SUPPLEMENTARY INFORMATION

Transcription

1 Supplementary Methods & Results. Model fitting of frequency distributions We tested the robustness of power law model fits to the move step frequency distributions for the seven species using two approaches: () comparison of the power law and exponential model fits to logarithmically binned move-step frequency distribution data, and (2) model fits to rank-frequency plot data. Power law versus exponential models. Support for a power-law fit to the observed data indicates move steps are best described by Lévy-like processes, whereas strong support for an exponential fit suggests where Brownian (random) motion probably dominates at the long-term limit. To determine the relative support of the power-law model versus an exponential model, we fitted a linear model to log N(x) ~ log x (power) and to log N(x) ~ x (exponential). We used an estimate of Kullback-Leibler (K-L) information loss to assign relative strengths of evidence to the different competing models, that is, Akaike s information criterion corrected for small sample sizes (AIC c ). This index of model parsimony identifies the relative evidence of model(s) from a set of candidate models. The relative likelihoods of candidate models were calculated using AIC c weights, with the weight (waic c ) of any particular model varying from (no support) to (complete support) relative to the entire model set. Model goodness-of-fit was assessed by calculating the % deviance explained (%DE) relative to the null (intercept) model. Models were run on the normalised average movestep values over all individuals (Supplementary Table 3) and with all individuals pooled (Supplementary Table 4). To account for variation among individuals for each species considered, we also fitted generalized linear mixed-effect models (GLMM) of the same form to the data using the lmer function (lme4 library) in the R Package 2. The term individual was coded as a random factor, and we used a Gaussian error distribution and the identity link function (Supplementary Table 5).

2 The results (Supplementary Tables 3-5) for all datasets (individual values averaged, individuals pooled, and GLMM) confirm that the power-law model to observed move-step frequency data is strongly supported for basking shark (Cetorhinus maximus), bigeye tuna (Thunnus obesus), cod (Gadus morhua), leatherback turtle (Dermochelys coriacea) and Magellanic penguin (Spheniscus magellanicus). In contrast, model fits for southern elephant seal (Mirounga leonina) data were best supported by the exponential model. This suggests the foraging diving movements of this marine mammal were not described purely by Lévy-like processes and that other movement types predominated, such as an ordinary random walk. Model fits to the small spotted catshark (Scyliorhinus canicula) logarithmically binned data supported the power law over the exponential model, however rank-frequency plotting of only move steps greater than the body length of the animal indicated the exponential model to be relatively better (see below). This suggests that the presence of power-law like patterns may be marginal for some species. Cumulative distributions. The second method we used to investigate the underlying form of the move step distributions of the seven species studied was to construct cumulative distribution histograms or rank-frequency plots 3,4. A rank-frequency plot in the context of this study displays a cumulative distribution of the number of move step lengths that are greater than or equal to any move step, s, each plotted against the respective move step length. This plotting method has the potential advantage of avoiding some histogram binning issues when determining the Lévy exponent 4 and takes account of all the data points during line fitting 3. However, a little realised weakness of using the rank-frequency plot method with animal movement data when testing for the presence of Lévy processes, is the assumption that a power law of any exponent will yield a straight line on a log-log plot of rank frequency versus move step length. Simulation studies show that when μ 2, curvilinear plots can predominate 4,5. Similarly, tests of empirical data identify strong putative power law forms in the heavy 2

3 tails of curvilinear rank-frequency plots 6 suggesting it should not be assumed that curvilinear rank-frequency plots do not possess Lévy-like properties. In the case of simulated data, this curvilinearity arises due to the realistic imposition of a maximum step length to the steps drawn from the Lévy power law. The maximum step length is realistic and an unavoidable feature of real animal movement data because it reflects physiological limits, such as the maximum swimming speed of an animal, which in turn determines how far it can move per unit time (Lévy walk) or between turns (Lévy flight). This is potentially an important problem for air-breathing species in particular because extremely long move step lengths are constrained by maximum diving depths and durations. For example, a species which regularly pushes itself to its physiological diving limits (e.g., southern elephant seal) is expected to have fewer deep and long dives than predicted by a purely Lévy distribution. Thus, a downward curving relationship using the rank-frequency method is expected. The maximum step length will also be limited by environmental factors such as the depth of water available to the animal during diving, where, for example, shallow depths reduce the maximum step length possible compared to deeper seabed depths. These influences have the effect of introducing considerable bias, for example, away from fewer data in a distribution s tail compared with the left-hand side of the distribution. This bias is not controlled for as it is when model fitting with normalised logarithmically binned data, where all data points are equidistant and have equal weighting. Consequently, some caution 5 is needed when model fitting to rank-frequency plots to ensure important components of the data set with fewer data are accounted for within a particular model; in this study, the heavy tails of the move-step frequency distribution have few data but are important for identifying the presence of Lévy-type walks. For air-breathers in particular, it may be prudent to define an upper percentile of the move step lengths to avoid rare extreme values. 3

4 It is apparent that curvilinear plots in addition to those with a straight-line form may approximate Lévy-like processes summarised in rank-frequency plots. Because of this we chose to contrast three models describing the relationship between the log rank and log move step distribution described above: () a linear model [log (y) ~ log (x)] indicative of a power law, (2) an exponential model [log (y) ~ (x)] indicating more Brownian-like movement and (3) a quadratic model [log (y) ~ log (x)+log(x) 2 ] describing intermediate behaviour. Although there is no particular statistical or biological justification for the quadratic model, it is the simplest (i.e., most parsimonious) alternative model that tests the explicit hypothesis that the relationship between rank and frequency is neither strictly linear nor exponential. We contrasted the three models using the dimension-consistent Bayesian Information Criterion (BIC) because the K-L prior used to justify AIC c weighting (see above) can favour more complex models when sample sizes are large 7, such as those used in the rank-frequency comparisons (n = 866 to 299 ranks). Relative likelihoods of the three candidate models were calculated using BIC weights. The log-log cumulative distribution plot of krill (Euphausia superba) density change showed a straight-line form that is consistent with a power law (Fig. 2c; Supplementary Table 6, Fig. d). There was no effect on μ of using log-transformed versus non-transformed raw data to derive krill density changes (Supplementary Fig. 2). Similar straight-line forms were seen in cumulative distributions for predators such as Atlantic cod (Gadus morhua) and leatherback turtle (Dermochelys coriacea), with some curvature emerging in the rank-frequency plots for basking sharks (Cetorhinus maximus), bigeye tuna (Thunnus obesus) and Magellanic penguin (Spheniscus magellanicus) (Supplementary Fig. a-c, e, g, h). According to BIC weights, all individuals (or krill density) except the catshark (Supplementary Fig. i) and southern elephant seal (Supplementary Fig. f) showed overwhelming support for the quadratic model, despite it having an extra parameter relative to the linear or exponential model 4

5 (Supplementary Table 7). There was virtually no support for the linear model in any individual (or krill density) relative to the other models, although in both prey and major predators (basking shark, Atlantic cod, leatherback turtle, Magellanic penguin) the percentage deviance explained was relatively high for the linear model, and in these species cases was higher than that obtained for the exponential model (Supplementary Table 6, 7). Catshark and elephant seal demonstrated strongest support for the exponential model, but for all other predators and prey the quadratic model was better supported, indicating intermediate behaviour and Lévy-like movement as assessed using rank-frequency plots. These results serve to demonstrate two important points. First, curvilinear rankfrequency plots can describe Lévy (power law) processes 4,5,6. In this study, scaleinvariant properties of movement patterns of basking shark, bigeye tuna, Atlantic cod, leatherback turtle and Magellanic penguin were identified using several methods (logarithmic binning with normalisation; root mean square fluctuation; power spectrum analysis) in addition to rank-frequency plots, thereby reducing the probability of not identifying Lévy-like behaviour even when they are present. The second point concerns model selection and whether Lévy behaviour is identified within movement (or density) time series. For example, it could have been concluded that the exponential model was relatively better than the linear (power law) model in some cases simply because other intermediate models were not tested. By assessing fits to only two models therefore, and if the exponential model was favoured in such a comparison, it would have been concluded that those time series of animal movements were purely random (exponential form to the rank-frequency plot) rather than displaying a Lévy pattern (power law straight or curvilinear form), even though other models, implying more complex and realistic properties, perhaps comprising elements of both move patterns, may be more appropriate. 5

6 Using these methods we show that the foraging movement patterns of the five species were power-law-like and therefore not purely random. Some curvature of the rank-frequency plot in some species may represent Lévy behaviour where μ 2, or may indicate the presence of Lévy-like movements in addition to other movement types and which for the case of rank-frequency plots were strongly supported by quadratic model fits to the data. The plotting results we show suggest that the different move-step distributions observed among marine vertebrate predators are probably not only due to different evolutionary and life history traits, but may also be a consequence of the complex movements these animals use during foraging in relation to complex and changeable prey landscapes. Hence, several distribution types may be possible within a species move-step time series if, as they seem to, individuals demonstrate flexibility in their foraging strategy 8. In this sense, moderate curvature in rank-frequency plots may be common among foragers and may resemble an intermediate form describing Lévylike processes.. Burnham, K. P. & Anderson, D. R. Model Selection and Multimodal Inference: A Practical Information-Theoretic Approach (Springer-Verlag, New York, USA, 22). 2. R Development Core Team. R: A language and environment for statistical computing ( 24). 3. Newman, M.E.J. Power laws, Pareto distributions and Zipf s law. Contemp. Phys. 46, (25). 4. Sims, D.W., Righton, D. & Pitchford, J.W. Minimizing errors in identifying Lévy flight behaviour of organisms. J. Anim. Ecol. 76, (27). 5. Bradshaw, C.J.A & Sims, D.W. Solutions to problems associated with identifying Lévy walk patterns in animal movement data. Ecol. Applic. (submitted). 6. Clauset, A., Shalizi, C.R. & Newman, M.E.J. Power-law distributions in empirical data. E-print (June 27). arxiv physics/

7 7. Link, W.A. & Barker, R.J. Model weights and the foundations of multimodel inference. Ecology 87, (26). 8. Hays, G.C. Hobson, V.J., Metcalfe, J.D., Righton, D., Sims, D.W. Flexible foraging movements of leatherback turtles across the North Atlantic Ocean. Ecology 87, (26). 7

8 Supplementary Methods & Results 2. Projections of 3D Lévy movements Our results refer to the distribution of vertical movements, whereas in reality the organisms are free to move in a three-dimensional (3D) ocean. Here we argue that, for a forager following an isotropic Lévy search pattern, analysis of depth data is sufficient to identify the existence of a scaling law and to recover the Lévy exponent. Suppose an organism performs a random walk in three dimensions (3D); at each time step dt it takes a move step of length R in a random direction (i.e. there is no correlation or bias), where R is distributed with some probability density f(x). We then ask what will be the observed distribution of move steps when this 3D movement is projected into some fixed vertical 2D plane. This question is answered in general by considering the absolute angle of elevation θ of each move step relative to that plane, and noticing that the projected move step is simply Rcos(θ), a product of two independent random variables. Since θ is uniform on [; π/2], the inverse transform method shows that cos(θ), which is necessarily non-negative since we are concerned with projected distances and absolute angles of elevation, is distributed with density function 2 g( y) =, π 2 y given by where y. The probability density for the projected move step, h(x), is then f y g x y h x ( ) ( / ) ( ) = dy. y If R has a power law distribution, so that α α x f ( x) =, xmin xmin 8

9 and the organism follows a Lévy flight in 3D, then the distribution of projected move steps is α α x Γ( α / 2) h ( x) = xmin xmin π Γ(( α + ) / 2) provided x x min and where Γ(.) is the standard gamma function and is independent of x. Therefore, the projected move step distribution follows a power law with an unchanged exponent at all scales greater than the minimum move step. Note that the projection from 3D into 2D can lead to observations of steps lengths smaller than x min x min ; these observations at the smallest scales do not obey a power law. Supplementary Fig. 7 provides a concrete illustration: million random 3D steps are taken from a power law distribution with α = 2.7 and =. and are projected into 2D (red crosses). The blue line shows the predicted distribution of the projected steps, and the parallel green line shows the original 3D distribution. x min Exactly the same argument can then be applied to show that the projection of these 2D Lévy random walk steps into a single vertical dimension also preserves the general scaling law with an unchanged exponent; again the isotropy of the original random walk ensures that the angle of elevation is distributed uniformly, and the mathematical argument can be repeated without modification. It is important to note that this invariance under projection is not a general feature of random walks. For example, if R is constant (i.e., all steps are of equal length in 3D) then the projection method shows that the projected 2D move steps are distributed as h( x) = Rπ 2 x R

10 This distribution is strongly biased in favour of small move steps and (of course) move step can never exceed R. It never resembles a power law distribution over any part of its range. Similarly, if the move step is exponentially distributed with parameter β then the projection distribution is βx 2 2β h( x) = β exp dv = Bk (, βx) v 2 πv v π where B k is a modified Bessel function of the second kind. Again, such a distribution does not exhibit power law behaviour; large move steps are underrepresented.. Rohatgi, V. K. An introduction to Probability Theory and Mathematical Statistics (Wiley, New York, 976).

11 Supplementary Methods & Results 3. Detailed description of the simulations output We simulated a predator s vertical diving movements described by a Lévy walk to encounter prey patches positioned within the water column according to Lévy or random distributions. Lévy searches (μ opt = 2.) reflecting marine predator movements within Lévy (fractal)-distributed prey fields (LL) were compared with encounter rates in random landscapes (LR) (defined here as a prey distribution resulting from a homogeneous spatial Poisson process). Our expectation is that the foraging success ratio LL:LR should not deviate substantially from. if adapting to a fractal prey field presents no particular foraging advantage to a Lévy searcher. Foraging performance varied between replicates but was on average 4 % greater when patch size was fixed, and 9 % greater for virtual foragers in seascapes where patch size varied according to a Lévy distribution (Supplementary Fig. a,b). The observed variance in LL:LR ratio was not evident between three repeats within the same prey field of each replicate (Supplementary Fig. ), or with overall biomass available within a prey field (Supplementary Fig. c). Instead, it appears that variance was related to the specific differences in distribution between different prey field types. To confirm whether distributional relationships of different prey patches influenced foraging success, we compared foragers in prey fields of the same type and overall biomass, but with differing amounts of prey patch dispersion (Supplementary Fig. d). Results show that when prey patches were highly clustered foraging success of both straight-line (ballistic) and Lévy walk foragers (Supplementary Fig. d: BL and LL ) was generally lower (by ~5 %) compared with Lévy foragers in more dispersed fractal fields (LL 2 and LL 3 ) (Supplementary Fig. 2a). As expected, we found simulated Lévy-type foraging in a random field yielded lower encounter success than in more natural fractal fields (LL 2 and LL 3 ).

12 We conducted a second set of simulations with random walkers searching in Lévy (fractal)-distributed prey fields (RL) compared with random prey distributions (RR) to examine whether a random walker would achieve a foraging advantage in fractal versus random prey fields (foraging success ratio, RL:RR). The random walk was parameterised with a maximum vertical step length of 25 which yielded path lengths of equivalent magnitude (that is, the total distance moved) to simulated Lévy walkers. Results showed that the RL:RR foraging success ratio was closer to. than those found for both the LL:LR cases simulated (RL:RR ratio x =.99 ±.3 S.D.; n = sets of simulations) (Table ). This indicates that the foraging advantage of a random walk in a fractal environment is similar to that in a random landscape and not equal to the encounter success identified in simulations examining the LL:LR case. A third set of simulations completed this analysis by comparing random walkers searching in Lévy (fractal)-distributed prey fields (RL) with Lévy searchers (μ opt = 2.) within fractal landscapes (LL). This direct comparison tests whether random searchers can outperform Lévy foragers in a fractal landscape. Results gave a RL:LL ratio of <. (RL:LL ratio x =.89 ±.4 S.D.; n = sets of simulations) indicating that a random searcher encounters about % less prey than a Lévy forager in the same fractal field. The results of the three simulation tests are therefore consistent with the hypothesis that a Lévy strategy may be more efficient in complex natural environments than an ordinary random strategy. 2

13 Supplementary Table Summary details of electronic tags deployed Species Area Tag type** Record Resolution n data Ref. (approx. body size*) ing interval (m) sets (min) Basking shark (TL, m) Northeas tatlantic WC PAT3/4. min.5 6 Small Northeas LTK spotted tatlantic LTD24 catshark (M,. kg) Bigeye tuna North WC Mk (M, 4 kg) Pacific NMT Atlantic cod North LTK (M, ~5 kg) Sea LTD2 Leatherback turtle (CL, ~.5 m) North Atlantic SO DST Milli Magellanic penguin (M, 4 kg) South Atlantic D & K DST.7. 7 R.P. Wilson, unpubl. data Southern Southern WC Mk7 & elephant Ocean TDR seal (M, ~45 kg) *Body size: TL, total length; M, mass; CL, curved carapace length. 3

14 ** Tag manufacturers: D & K, Driesen and Kern; LTK, Lotek Wireless; NMT, Northwest Marine Technologies; SO, Star Oddi; WC, Wildlife Computers. Tag type: PAT, pop-up archival transmitting tag; LTD, light-temperature-depth datalogger; DST, data storage tag; TDR, timedepth recorder.. Sims, D.W., Southall, E.J., Richardson, A.J., Reid, P.C., Metcalfe, J.D. Seasonal movements and behaviour of basking sharks from archival tagging: no evidence of winter hibernation. Mar. Ecol. Prog. Ser. 248, (23). 2. Sims, D.W., Wearmouth, V.J., Southall, E.J., Hill, J., Moore, P., Rawlinson, K., Hutchinson, N., Budd, G.C., Righton, D., Metcalfe, J.D, Nash, J.P. & Morritt, D. Hunt warm, rest cool: bioenergetic strategy underlying diel vertical migration in a benthic shark. J. Anim. Ecol. 75, 76-9 (26). 3. Musyl, M.K., Brill, R.W., Boggs, C.H., Curran, D.S., Kazama, T.K. & Seki, M.P. Vertical movements of bigeye tuna (Thunnus obesus) associated with islands, buoys, and seamounts near the main Hawaiian Islands from archival tagging. Fish. Oceanogr. 2, (23). 4. Righton, D., Metcalfe, J.D. & Connolly, P. Fisheries: Different behaviour of North and Irish Sea cod. Nature 4,56 (2). 5. Myers, A.E. & Hays, G.C. Do leatherback turtles (Dermochelys coriacea) forage during the breeding season? A combination of novel and traditional data logging devices provide new insights. Mar. Ecol. Prog. Ser. 322, (26). 6. Bradshaw, C.J.A., Hindell, M.A., Sumner, M.D. & Michael, K.J. Loyalty pays: potential life history consequences of fidelity to marine foraging regions by southern elephant seals. Anim. Behav. 68, (24). 4

15 Supplementary Table 2 Summary of the Lévy exponent data for five species Species Lévy Upper Lower Regression Step length range (m) exponent 95% CI 95% CI coefficient (r 2 ) Lower Upper Cetorhinus maximus Thunnus obesus Gadus morhua Dermochelys coriacea Spheniscus magellanicus Parameters were derived from least-squares linear regression of normalised log-log move steplength frequency distribution plots of pooled species data (shown in Fig. ) for which there was reliable support for power-law-like move step distributions (see Supplementary Methods and Discussion). Lower move step length in the range was taken as the lowest value bin in the histogram. 5

16 Supplementary Table 3 Comparison of power law and exponential models for individual values averaged Species Power law %DE-PL Exponential %DE-EXP Cetorhinus maximus Dermochelys coriacea > < Gadus morhua Mirounga leonina Scyliorhinus canicula Spheniscus magellanicus > <. 83. Thunnus obesus Akaike s information criterion corrected for small sample sizes weights (waic c ) and % deviance explained (%DE) for two contrasted models: power-law (PL) and exponential (EXP) for the move step distributions of the seven species analysed (average values over all individuals). 6

17 Supplementary Table 4 Comparison of power law and exponential models for individual pooled values Species Power law %DE-PL Exponential %DE-EXP Cetorhinus maximus > <. 66. Dermochelys coriacea > < Gadus morhua > < Mirounga leonina <. 9. > Scyliorhinus canicula Spheniscus magellanicus > < Thunnus obesus > < Akaike s information criterion corrected for small sample sizes weights (waic c ) and % deviance explained (%DE) for two contrasted models: power-law (PL) and exponential (EXP) for the move step distributions of the seven species analysed (average values over all individuals). 7

18 Supplementary Table 5 Comparison of power law and exponential models Species Power law Exponential Cetorhinus maximus >.999 <. Dermochelys coriacea >.999 <. Gadus morhua >.999 <. Mirounga leonina <. >.999 Scyliorhinus canicula Spheniscus magellanicus >.999 <. Thunnus obesus >.999 <. Akaike s information criterion corrected for small sample sizes weights (waic c ) and % deviance explained (%DE) for two contrasted generalised linear mixed-effects models (GLMM): powerlaw (PL) and exponential (EXP), coding individual as a random factor, for the move step distributions of the seven species analysed. 8

19 Supplementary Table 6 Contrast of linear (power law), exponential and quadratic models of rank-frequency plots using Bayesian Information Criteria (BIC) weights Species Linear Exponential Quadratic Euphausia superba <. <. ~. Cetorhinus maximus <. <. ~. Cetorhinus maximus 2 <. <. ~. Dermochelys coriacea <. <. ~. Gadus morhua <. <. ~. Mirounga leonina <. ~. <. Scyliorhinus canicula <. ~. <. Spheniscus magellanicus <. <. ~. Thunnus obesus <. <. ~. Numerals following Cetorhinus maximus denote different individuals. 9

20 Supplementary Table 7 Maximum log-likelihood and % deviance explained for linear (power law), exponential and quadratic models Species LL-lin LL-exp LL-quad %DElin %DEexp %DEquad Euphausia superba Cetorhinus maximus Cetorhinus maximus Dermochelys coriacea Gadus morhua Mirounga leonina Scyliorhinus canicula Spheniscus magellanicus Thunnus obesus Numerals following Cetorhinus maximus denote different individuals. 2

21 Supplementary Figure Supplementary Figure Rank-frequency plots for a prey and seven predator species. a, Three model fits to the cumulative distributions (circles) for (a,b) move steps of two basking shark individuals (Cetorhinus maximus), (c) leatherback turtle (Dermochelys coriacea), (d) krill density (Euphausia superba), (e) move steps of Atlantic cod (Gadus morhua), (f) southern elephant seal (Mirounga leonina), (g) Magellanic penguin (Spheniscus magellanicus), (h) bigeye tuna (Thunnus obesus), and (i) small spotted catshark (Scyliorhinus canicula). The rank-frequency plots of prey (krill density) and major predators (basking shark, leatherback turtle, Atlantic cod and Magellanic penguin) were of 2

22 a straight line form, but were most strongly supported by the quadratic model (green line) relative to the linear (power law) (black line) or exponential models (red line). This indicates predator movements and prey density were composed of Lévy-like processes. 22

23 Supplementary Figure 2 Log N (x) (normalised frequency) Log number of samples d Raw data untransformed a Log x (density, g m -2 ) Log x (density, log ([g m -2 ]+)) b μ =.56 r 2 =.98 μ =.62 r 2 = Raw data log-transformed Log krill density, d (g m -2 ) Log krill density, d (log ([g m -2 ]+)) c d μ =.66 r 2 =.99 μ =.64 r 2 =.97 Supplementary Figure 2 Comparison of μ for krill using untransformed and logtransformed raw data. Here we show that whether using untransformed (a, b) or log-transformed data (c, d) of krill density change, the effect on the estimation of the slope μ (black line, upper panels; blue line, lower panels) is small for both the histogram (a, c) and rank-frequency (b, d) techniques. 23

24 Supplementary Figure 3 a Log N(x) μ = 2.4 r 2 =.9-2 d Log N(x) μ =.9 r 2 = b Log N(x) μ = 2.4 r 2 = e Log N(x) μ =.7 r 2 =.9-2 c Log N(x) Log x μ = 2.2 r 2 =.95 f Log N(x) Log x Supplementary Figure 3 General Lévy-like scaling law among diverse marine vertebrates. Normalised log-log plots of the step-length frequency distribution P(l j ) ~ l -μ j, where l j is the step length and μ the power law exponent, for individuals of (a) sub-adult and adult basking shark (Cetorhinus maximus), (b) bigeye tuna (Thunnus obesus), (c) Atlantic cod (Gadus morhua), (d) leatherback turtle (Dermochelys coriacea) and (e) Magellanic penguin (Spheniscus magellanicus). Linear regression analysis for all individuals grouped by species gave power law exponents within ideal Lévy flight limits ( < 24

25 μ 3) and were close to the theoretical optimum μ ~ 2; (f) normalised log-log plot of step-length frequency distribution for a 2.5-m long young-of-the-year basking shark that shows a poor fit to a linear model (r 2 =.26). A frequency distribution of Lévy exponents for 24 individuals is given in Supplementary Fig

26 Supplementary Figure 4 Frequency Lévy exponent Supplementary Figure 4 Lévy-exponent frequency distribution for non-juvenile individuals of five species shown in Figure. Frequency distribution of calculated Lévy exponents for the 24 individual (non-juvenile) animals analysed. Note all exponents lie within Lévy limits with higher frequencies near the theoretical optimum μ ~

27 Supplementary Figure 5 a b Log F(t) Log F(t) α =.4 r 2 = c d e Log F(t) Log t Log t Supplementary Figure 5 Detection of long-range correlations in vertical movement time series. Root mean square (RMS) fluctuation plots averaged across individuals for (a) basking shark (Cetorhinus maximus), (b) bigeye tuna (Thunnus obesus) (α =.9), (c) Atlantic cod (Gadus morhua) (α =.8), (d) leatherback turtle (Dermochelys coriacea) (α =.24) and (e) magellanic penguin (Spheniscus magellanicus) (α =.5). Least squares linear regression was used to determine α for each mean RMS of each species. All species examined have RMS α >.5 indicating the presence of long-range correlations typical of scale-invariant systems where α

28 Supplementary Figure 6 a b Log S(f) Log S(f) β = Log f β = Log f Supplementary Figure 6 Spectral analysis of animal movement and a prey field. Sum of the spectra against frequency for (a) leatherback turtle movements and (b) krill density changes both show β > in the low frequency regime indicating presence of long-range correlations typical of Lévy-like motion and scaleinvariant systems. The plateau in the spectra at intermediate frequencies in both a and b appears to separate low and high frequency domains, both with negative gradients. The plateau in a may signify the transition from short-term correlations in dive behaviour to long-term searching behaviour. In b, the 28

29 transition may be more closely related to long-term, large-scale patterns in density distributions in the low-frequency regime, whereas over small distances and time scales behavioural or fluid turbulent effects on krill density may become important. In support of this, a 2 h periodicity in the krill density data was found in a previous study (ref. 27 in the main paper) suggesting a tidal influence. 29

30 Supplementary Figure 7 Supplementary Figure 7 The distribution of simulated move steps from a powerlaw distribution, α = 2.7 and x min =.. The red crosses show the projected 2D step length distribution of million 3D steps. Notice that the projected length can be less than x min. The green line shows the original 3D step length distribution. The blue line shows the predicted distribution of the projected steps, the prediction of power law distributed lengths holds for all step lengths above x min. 3

31 Supplementary Figure 8 a Total zooplankton change (individuals m -3 ) Time series (sample interval, ~4 d) b Frequency Log N(x) Log x μ = 2. r 2 = Total zooplankton change (individuals m -3 ) Supplementary Figure 8 Macroscopic properties of total zooplankton density distribution. (a) Change in zooplankton density in a time series of vertical (oblique tow) samples (see below). (b) Step frequency distribution of change in zooplankton density follows a heavy-tailed power law with an exponent within Lévy limits and matching the theoretical optimum μ ~ 2. Note the similarity of intermittent pattern between zooplankton in (a) and krill densities in Fig. 2a. Total zooplankton sampled vertically by oblique-towed nets at station L4 was collected approximately every 4 days between January 988 and December 2 (L4 dataset at and was used to derive a time series of change in zooplankton abundance. 3

32 Supplementary Figure 9 Frequency Log N(x) Total zooplankton (g m -3 ) Log x μ =.8 r 2 =.92 Supplementary Figure 9 Zooplankton prey field sampled from basking shark feeding paths. The frequency distribution of change in zooplankton density between consecutive samples resembles a power law with a heavy tail reminiscent of a Lévy distribution. Inset: normalised log-log plot of the frequency distribution is within Lévy limits with an exponent close to the optimum μ ~ 2. The feeding path of the predator represents a horizontal series of samples of the prey environment, but that a power law-like pattern of prey densities emerges suggests: () the prey field sampled has a fractal-like pattern of densities, and (2) that the points visited by a Lévy-like predator such as the basking shark yield a power law-like pattern of prey densities which might be expected if a Lévy walk was adopted to find target sites varying in prey density. Zooplankton sampled from basking shark feeding paths were taken within a restricted area ( ºN and ºW) over 7 days between 26 May 2 June 23 in the western English Channel. The data set used comprised < samples so the apparent power law is under-sampled. Caution is needed in ascribing fractal processes to this particular prey field; nonetheless, the result is consistent with those found for the two other prey fields. 32

33 Supplementary Figure a b Foraging success (LL:LR ratio) Replicate number c d Foraging success (LL:LR ratio) Foraging success (biomass consumed /path length x 3 ) BL Biomass units LL LL 2 LL 3 LR Increasing dispersion Supplementary Figure Optimal foraging simulations. Lévy foraging movements were simulated in Lévy-like (fractal) prey fields (LL) and across landscapes with randomly distributed patches (LR). The ratio LL:LR defines the relative foraging success of Lévy foragers in natural (LL) compared with unnatural (LR) fields (N = 5 patches) with deviation above. signifying a potential selective advantage of predators in complex prey environments. a, Foraging success ratio of LL vs LR across replicates each comprising 3 repeats of, LL foragers vs, LR foragers when size of patches remains constant, and b, when prey patch varies according to a Lévy distribution. c, The variation in foraging success ratio evident in b and c was not attributable to the overall biomass units within prey patches. d, The effect on the 33

34 mean foraging success (± S.D.) of increasing dispersion altering the distribution characteristics in prey fields populated with the same number of prey patches and overall biomass. BL denotes a ballistic (straight line) forager in a more clustered Lévy-like field (see a schematic representation of the prey field above each histogram bar) compared with LL foragers in increasingly dispersed Lévy-like fields and in a random prey landscape. 34

35 Supplementary Figure.25 Foraging success ratio (LL:LR) A A2 A3 B B2 B3 C C2 C3 Repeat number Supplementary Figure Effect of prey patch distributional changes on foraging success. Foraging success of simulated Lévy foragers in fractal (Lévy-like) prey distributions (LL) compared with Lévy foragers in random prey fields (LR). Each repeat comprises, LL versus, LR with total biomass held constant between each of 3 runs (A-C). Total biomass (arbitrary units) in prey field: A, ; B,. 6 ; C,.2 6. Note the variation between repeats of each run is minimal but the variation between runs as a consequence of prey field distributional change (when biomass is held constant) is greater by comparison. 35

36 Supplementary Figure 2 a Inter-patch distance (arbitrary units) LL LL 2 LL 3 Random Fractal (Lévy) prey fields b Frequency (interpatch distance) Frequency (spatial density, patches per cell) Supplementary Figure 2 Patch geometry in simulated prey fields and relationship to spatial prey density. a, Plot to show how the distances between patches (N = 5) from left to right across a prey field of fixed overall biomass varies with different levels of dispersion referred to in Supplementary Figure. The fractal prey field LL is generated with a dispersion factor of 5 so interpatch distances are generally short with a few long distances between the highly clustered patches. LL 2 is generated with a dispersion factor of 2 (which 36

37 was used in all other simulation runs) and LL 3 with a factor of, each showing progressively longer inter-patch distances dominating the Lévy distribution. For comparison, the random prey field shows an average interpatch distance that visually but not statistically resembles the LL 3 prey field in 2- D. Summary statistics (n = 49 steps per field): LL, x = 6.4 ± S.D.; LL 2, x = 3.62 ± S.D.; LL 3, x = 46.8 ± S.D.; Random, x = 7.23 ± S.D. b, The frequency distribution of inter-patch distances is positively correlated with the frequency of patch spatial density for simulated fractal prey field LL 2 (Pearson correlation coefficient, r =.98, P <.). For field LL 2, a frequency distribution of inter-patch distances (with six equidistant bins) was correlated with a frequency distribution of spatial patch density with the same bin number. The highest frequency of one method correlated with the highest frequency of the other (i.e. lowest spatial density vs smallest inter-patch distances), and so on to the lowest frequency bins. This shows there is congruence between the inter-patch distance method we used to derive prey fields in the simulation program, and the empirical measurements of prey density analysed. Therefore, comparing empirical and simulated results is reasonable as either method can summarise adequately for our purposes the underlying pattern in a prey field. 37