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 Fisheries, 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
14. Present address: Research Institute for Climate Change and Sustainability, School of Earth and Environmental Sciences, University of Adelaide, Adelaide, South Australia 5005, Australia.

Correspondence to: David W. Sims^1,2 Correspondence and requests for materials should be addressed to D.W.S. (Email: dws@mba.ac.uk).

Predators can sometimes fine-tune their foraging by using sensory information of resource abundance and distribution at near-distance scales dominated by proximal clues¹⁰, and at very broad scales some may have awareness of seasonal and geographical prey distributions¹¹. 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 probabilistic or 'blind' hunters than deterministic foragers. Fully aquatic marine vertebrates that feed on ephemeral resources like zooplankton 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, 11, 12} over a broad range of spatio-temporal scales^{13, 14, 15}.

Probabilistic search patterns described by a category of random-walk models known as Lévy walks⁴ appear to describe foraging movements of some species³. 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 trajectories with fractal patterns³. Lévy-walk move steps are drawn from a probability distribution with a power-law tail: P(l_j)  l_j^-, with 1 <   3, where l_j is the move-step length and is the power-law (Lévy) exponent (here '' means 'distributed as'). Theoretical studies^{2, 3, 16} show that Lévy walks and Lévy flights (the turning points in a Lévy walk⁴) across random prey distributions increase new-patch encounter probability compared with simple brownian motion, with an optimal search having an exponent   2. Recent studies^{17, 18, 19} contend that Lévy walks or flights have been wrongly ascribed to some species through use of incorrect methods, while others indicate Lévy-like behaviour with optimal power-law exponents^{3, 20, 21} for highest-efficiency searches, supporting the hypothesis that Lévy behaviour may represent an evolutionary optimal value of the Lévy 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, 14, 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 tags²³. 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 zones²⁴. 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 distributions for five species across diverse taxa (shark, teleosts, sea turtle, penguin) closely resembled an inverse-square power law²⁵ with a heavy tail of increasingly longer steps intermittently distributed within the time series that is typical of ideal Lévy walks^{3, 4} (Fig. 1; Supplementary Information). Lévy exponents derived from Lévy-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   2 exponent (  s.d. = 2.12  0.31, n = 24; species range: 1.34    2.91) (Fig. 1; Supplementary Information). Relative likelihood estimates of model fits to the move-step-length frequency distributions for all species supported only an exponential function typifying random motion for two species (catshark, elephant seal), confirming that Lévy-like processes may not predominate within vertical search strategies in all species (see Supplementary Information).

Figure 1: Lévy-like scaling law among diverse marine vertebrates.

a, Movement time series recorded by electronic tags were analysed to determine the Lévy exponent to the heavy-tailed power-law distribution. Left, time series of vertical move (dive) steps (n = 5,000) 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 within ideal Lévy limits ( = 2.3). The Lévy exponent is conserved across longer temporal scales, for example, expanding the time series for this individual to 30 days (n = 43,200 steps) maintains = 2.3 (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 (n = 503,447 move steps); c, bigeye tuna (Thunnus obesus) (n = 222,282 steps); d, Atlantic cod (Gadus morhua) (n = 94,314 steps); e, leatherback turtle (Dermochelys coriacea) (n = 4,393 steps); and f, Magellanic penguin (Spheniscus magellanicus) (n = 9,727 steps). Lévy 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 Lévy-like and not purely random (Supplementary Information).

High resolution image and legend (168K)

To test for the presence of long-term correlations that also characterize scale-invariant Lévy walks³, 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 = 0.5 for the relationship F(t)  t (ref. 26); in contrast, weighted means of for each species tested here were between 0.80 and 1.24 (  s.d. = 1.08  0.17, n = 5), confirming the presence of long-range correlations in diving time series across the five species (Supplementary Information). The scaling exponent of 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 systems²⁶ because   0 when behaviour is temporally 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 Lévy-like motion²¹ over the full range of underwater movements; however, considering the data as 2D projections of three-dimensional (3D) movements presents no obstacle to their statistical treatment. The projection of spatially homogeneous 3D Lévy 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 Lévy-like vertical movements described here, therefore, reflect the more complex 3D movements made by a range of phylogenetically distinct marine vertebrate species, implying that Lévy-like walks may be a common strategy employed by open-ocean foragers.

Lévy-walk-like behaviour of foragers may show mechanistic links with natural prey fields if the search pattern emerges from the underlying pattern of food distribution²⁰, or if the strategy evolved to enhance foraging success in particular prey distributions. To examine these contrasting hypotheses we analysed the structure of two different 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 echosounder²⁷ were measured throughout the 200m water column at consecutive, equally spaced time intervals. Krill densities showed extreme variance in amplitude through time, with an intermittent structure of large 'jumps' (Fig. 2a). The frequency distribution of krill density changes also closely resembled a power law with a heavy tail, giving a Lévy exponent of 1.7 (Fig. 2b, c). Root-mean-square fluctuations gave = 0.9 for krill and spectral analysis revealed low-frequency changes at = 0.3 (Supplementary Information). Similar results were found for the zooplankton time series ( = 1.8 and 2.0, = 0.9; 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 Lévy 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 Lévy 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 movement response and natural prey assemblages, but the close resemblance does indicate that Lévy 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 demonstrating fractal properties.

Figure 2: Macroscopic properties of a prey field.

a, Krill (Euphausia superba) density d (in g m^-2) (n = 1,215 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 Lévy-walk-like predators, and as for the predators, it shows an exponent within Lévy limits and close to the optimum   2. c, Cumulative distribution or rank-frequency plot¹⁷ of krill density change showing a straight-line form consistent with power-law-like and Lévy-like processes. This plot gives a Lévy exponent of 1.64. The relative likelihood model fit to the rank-frequency plot also indicated that the density-change distribution was Lévy-like and not purely random (see Supplementary Information).

High resolution image and legend (158K)

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 study²⁰ support the latter explanation by showing that scale-free foraging patterns (Lévy walks) emerge from the interaction of animals with a particular resource distribution. Likewise, field observations of primates moving 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 steps²⁰. 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, behavioural kineses to prey are limited to relatively small vertical distances in the ocean²⁴, so when threshold prey densities are reached, a predator 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 Lévy-like behaviour (Fig. 1g). We suggest that this striking difference in search pattern from those of mature individuals reflects ontogenetic behavioural development²⁸, 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 Lévy 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 acquisition²⁹. To test this, we investigated the foraging success (total energy acquisition) of a Lévy searcher adapted to a natural-like, fractal prey field by simulating a Lévy-walk predator's vertical diving movements within a virtual prey field defined by either a Lévy or random distribution. Lévy searches ( = 2.0) reflecting marine predator movements within Lévy (fractal)-distributed prey fields (LL, denoting a Lévy searcher in a Lévy 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 Lévy 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 Lévy searchers in fractal fields (RL, denoting a random searcher in a Lévy prey field) and found that the RL:LL ratio, by contrast, showed a negative foraging gain (-10%), 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 Lévy-like searches may represent an adaptation to complex prey distributions by evolving optimal search strategies.

Table 1: Comparison of foraging success for simulated searchers

Full table

Our findings indicate that animals in stochastic environments necessitating probabilistic foraging may derive benefits from adapting movements described by Lévy processes. Lévy models express behavioural minimalism³, 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 addition, it is unlikely that animals search with Lévy-like motion at all times, especially if, for some species, foraging decisions are predominantly deterministic within stable environments. However, evidence that Lévy-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, Lévy-like walks may be useful for developing more realistic models of how animals redistribute themselves in response to shifting resources as a consequence of climate change, fisheries extractions and other habitat modifications³⁰. Such general and simple laws of movement as optimal Lévy 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.

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Supplementary Information

Supplementary information accompanies this paper.

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 Lévy 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.

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