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LETTERS
Scaling laws of marine predator search behaviour
David W. Sims
1,2
, Emily J. Southall
1
, Nicolas E. Humphries
1
, Graeme C. Hays
4
, Corey J. A. Bradshaw
5
{,
Jonathan W. Pitchford
6
, Alex James
6,7
, Mohammed Z. Ahmed
3
, Andrew S. Brierley
8
, Mark A. Hindell
9
,
David Morritt
10
, Michael K. Musyl
11
, David Righton
12
, Emily L. C. Shepard
4
, Victoria J. Wearmouth
1
, Rory P. Wilson
4
,
Matthew J. Witt
13
& Julian D. Metcalfe
12
Many free-ranging predators have to make foraging decisions with
little, if any, knowledge of present resource distribution and avail-
ability
1
. The optimal search strategy they should use to maximize
encounter rates with prey in heterogeneous natural environments
remains a largely unresolved issue in ecology
1–3
.Le
´vy walks
4
are
specialized random walks giving rise to fractal movement trajec-
tories that may represent an optimal solution for searching com-
plex landscapes
5
. However, the adaptive significance of this
putative strategy in response to natural prey distributions remains
untested
6,7
. Here we analyse over a million movement displace-
ments recorded from animal-attached electronic tags to show that
diverse marine predators—sharks, bony fishes, sea turtles and
penguins—exhibit Le
´vy-walk-like behaviour close to a theoretical
optimum
2
. Prey density distributions also display Le
´vy-like fractal
patterns, suggesting response movements by predators to prey
distributions. Simulations show that predators have higher
encounter rates when adopting Le
´vy-type foraging in natural-like
prey fields compared with purely random landscapes. This is con-
sistent with the hypothesis that observed search patterns are
adapted to observed statistical patterns of the landscape. This
may explain why Le
´vy-like behaviour seems to be widespread
among diverse organisms
3
, from microbes
8
to humans
9
, as a ‘rule’
that evolved in response to patchy resource distributions.
Predators can sometimes fine-tune their foraging by using sensory
information of resource abundance and distribution at near-distance
scales dominated by proximal clues
10
, and at very broad scales some
may have awareness of seasonal and geographical prey distribu-
tions
11
. However, across the broad range of mesoscale boundaries
(a few to hundreds of kilometres), the necessary spatial knowledge
required for successful foraging will depend largely on the search
strategy used. Over these scales some predators are more like prob-
abilistic or ‘blind’ hunters than deterministic foragers. Fully aquatic
marine vertebrates that feed on ephemeral resources like zooplank-
ton and small pelagic fish typify this type of predator because they
have sensory detection ranges limited by the seawater medium and
experience extreme variability in food supply
7,10–12
over a broad range
of spatio-temporal scales
13–15
.
Probabilistic search patterns described by a category of random-
walk models known as Le
´vy walks
4
appear to describe foraging
movements of some species
3
. These specialized random walks
have super-diffusive properties comprising ‘walk clusters’ of short
move step lengths (distance moved per unit time) with longer re-
orientation jumps between them. This pattern is repeated across
all scales, with the resultant scale-invariant clusters creating trajec-
tories with fractal patterns
3
.Le
´vy-walk move steps are drawn from a
probability distribution with a power-law tail: P(l
j
),l
j
2m
, with
1,m#3, where l
j
is the move-step length and mis the power-law
(Le
´vy) exponent (here ‘,’ means ‘distributed as’). Theoretical
studies
2,3,16
show that Le
´vy walks and Le
´vy flights (the turning
points in a Le
´vy walk
4
) across random prey distributions increase
new-patch encounter probability compared with simple brownian
motion, with an optimal search having an exponent m>2. Recent
studies
17–19
contend that Le
´vy walks or flights have been wrongly
ascribed to some species through use of incorrect methods, while
others indicate Le
´vy-like behaviour with optimal power-law expo-
nents
3,20,21
for highest-efficiency searches, supporting the hypothesis
that Le
´vy behaviour may represent an evolutionary optimal value of
the Le
´vy exponent
3,5,22
.
We hypothesized that fully aquatic (non-aerial) marine predators
should adopt a movement (search) strategy that optimises prey-
patch encounter rates, thus conferring an advantage when foraging
within naturally non-random prey distributions
13–15
. Long-term
movements of large marine predators can be recorded accurately at
fine temporal resolution (seconds) for long periods (months) using
electronic data-logging tags
23
. In the largest such analysis yet
attempted, we collated the vertical movements resulting from
recorded diving activity within the foraging range of seven large
vertebrate species that feed on patchily distributed prey (for example,
zooplankton, small fish) (see Methods). Numerous investigations
have tracked a predator’s horizontal movements but none have
studied vertical movements for which the same considerations of
‘blind’ hunting probably hold over much shorter vertical spatial
scales (tens of metres), particularly outside the well-lit near-surface
zones
24
. We analysed a total of 1,209,088 vertical move steps for 31
individual predators from seven species and found that the large-
scale structure of vertical movement was similar for the majority of
species (Fig. 1). Model fits to move-step-length frequency distribu-
tions for five species across diverse taxa (shark, teleosts, sea turtle,
penguin) closely resembled an inverse-square power law
25
with a
heavy tail of increasingly longer steps intermittently distributed
within the time series that is typical of ideal Le
´vy walks
3,4
(Fig. 1;
Supplementary Information). Le
´vy exponents derived from
1
Marine Biological Association of the United Kingdom, The Laboratory, Citadel Hill, Plymouth PL1 2PB, UK.
2
Marine Biology and Ecology Research Centre, School of Biological Sciences,
3
School of Computing, Communications and Electronics, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK.
4
Department of Biological Sciences, Institute of Environmental
Sustainability, Swansea University, Singleton Park, Swansea SA2 8PP, UK.
5
School for Environmental Research, Charles Darwin University, Darwin, Northern Territory 0909, Australia.
6
Department of Biology and York Centre for Complex Systems Analysis, University of York, York YO10 5YW, UK.
7
Department of Mathematics and Statistics, University of
Canterbury, Christchurch, New Zealand.
8
Gatty Marine Laboratory, School of Biology, University of St Andrews, Fife KY16 8LB, UK.
9
School of Zoology, University of Tasmania, Private
Bag 05, Hobart, Tasmania 7001, Australia.
10
School of Biological Sciences, Royal Holloway, University of London, Egham TW20 0EX, UK.
11
Joint Institute for Marine and Atmospheric
Research, Pelagic Fisheries Research Programme, University of Hawaii at Manoa, Kewalo Research Facility/NOAA Fisheri es, 1125-B Ala Mona Boulevard, Honolulu, Hawaii 96814,
USA.
12
Centre for Environment, Fisheries and Aquaculture Science, Lowestoft Laboratory, Pakefield Road, Lowestoft NR33 0HT, UK.
13
Centre for Ecology and Conservation, University
of Exeter in Cornwall, Tremough TR10 9EZ, UK. {Present address: Research Institute for Climate Change and Sustainability, School of Earth and Environmental Science s, University of
Adelaide, Adelaide, South Australia 5005, Australia.
Vol 451
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Publishing Group
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Le
´vy-like move-step-length frequency distributions for the five
species were maintained for individuals, and it is striking that they
were close to a theoretically optimal m<2 exponent (
mm 6s.d. 5
2.12 60.31, n524; species range: 1.34 #m#2.91) (Fig. 1; Supple-
mentary Information). Relative likelihood estimates of model fits
to the move-step-length frequency distributions for all species sup-
ported only an exponential function typifying random motion for
two species (catshark, elephant seal), confirming that Le
´vy-like pro-
cesses may not predominate within vertical search strategies in all
species (see Supplementary Information).
To test for the presence of long-term correlations that also
characterize scale-invariant Le
´vy walks
3
, we used the root-mean-
square fluctuation of the displacement, F(t), in each time series.
Uncorrelated time series arise from uncorrelated random walks for
which a50.5 for the relationship F(t),t
a
(ref. 26); in contrast,
weighted means of afor each species tested here were between 0.80
and 1.24 (
aa 6s.d. 51.08 60.17, n55), confirming the presence of
long-range correlations in diving time series across the five species
(Supplementary Information). The scaling exponent bof the sum of
the spectra against frequency in the dive time series was 0.8 in the
low-frequency regime, also consistent with long-range correlations
in scale-invariant systems
26
because b>0 when behaviour is tem-
porally uncorrelated (Supplementary Information). We considered
vertical foraging movements only in one dimension (depth) through
time (that is, the total dimensionality is two dimensional, 2D), so we
were unable to determine randomness in turning angles which would
further confirm the existence of Le
´vy-like motion
21
over the full range
of underwater movements; however, considering the data as 2D pro-
jections of three-dimensional (3D) movements presents no obstacle
to their statistical treatment. The projection of spatially homo-
geneous 3D Le
´vy movements into 2D preserves the power-law
relationship with an unchanged exponent at all length scales greater
than the minimum move step of the original 3D trajectory. This
invariance under projection does not hold for other move-step
distributions (Supplementary Information). The Le
´vy-like vertical
movements described here, therefore, reflect the more complex 3D
movements made by a range of phylogenetically distinct marine
vertebrate species, implying that Le
´vy-like walks may be a common
strategy employed by open-ocean foragers.
Le
´vy-walk-like behaviour of foragers may show mechanistic links
with natural prey fields if the search pattern emerges from the under-
lying pattern of food distribution
20
, or if the strategy evolved to
a
0
10
20
30
40
50
0 1,000 2,000 3,000 4,000 5,000
Time (min)
Move-step length (m)
0
200
400
600
800
1,000
3
9
15
21
27
33
39
45
Frequency
Move-step length (m)
4,353
log10[Move step, x (m)]
log10[Move step, x (m)]
log10[Normalized
frequency, N(x)]
–5
–0.5 0 0.5 1 1.5
–4
–3
–2
–1
0
1
m = 2.3
r2
= 0.95
b
–6
–5
–4
–3
–2
–1
0
–6
–5
–4
–3
–2
–1
0
1
–1
–1 –1 –0.5 0 0.5 1 1.5 20120123
–2 –1 0 1 20 012312
m = 2.4
r2 = 0.90
m = 1.9
r2 = 0.91
m = 1.7
r2 = 0.91
m = 2.4
r2 = 0.94
m = 2.0
r2 = 0.95
–8
–7
–6
–5
–4
–3
–2
–1
0
–7
–6
–5
–4
–3
–2
–1
0
1
2
-4
-3
-2
-1
0
c d
ge
log10[Normalized frequency, N(x)]
f
–6
–5
–4
–3
–2
–1
0
1
Figure 1
|
Le
´vy-like scaling law among diverse marine vertebrates.
a, Movement time series recorded by electronic tags were analysed to
determine the Le
´vy exponent to the heavy-tailed power-law distribution.
Left, time series of vertical move (dive) steps (n55,000) of an individual
basking shark (Cetorhinus maximus) over 3.5 days, showing an intermittent
structure of longer steps. Right, the move-step-length frequency distribution
for the same data. Inset, the normalized log–log plot of move-step frequency
versus move-step length, giving an exponent mwithin ideal Le
´vy limits
(m52.3). The Le
´vy exponent is conserved across longer temporal scales, for
example, expanding the time series for this individual to 30 days (n543,200
steps) maintains m52.3 (data not shown), indicating scale-invariance in
move-step distribution. b
–
g, Normalized log–log plots of the move-step
frequency distributions for: b, sub-adult and adult basking shark
(n5503,447 move steps); c, bigeye tuna (Thunnus obesus)(n5222,282
steps); d, Atlantic cod (Gadus morhua)(n594,314 steps); e, leatherback
turtle (Dermochelys coriacea)(n54,393 steps); and f, Magellanic penguin
(Spheniscus magellanicus)(n59,727 steps). Le
´vy exponents were
maintained at the individual level for 24 sub-adult and adult animals (see
Supplementary Information). g, Normalized log–log plot of move-step-
length frequency distribution for a 2.5-m-long, ,1-year-old basking shark
showing nonlinear form. Relative likelihood model fits to rank-frequency
plots for the five species also indicated that move-step distributions were
Le
´vy-like and not purely random (Supplementary Information).
NATURE
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28 February 2008 LETTERS
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Publishing Group
©2008
enhance foraging success in particular prey distributions. To examine
these contrasting hypotheses we analysed the structure of two differ-
ent prey fields (krill, total zooplankton) consumed by some predators
considered here (planktivorous shark, penguin). Krill (Euphausia
superba) densities occurring horizontally in a current passing a
moored echosounder
27
were measured throughout the 200m water
column at consecutive, equally spaced time intervals. Krill densities
showed extreme variance in amplitude through time, with an inter-
mittent structure of large ‘jumps’ (Fig. 2a). The frequency distri-
bution of krill density changes also closely resembled a power law
with a heavy tail, giving a Le
´vy exponent of 1.7 (Fig. 2b, c). Root-
mean-square fluctuations gave a50.9 for krill and spectral analysis
revealed low-frequency changes at b50.3 (Supplementary Infor-
mation). Similar results were found for the zooplankton time series
(m51.8 and 2.0, a50.9; Supplementary Information). Therefore,
the presence of long-range correlations and scale invariance in the
spatial changes in prey density are properties in common with
marine predator movements. The Le
´vy exponents describing the
slopes of the power-law-like distributions were also similar, as were
frequency spectra of predator movements and prey distribution. The
similarity of Le
´vy exponents and frequency spectra between predator
movements and prey distributions does not necessarily prove the
existence of a mechanistic link between a predator’s foraging move-
ment response and natural prey assemblages, but the close resemb-
lance does indicate that Le
´vy properties (describing fractal processes)
underlie both predator movements and prey distribution. Thus, for
these specific ecological cases, the exponent of the searcher may
represent an optimization to the heterogeneous prey fields dem-
onstrating fractal properties.
There are, however, two competing hypotheses to explain the
predator–prey interactions we propose: (1) animal search patterns
are adapted stochastically to their prey field structures because their
environment is so heterogeneous (predators actively search following
rules of optimality), or (2) apparently ‘optimal’ search patterns may
arise simply as a function of the underlying distribution of the prey
field (a predator’s patterns are a by-product of the prey distribution it
encounters). The results of a recent modelling study
20
support the
latter explanation by showing that scale-free foraging patterns (Le
´vy
walks) emerge from the interaction of animals with a particular
resource distribution. Likewise, field observations of primates mov-
ing between fruiting trees fit the expected pattern probably because
primates possess complex mental maps of resource location; hence,
the underlying resource landscape determines the distribution of
move steps
20
. However, this process is unlikely to account for the
move-step distributions of marine species we measured because they
have an incomplete knowledge of resource location. First, beha-
vioural kineses to prey are limited to relatively small vertical distances
in the ocean
24
, so when threshold prey densities are reached, a pred-
ator should initiate searches aimed at traversing distances exceeding
the sensory detection range
2,7,10
. Second, strict fidelity of a marine
predator to small target locations (analogous to the trees visited by
primates) will be ineffective because locations of prey such as
zooplankton, squid and shoaling fish often change rapidly and
dynamically across a range of spatio-temporal scales
7,10,14
. So our
empirical results favour the first explanation—predators feeding
on patchy, heterogeneous prey should adapt the best probabilistic
search strategy given that they are essentially ‘blind’ hunters at the
spatial and temporal scales over which they typically forage. This
conclusion regarding adaptation is strengthened by the vertical
move-step-length frequency distribution of a 2.5-m-long, ,1-year-
old basking shark that we tracked for 7 months that did not conform
to Le
´vy-like behaviour (Fig. 1g). We suggest that this striking diffe-
rence in search pattern from those of mature individuals reflects
ontogenetic behavioural development
28
, that is, juveniles learn about
the underlying structure of prey distributions as they gain foraging
experience.
Further support for the hypothesis that movement processes
are linked to prey distributions could be inferred if there were an
advantage to predators adopting Le
´vy walks in fractal landscapes
compared to other distribution types. For a particular search
strategy to evolve, it must confer an advantage in terms of higher
fitness resulting from greater efficiency in energy acquisition
29
.To
test this, we investigated the foraging success (total energy acquisi-
tion) of a Le
´vy searcher adapted to a natural-like, fractal prey field
by simulating a Le
´vy-walk predator’s vertical diving movements
within a virtual prey field defined by either a Le
´vy or random
distribution. Le
´vy searches (m52.0) reflecting marine predator
a
Time series with 4-min sample interval
0
0.5
1
1.5
2
2.5
3
3.5
0 200 400 600 800 1,000 1,200
log10[Krill density, d (g m
–2
)+1]
log10[Krill density, d (g m
–2
)+1]
log10[Krill density, d (g m
–2
)+1]
Frequency
0
100
200
300
400
500
0.25
0.45
0.65
0.85
1.05
1.25
1.45
1.65
1.85
2.05
2.25
2.45
log
10
[N(x)]
log
10
[x]
–6
–2 0 2
–5
–4
–3
–2
–1
0
1
m = 1.7
r
2
= 0.99
1,172
c
b
0.1
1
10
100
1,000
10,000
0.01 0.1 1 10 100 1,000
Number of samples ≥ d
Figure 2
|
Macroscopic properties of a prey field. a, Krill (Euphausia
superba) density d(in g m
22
)(n51,215 samples) occurring horizontally
and integrated vertically within the current flowing past a moored, upward-
looking echosounder off South Georgia. b, The move-step frequency
distribution of krill density follows a heavy-tailed power law reminiscent of
those obtained for Le
´vy-walk-like predators, and as for the predators, it
shows an exponent within Le
´vy limits and close to the optimum m<2.
c, Cumulative distribution or rank-frequency plot
17
of krill density change
showing a straight-line form consistent with power-law-like and Le
´vy-like
processes. This plot gives a Le
´vy exponent of 1.64. The relative likelihood
model fit to the rank-frequency plot also indicated that the density-change
distribution was Le
´vy-like and not purely random (see Supplementary
Information).
LETTERS NATURE
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28 February 2008
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Nature
Publishing Group
©2008
movements within Le
´vy (fractal)-distributed prey fields (LL,
denoting a Le
´vy searcher in a Le
´vy prey field) were compared with
encounter rates in ordinary, random prey fields (LR) (defined as a
prey distribution resulting from a homogeneous spatial Poisson
point process) (see Methods). Our expectation was that the
foraging success ratio LL:LR should not deviate substantially from
1.0 (zero difference) if adapting to a fractal prey field presents
no particular foraging advantage to a Le
´vy searcher. However, the
LL:LR ratio always exceeded 1.0, and LL was 14% higher on average
than LR (Table 1), which thus supports the optimality hypothesis.
We next compared random with Le
´vy searchers in fractal fields
(RL, denoting a random searcher in a Le
´vy prey field) and found that
the RL:LL ratio, by contrast, showed a negative foraging gain
(210%), whereas comparing RL with a random forager in a
random field (RR) indicated similar levels of search performance
(RL:RR >1.0; Table 1 and Supplementary Information). These
results are consistent with the hypothesis that Le
´vy-like searches
may represent an adaptation to complex prey distributions by
evolving optimal search strategies.
Our findings indicate that animals in stochastic environments
necessitating probabilistic foraging may derive benefits from adapt-
ing movements described by Le
´vy processes. Le
´vy models express
behavioural minimalism
3
, so not all movements made by marine
vertebrates and other animals will be associated with optimal
foraging (for example, resting, breeding and migration) and, in addi-
tion, it is unlikely that animals search with Le
´vy-like motion at all
times, especially if, for some species, foraging decisions are predomi-
nantly deterministic within stable environments. However, evidence
that Le
´vy-walk search patterns apply to a diverse range of taxa
3,8,9,21
,
together with our results, suggest that foragers are adapted to optimal
behaviour in complex landscapes. Hence, Le
´vy-like walks may be
useful for developing more realistic models of how animals redistri-
bute themselves in response to shifting resources as a consequence of
climate change, fisheries extractions and other habitat modifica-
tions
30
. Such general and simple laws of movement as optimal Le
´vy
walks could prove useful in robotics—for example, in an algorithm
controlling the movements of autonomous robots designed to
sample optimally in hostile and heterogeneous environments such
as the deep sea, active volcanoes or on other planets.
METHODS SUMMARY
Electronic tagging. Pressure (depth)-sensitive data-logging tags were attached
to basking sharks Cetorhinus maximus (n56 individuals) and small spotted
catshark Scyliorhinus canicula (n53) in the northeast Atlantic Ocean, bigeye
tuna Thunnus obesus (n53) in the North Pacific near Hawaii, Atlantic cod
Gadus morhua (n55) in the North Sea, leatherback turtles Dermochelys
coriacea (n54) in the Atlantic Ocean, Magellanic penguins Spheniscus magella-
nicus (n57) off Patagonia, Argentina, and southern elephant seals Mirounga
leonina (n53) in the Pacific sector of the Southern Ocean. Full details of
deployments, animal body sizes, tag types and data sources are given in
Supplementary Table 1.
Prey sampling. Antarctic krill (Euphausia superba) in the top 200 m were
detected at 4-min intervals within the current flowing past a moored, upward-
looking data-logging echosounder at South Georgia, South Atlantic Ocean
27
.
Logged data were processed to provide a prey-field time series of horizontal
changes in krill density at a point location integrated vertically in the water
column, and scaled to account for variable current flow over time. Two
zooplankton time series were also analysed (see Supplementary Information).
Simulation program. The purpose of simulating searches was to test the hypo-
thesis that foraging success (biomass consumed per distance moved) by optimal
Le
´vy walkers (m
opt
52.0) in fractal (natural) prey distributions exceeded prey
acquisition rates within random prey fields. We developed a simulation where
vertical trajectories (y, time) of Le
´vy foragers were routed through seascapes with
heterogeneous prey patches distributed according to Le
´vy (describing fractal
processes) or random distributions. This simulated a predator searching verti-
cally for patchy resources.
Full Methods and any associated references are available in the online version of
the paper at www.nature.com/nature.
Received 17 October; accepted 29 November 2007.
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19. Edwards, A. M. et al. Revisiting Le
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20. Boyer, D. et al. Scale-free foraging by primates emerges from their interaction with
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22. Bartumeus, F. Le
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24. Myers, A. E. & Hays, G. C. Do leatherback turtles Dermochelys coriacea forage
during the breeding season? A combination of data-logging devices provide new
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25. Bradshaw, C. J. A. & Sims, D. W. Solutions to problems associated with
identifying Le
´vy walk patterns in animal movement data. J. Anim. Ecol.
(submitted).
26. Viswanathan, G. M. et al. Le
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27. Brierley, A. S. et al. Use of moored acoustic instruments to measure short-term
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Table 1
|
Comparison of foraging success for simulated searchers
Ratio Mean foraging success (% difference) Standarderror of the mean
LL:LR 14.47 2.49
RL:RR 20.52 0.73
RL:LL 210.89 1.32
See text for explanation of the ratio and Supplementary Information for further details of
simulation results. The mean foraging success ratio was calculated from ten replicate simulation
sets, with each replicate comprising three runs of each forager type per prey field type, with each
run estimating foraging success (prey encountered per distance moved) for each of 100,000
foragers. Hence, each pair of sets making up a replicate summarized searching by 600,000
foragers.
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28. Field, I. C., Bradshaw, C. J. A., Burton, H. R., Sumner, M. D. & Hindell, M. A.
Resource partitioning through oceanic segregation of foraging juvenile southern
elephant seals (Mirounga leonina). Oecologia 142, 127
–
135 (2005).
29. Perry, G. & Pianka, E. R. Animal foraging: past, present and future. Trends Ecol. Evol.
12, 360
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364 (1997).
30. McMahon, C. R. & Hays, G. C. Thermal niche, large-scale movements and
implications of climate change for a critically endangered marine vertebrate. Glob.
Change Biol. 12, 1330
–
1338 (2006).
Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements This research was facilitated through the European Tracking
of Predators in the Atlantic (EUTOPIA) programme in the European Census of
Marine Life (EuroCoML). Funding was provided by the UK Natural Environment
Research Council (NERC) grant-in-aid to the Marine Biological Association of the
UK (MBA), the NERC ‘Oceans 2025’ Strategic Research Programme (Theme 6
Science for Sustainable Marine Resources), UK Defra, The Royal Society and the
Fisheries Society of the British Isles. D.W.S. thanks G. Budd, P. Harris, N. Hutchinson
and D. Uren for field assistance. G.C.H. thanks Ocean Spirits Incorporated,
J. Houghton and A. Myers for logistical help in the field. A.S.B. thanks E. Murphy,
M. Brandon, R. Saunders, D. Bone and P. Enderlein for their contributions to
mooring sampling at South Georgia. NERC Plymouth Marine Laboratory provided
L4 zooplankton data. This research complied with all animal welfare laws of the
countries or sovereign territories in which it was conducted. D.W.S. was supported
by a NERC-funded MBA Research Fellowship.
Author Contributions D.W.S. conceived and planned the study, led the data
analysis and wrote the manuscript. All co-authors contributed to subsequent
drafts. Field data of animal movements and/or prey distributions were collected by
D.W.S., E.J.S., J.D.M., G.C.H., C.J.A.B., A.S.B., M.A.H., D.M., M.K.M., D.R., V.J.W.
and R.P.W. The simulation model was conceived by N.E.H. and D.W.S. with N.E.H.
writing the programming code. J.W.P. and A.J. were responsible for analysis of
projections of 3D Le
´vy movements, M.Z.A. and E.L.C.S. coded the power spectrum
analysis, C.J.A.B. completed the relative likelihood modelling, and G.C.H. and
M.J.W. provided additional data analysis.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. Correspondence and requests for materials should be
addressed to D.W.S. (dws@mba.ac.uk).
LETTERS NATURE
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METHODS
Movement analysis. For predatory fish movements (sharks, tunas, cod), the
change in selected water-column depth between consecutive time intervals,
u(t), was calculated to derive a time series of vertical displacement (move) steps
for each individual. For air-breathers (turtles, penguins, seals) the change in
chosen maximum depth between successive dives in single foraging trips, u(t),
was used as a proxy for searching to remove the anomalous effect on the move-
step-length frequency distribution caused by necessity of leaving and returning
to the surface to breathe air. Time series of chosen depth changes represent
short-term search decisions of predators extending across long temporal
scales, enabling robust analysis of macroscopic properties
3,17
. Each entry in the
time series u(t) is a vertical step in metres, with time measured in minutes
for fish (t51, 2…t
max
) and relative elapsed time for air-breathers (t
1
5d
1
,
t
2
5d
2
…t
max
5d
x
where dis a dive number in the series). Elapsed time was
used because breathing-corrected steps traversed boundaries of fixed time
intervals and was justified because animals dived continuously and variation
in dive duration was relatively small within individuals. We calculated
26
the
net displacement y(t) of each time series u(t) defined by the running sum
y(t):Pt
i~1u(i)and determined the root-mean-square fluctuation of the
displacement F(t):ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(Dy(t))2
{Dy(t)
hi
2
qwhere Dy(t):y(t0zt){y(t0).
For power spectrum analysis
31
of leatherback turtle, krill and zooplankton time
series, we calculated the sum of power spectra S(f) plotted against the period 1/f.
Histogram plots of power laws were calculated using logarithmically increasing
move-step-length bins, with each bin width k(for example, 1, 2, 3…) increasing
by 2
k
(for example, 2, 4, 8…). The frequency per logarithmic bin was normalized
by dividing by total frequency Nand bin width to obtain the probability density
of each bin
17
. Relative likelihoods of models fitted to rank-frequency plots were
compared using Akaike’s and Bayesian Information Criterion weights (see
Supplementary Information).
Optimal foraging simulation. A prey patch generator (see below) is used to
create nunique, randomly generated variable density patches which are then
‘pasted’ into a 2D seascape following either a Le
´vy or random distribution. The
initial patch is positioned using a uniform random number generator. For a
random patch distribution, the relative position of the second patch (dx,dy)is
again calculated using a uniform distribution. For a Le
´vy patch distribution the
direct distance to the second patch is calculated using a Le
´vy random number
generator (see below) with a uniform distribution giving the angle between the
two patches. Patches are thereafter positioned iteratively, with the position of the
next patch based on the position of the current patch until the desired number of
patches has been created. The seascape is treated as a torus; if positions exceed the
preset dimensions of 2,500 35,000 it wraps around from top to bottom and left
to right. Spacing patches by distances (step lengths) drawn from a Le
´vy distri-
bution yielded an underlying pattern congruent with the spatial density of
patches (see Supplementary Information).
A single foraging run though the prey field starts at a random depth on one
side (x50) and proceeds to the other side (x55,000) in a series of horizontal
steps of fixed distance (in this case 1, giving 5,000 steps per foraging run). The
vertical displacement at each time step is generated from either a uniform
distribution or from the Le
´vy random number generator. The resulting diagonal
path is traversed by calculating an interpolated movement such that every cell
along the path is visited by the forager. Prey biomass (density values) encoun-
tered in each grid cell are accumulated to give total biomass consumed for the
foraging run.
Prey patch generator. The purpose of the prey patch generator is simply to
generate a quasi-realistic patch rather than a square block of uniform density.
A prey patch (for example, zooplankton patch) of a specified area in a 2D grid of
size 100 3100 is created. The patch is created by building up a number of
superimposed random walks over this grid until the specified number of grid
cells has been occupied. The length of each random walk, L, is computed as the
square root of the required patch area and each walk begins at the approximate
centre of the grid (50, 50). This first cell is given the value L, the second L21,
until the final cell has value 1. Grid cells reached that are already occupied are not
renumbered; however, the step is still counted so that some random walks will
occupy no new grid cells and cells at the edge of the growing patch will always
have low density values. Random walks are repeated, building up the patch, until
the required number of grid cells have been filled (that is, the required patch area
has been achieved). Each completed patch is then pasted into the simulation
seascape as described above.
Le
´vy random number generator. The Le
´vy random number generator returns
an integer value between 1 and a specified maximum value, with a probability
density that approximates to the power distribution P(n),n
2m
,where Prepre-
sents the probability of the value nbeing returned by the random number
generator and m52.0 (to reflect the theoretical optimal Le
´vy exponent
5
). An
integer array is generated, with each element taking a value between 1 and n
(where nis the specified maximum value) such that the proportion of array
elements populated by any given value is equal to the probability of that value
arising from the power function. For example, if the generator has to return a
maximum value of 1,000 then the array is created such that the value 1 will
occupy 1,000,000 array elements, the value 2 will occupy 250,000 elements,
and so on, until the value 1,000, which will occupy 1 array element:
P(1,000) 50.000001. When the generator is called to provide a uniform random
number, a uniform random number generator is used to select an array element
and the value of that element is returned as the Le
´vy random number. In the
simulation, a maximum of 2,500 (the modelled depth of the sea, in metres) is
used.
Theoretically, the power function should allow some rare large values to
occur. Such steps are meaningless in the context of this study of vertical move-
ments; they would take the virtual forager outside the limits of the seascape or, in
the case of air breathers, beyond their physiological diving limits. When large
values occur in the simulations, they are simply wrapped around (as with the
prey patch generator). This method allows fast random number generation that
can be done on PC platforms, but a necessary consequence is that the lack of
some rare large value causes the generator to return values in a distribution
nearer to m51.95 than to m52.0.
31. Chatfield, C. The Analysis of Time Series 6th edn (Chapman & Hall, London, 1996).
doi:10.1038/nature06518
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Publishing Group
©2008
