as with all methods in ecology, must be used properly, and that claims of Lévy movements by animals do not always hold up to scrutiny. The prevalence of important methodological errors in high-profile papers that test for Lévy movement patterns is problematic, leading to incorrect biological conclusions. This negatively impacts the general field of movement ecology, and could have undesirable consequences if conclusions from such studies influence management decisions concerning marine ecosystems.

All computations used R version 2.9.2 or later

In

We first re-analyse the bigeye tuna (

(A) Logarithmic axes. Black circles are the 29,900 data points, as shown in Supplementary Figure 1(h) of

To compare models in

However, the resulting

Note that two further methods (involving root-mean square fluctuations and power-spectrum analysis) were used in

First, we compare the support for four models using likelihood functions. Full R code for these calculations is given in the Supporting Information (

The Lévy flight hypothesis is that the distribution of movements has a power-law tail with

The bounded versions of the two distributions are tested here due to previous lack of support for the unbounded power-law model

We use the unique likelihood functions of the respective probability distributions to find the maximum likelihood estimates for the parameters, which are used to plot the distributions and compute standard Akaike weights

The distributions shown in

From the maximum likelihood estimates, we calculate standard Akaike weights

Method | Power-law model | Exponential model | Quadratic model |

0 | 1 | – | |

0.769 | 0.231 | – | |

>0.999 | <0.001 | – | |

>0.999 | <0.001 | – | |

<0.0001 | <0.0001 | ∼1.000 | |

0 | 1 | – |

Our result contradicts the Akaike weights calculated in

Note that Methods

For the rank/frequency method (

However, this log-likelihood calculation is based on the standard assumption of Gaussian errors when fitting a straight line. Since

Calculations from the data give

Thus, 29,016.7 is not the log-likelihood of the exponential

The above regression approach is fitting a model for which the associated probability density function is

Graphically, this can be seen in

(A) Functions start from the value

So the reported log-likelihood from the incorrect regression method (reproduced above) relates to a function that is not an exponential distribution. The estimated value of

The weights for the power-law and exponential models were not directly compared for the rank/frequency method

However, this model also corresponds to an invalid probability density function. Similar calculations to those described for (7) give the resulting probability density function

The curves given by (7) and (8) (and (9) which is defined below) are plotted in

We agree that the quadratic model “has no particular statistical or biological justification” (page 4 of Supplementary Information of

Had the quadratic model been a valid model (i.e., a properly normalised non-negative probability density function), and been justified as an intermediate model between the power law and exponential models, then the support found for this model should also have implied no support for the Lévy hypothesis (because it is an intermediate model). However,

Issues one and two also apply to the power-law model. Setting

To be the normalised power-law probability density function (1) requires

By attempting to reproduce the original results we realised the regression intercept parameter,

Issues one and two also apply to the binning methods (

To conclude that Lévy walks evolve through interaction between movement and environmental complexity

The unnecessary specification of a plotting method when using likelihood suggests that some of the aforementioned problems may again be applicable. Ref.

Examination of the detailed Supporting Online Material of

For the bounded power-law (PLB) model (3), only discrete values of the exponent

Issue one occurs – AIC calculations were again based on linear fits of models. This is because AIC was calculated in R using the command

A Rayleigh distribution was also analysed in the R code from

In a recent Erratum

We agree that the truncated Lévy walk model is indeed more supported by the data than an exponential model (overwhelmingly so, given our calculated respective Akaike weights of 1 and 0). However, as emphasised by

We therefore conduct goodness-of-fit tests on the corrected mussel data set. Our results decisively reject (

In February 2012, a Technical Comment

In their Response

However, their

In

Ref.

So

Related to this,

The issues we have highlighted are not solely confined to work whose conclusions support the Lévy idea, or to marine ecology. Recently,

We have identified three methodological issues that each occurred in one or more recent studies. The studies made similar conclusions regarding animal movements. Likelihood was calculated incorrectly in

When applying proper likelihood methods to an example data set from

Although we found the power-law distribution to have no support compared to the exponential for the bigeye tuna data set of

With regard to the exponential distribution, there seems to be a misunderstanding concerning Brownian motion. We previously

The above examples are correct and consistent with each other. The Exp model (2) represents a simple hypothesis. It gives rise to Brownian motion, as does the PL model (1) with

We explained a concern regarding the estimation of lower and upper bounds of the tested distributions. This raises a fundamental issue that the whole idea of Lévy flights is only concerned with the

As noted in the

One solution to the aforementioned problems is in the framework of mechanistic state-space models

For example, the use of state-space models to analyse location data from satellite transmitters fitted to grey seals revealed that the seals focussed foraging efforts on a smaller fraction of the continental shelf area than was previously thought

Given our findings, we caution against the idea

Here we briefly discuss Bayesian weights, give the derivations for

For method

Also, for the small sample

As outlined in the main text, movement steps,

To derive (7), first note that

We now derive the probability function

So a quadratic model was fitted to the data plotted on the rank/frequency plots with

Using the fact that

Substituting into (31) results in

To see if this is indeed somehow intermediate between power-law and exponential distributions, we now cast it in terms of a power-law term multiplied by a (complicated) exponential term and then multiplied by a

The normalisation condition can be most simply checked by using the fact that

Examination of the detailed Supporting Online Material of

Load in the data of step lengths and sort into ascending order.

First consider the bounded power-law (PLB) model, as given in (3).

Fix the lower bound

Create a vector of values of the exponent

Set a value of the upper bound

For each value of

Find the value of

Calculate the fitted inverse cumulative frequency distribution (evaluated for each step in the data set) using the values of

Repeat (c)-(f) for incrementally increasing values of

Each value of

Calculate the maximum likelihood estimate for

Calculate the maximum likelihood estimate for

Calculate an AIC value for each model. The AIC for the PLB model, for example, was calculated in R using the command

where

Calculate Akaike weights to compare models. This was done using the following R code, where

Comparisons were also made using G-statistics and the sum of squared differences between the fitted distributions and the observed distribution.

We now highlight some problems with the above methods, referencing by step number.

2(b). Only testing discrete values of

2(e). Rather than find the closest value of

2(h). Selecting the

The equation in the code incorrectly assumed the exponential distribution to reach 0, but if the power-law distributions are assumed to start at

The AIC calculation is based on linear fits of models – this is Issue one discussed above with respect to

Even if the above issues did not hold, the code for the Akaike weights is incorrect (e.g. see

The use of the additional methods involving G-statistics and sums of squares is not justified. In Issue one,

Some of the above problems (and others) were independently raised in

Regarding the above Step 2(h) of the methods of

Given a data set

For

On page 39 of the Supplementary Information of

To test whether the corrected mussels data set from

Note that although AME was thanked for “comments and suggestions” in

(R)

(R)

(R)

(R)

(R)

(TXT)

(TXT)

We thank David Sims for sharing data with us and for interesting discussions, and Monique de Jager and Johan van Koppel for providing data, interesting discussions and making code publicly available. We also thank Ian Perry, Nick Watkins and Rowan Haigh for useful conversations. We thank the three reviewers for their insightful reviews, as well as Marie Auger-Méthé for her detailed independent review, all of which have helped improve this work.