Content uploaded by Sergei Petrovskii

Author content

All content in this area was uploaded by Sergei Petrovskii on Apr 05, 2019

Content may be subject to copyright.

DOI: 10.1126/science.1215747

, 918 (2012);335 Science , et al.Vincent A. A. Jansen

Movement and Environmental Complexity''

Comment on ''Lévy Walks Evolve Through Interaction Between

This copy is for your personal, non-commercial use only.

clicking here.colleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others

here.following the guidelines can be obtained byPermission to republish or repurpose articles or portions of articles

): February 27, 2012 www.sciencemag.org (this infomation is current as of

The following resources related to this article are available online at

http://www.sciencemag.org/content/335/6071/918.3.full.html

version of this article at: including high-resolution figures, can be found in the onlineUpdated information and services,

http://www.sciencemag.org/content/suppl/2012/02/22/335.6071.918-c.DC1.html

can be found at: Supporting Online Material

http://www.sciencemag.org/content/335/6071/918.3.full.html#related

found at: can berelated to this article A list of selected additional articles on the Science Web sites

http://www.sciencemag.org/content/335/6071/918.3.full.html#ref-list-1

, 5 of which can be accessed free:cites 8 articlesThis article

http://www.sciencemag.org/content/335/6071/918.3.full.html#related-urls

1 articles hosted by HighWire Press; see:cited by This article has been

http://www.sciencemag.org/cgi/collection/tech_comment

Technical Comments http://www.sciencemag.org/cgi/collection/ecology

Ecology subject collections:This article appears in the following

registered trademark of AAAS. is aScience2012 by the American Association for the Advancement of Science; all rights reserved. The title CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005.

(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

on February 27, 2012www.sciencemag.orgDownloaded from

Comment on “Lévy Walks Evolve

Through Interaction Between Movement

and Environmental Complexity”

Vincent A. A. Jansen,

1

*Alla Mashanova,

1

Sergei Petrovskii

2

de Jager et al. (Reports, 24 June 2011, p. 1551) concluded that mussels Lévy walk. We confronted

a larger model set with these data and found that mussels do not Lévy walk: Their movement is

best described by a composite Brownian walk. This shows how model selection based on an

impoverished set of candidate models can lead to incorrect inferences.

ALévy walk is a form of movement in

which small steps are interspersed with

very long ones, in such a manner that

the step length distribution follows a power law.

Movement characterized by a Lévy walk has no

characteristic scale, and dispersal is superdiffu-

sive so that individuals can cover distance much

quicker than in standard diffusion models. de Jager

et al.(1) studied the movements of individual

mussels and concluded that mussels move

according to a Lévy walk.

The argument of (1) is based on model se-

lection, a statistical methodology that compares

a number of models—in this case, different step

length distributions—and selects the model that

describes the data best as the most likely model to

explain the data (2). This methodology is used to

infer types of movements of animals (3) and has

led to a number of studies that claim Lévy walks

are often encountered in the movement of ani-

mals. The methodology in (1) contrasts a power-

law distribution, which is indicative of a Lévy

walk, with an exponential distribution, which

indicates a simple random walk. If one has to

choose between these alternatives, the power-law

distribution gives the best description. However,

if a wider set of alternatives is considered, this

conclusion does not follow.

Heterogeneity in individual movement be-

havior can create the impression of a power law

(4–6). Mussels’movement is heterogeneous as

they switch between moving very little or not at

all, and moving much farther (1,7). If mussels

switch between different modes, and in each mode

display Brownian motion, this suggests the use of

a composite Brownian walk, which describes the

movement as a sum of weighted exponential dis-

tributions. We confronted this plausible model

with the mussel movement data (8).

Visual inspection of the data shows that the

cumulative distribution of step lengths has a humped

pattern that is indicative of a sum of exponentials

(Fig. 1A). We applied a model selection pro-

cedure based on the Akaike information criterion

(AIC) (2,3). We compared six different step

length distributions: an exponential distribution,

a power law, a truncated power law, and three

hyperexponential distributions (a sum of two,

three, or four exponentials to describe composite

Brownian walks). We did this for the data trun-

cated as in (1) (Fig. 1A) as well as all the full,

untruncated data set (Fig. 1B). In both cases, we

found that the composite Brownian walk con-

sisting of the sum of three exponentials was the

best model (Fig. 1 and Table 1). This convinc-

ingly shows that the mussels described in (1)do

not do a Lévy walk. Only when we did not take

the composite Brownian walk models into

account did the truncated power law model

perform best and could we reproduce the result

in (1).

Mussel movement is best described by a

composite Brownian walk with three modes of

movement with different characteristic scales be-

tween which the mussels switch. The mean move-

ment in these modes is robust to truncation of the

data set, in contrast to the parameters of the power

law, which were sensitive to truncation [Table 1;

TECHNICAL COMMENT

1

School of Biological Sciences, Royal Holloway, University of

London,SurreyTW200EX,UK.

2

Department of Mathematics,

University of Leicester, Leicester LE1 7RH, UK.

*To whom correspondence should be addressed. E-mail:

vincent.jansen@rhul.ac.uk

Fig. 1. The step length

distribution for mussel

movement [as in (10)]

and curves depicting some

of the models. The circles

represent the inverse cu-

mulative frequency of step

lengths, The curves repre-

sent Brownian motion

(blue), a truncated power

law (red), and a composite

Brownian walk consisting

of a mixture of three ex-

ponentials (blue-green). (A)

Data as truncated in Fig.

1in(1,10) (2029 steps).

(B) The full untruncated

data set (3584 steps).

24 FEBRUARY 2012 VOL 335 SCIENCE www.sciencemag.org918-c

on February 27, 2012www.sciencemag.orgDownloaded from

also see supporting online material (SOM)]. This

analysis does not tell us what these modes are,but

we speculate that it relates to the stop-move

behavior that mussels show, even in homoge-

neous environments (1). We speculate that the

mode with the smallest average movement

(~0.4 mm) is related to nonmovement, combined

with observational error. The next mode (average

movement ~1.5 mm) is related to mussels moving

their shells but not displacing, and the mode with

the largest movements (on average 14 mm, about the

size of a small mussel) is related to actual dis-

placement. This suggests that in a homogeneous

environment, mussels are mostly stationary, and

if they move, they either wobble or move about

randomly. Indeed, if we remove movements smaller

than half the size of a small mussel (7.5 mm), the

remaining data points are best described by

Brownian motion. This shows that mussel move-

ment is not scale invariant and not superdiffusive.

de Jager et al.’sanalysis(1) does show that

mussels do not perform a simple random walk

and that they intersperse relatively long displace-

ments with virtually no displacement. However,

one should not infer from that analysis that the

movement distribution therefore follows a power

law or that mussels move according to a Lévy

walk, and there is no need to suggest that mussels

must possess some form of memory to produce a

power law–like distribution (9). Having included

the option of a composite Brownian walk, which

was discussed in (1) but not included in the set of

models tested, one finds that this describes

mussels’movement extremely well.

Our analysis illustrates why one has to be

cautious with inferring that animals move accord-

ing to a Lévy walk based on too narrow a set of

candidate models: If one has to choose between a

powerlawandBrownianmotion,oftenthepower

law is best, but this could simply reflect the

absence of a better model. To make defensible

inferences about animal movement, model selec-

tion should start with a set of carefully chosen

models based on biologically relevant alterna-

tives (2). Heterogeneous random movement often

provides such an alternative and has the addition-

al advantage that it can suggest a simple mech-

anism for the observed behavior.

References and Notes

1. M. de Jager, F. J. Weissing, P. M. J. Herman, B. A. Nolet,

J. van de Koppel, Science 332, 1551 (2011).

2. K. P. Burnham, D. R. Anderson, Model Selection and

Multimodel Inference: A Practical Information-Theoretic

Approach (Springer-Verlag, New York, ed. 2, 2002).

3. A. M. Edwards et al., Nature 449, 1044 (2007).

4. S. Benhamou, Ecology 88, 1962 (2007).

5. S. V. Petrovskii, A. Y. Morozov, Am. Nat. 173, 278

(2009).

6. S. Petrovskii, A. Mashanova, V. A. A. Jansen, Proc. Natl.

Acad. Sci. U.S.A. 108, 8704 (2011).

7. J. van de Koppel et al., Science 322, 739 (2008).

8. We found that the results published in (1) were

based on a corrupted data set and that there were

errors in the statistical analysis. [For details, see our

SOM and the correction to the de Jager paper (10).]

Here, we analyzed a corrected and untruncated data

set provided to us by M. de Jager on 20 October 2011.

This data set has 3584 data points, of which 2029

remain after truncation. Since doing our analysis, an

amended figure has been published (10), which

appears to be based on ~7000 data points

after truncation.

9. D. Grünbaum, Science 332, 1514 (2011).

10. M. de Jager, F. J. Weissing, P. M. J. Herman, B. A. Nolet,

J. van de Koppel Science 334, 1641 (2011).

Acknowledgments: We thank M. de Jager for supplying

the data to do this analysis and the authors of (1)for

their constructive comments. This work was funded by

Biotechnology and Biological Sciences Research Council

Grant BB/G007934/1 (to V.A.A.J.) and Leverhulme Trust

Grant F/00 568/X (to S.P.).

Supporting Online Material

www.sciencemag.org/cgi/content/full/335/6071/918-c/DC1

Materials and Methods

SOM Text

References

25 October 2011; accepted 13 January 2012

10.1126/science.1215747

Table 1. Model parameters and Akaike weights. The maximum likelihood parameter

estimates, log maximum likelihoods (ML), AIC values, and Akaike weights are

calculated (for details, see SOM) for the datashowninFig.1,AandB.TheAkaike

weights without the composite Brownian walks are given in brackets. We analyzed the

full data set (*) with x

min

= 0.02236 mm, and the data set truncated as in (1)(†)with

x

min

= 0.21095 mm. For x

max

, the longest observed step length (103.9mm) was

used. The mix of four exponentials is not the best model according to the AIC

weights. It gives a marginally, but not significantly, better fit and is overfitted.

Models Formula Parameters* Parameters†ML AIC Weight

Exponential

(Brownian

motion)

P(X=x)=le−l(x−xmin)l= 1.133 l= 0.770 –3136.89*

–2558.67†

6275.78*

5119.37†

0 (0)*

0(0)†

Power law

(Lévy walk)

P(X=x)= m−1

x1−m

min

x−mm= 1.397 m= 1.975 –2290.10*

–1002.32†

4582.20*

2006.64†

0(0)*

0 (0.006)†

Truncated

power law

(Lévy walk)

P(X=x)= m−1

x1−m

min

−x1−m

max

x−mm= 1.320 m= 1.960 –2119.55*

–997.29†

4241.10*

1996.58†

0(1)*

0 (0.994)†

Mix of two

exponentials

(Composite

Brownian walk)

P(X=x)=X

i=1

2

pilie−li(x−xmin)

with X

i=1

2

pi=1

p = 0.073,

l

1

= 0.122,

l

2

= 3.238

p = 0.127,

l

1

= 0.123,

l

2

= 3.275

–906.15*

–1022.44†

1818.31*

2050.87†

0*

0†

Mix of three

exponentials

(Composite

Brownian walk)

P(X=x)=X

i=1

3

pilie−li(x−xmin)

with X

i=1

3

pi=1

p

1

= 0.034,

p

2

= 0.099,

l

1

= 0.069,

l

2

= 0.652,

l

3

= 3.613

p

1

= 0.063,

p

2

= 0.210,

l

1

= 0.072,

l

2

= 0.832,

l

3

= 4.309

–861.55*

–966.70†

1733.11*

1943.40 †

0.881*

0.873†

Mix of four

exponentials

(Composite

Brownian walk)

P(X=x)=X

i=1

4

pilie−li(x−xmin)

with X

i=1

4

pi=1

p

1

= 0.014,

p

2

=0.034,

p

3

= 0.085,

l

1

= 0.656,

l

2

= 0.069,

l

3

= 0.652

l

4

= 3.613

p

1

= 0.017,

p

2

= 0.060,

p

3

= 0.202,

l

1

= 0.377,

l

2

= 0.070,

l

3

= 0.902,

l

4

= 4.345

–861.55*

–966.63†

1737.11*

1947.26†

0.119*

0.127†

www.sciencemag.org SCIENCE VOL 335 24 FEBRUARY 2012 918-c

TECHNICAL COMMENT

on February 27, 2012www.sciencemag.orgDownloaded from