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Ecology

Fish in Lévy-flight foraging

Nature volume 465, pages 10181019 (24 June 2010) | Download Citation

Lévy flights are a theoretical construct that has attracted wide interdisciplinary interest. Empirical evidence shows that the principle applies to the foraging of marine predators.

Ecologists have long drawn inspiration from mathematics. Theory applied to population dynamics is one example. Another is the Lévy-flight foraging hypothesis1, which has been invoked to explain the strategies of organisms searching for food. Lévy flights, or walks, are characterized by rare but extremely long 'step' lengths, and the same sites are revisited much less frequently than in a normal diffusion process. The existence of Lévy flights as a foraging strategy has been the subject of controversy, but Humphries et al.2 (page 1066 of this issue) now report a notable advance. They have carried out a direct empirical test of the Lévy-flight foraging hypothesis, using the largest animal-movement data set assembled for this purpose.

How organisms move and disperse is of central importance in several fields. The classic paradigm of simple diffusion is used to describe a wide variety of phenomena, ranging from how humans dispersed out of Africa to how pollen spreads. Until the twentieth century, Fick's laws were thought to be universally valid for describing diffusion. The physiologist Adolf Fick introduced the idea that diffusion is proportional to the gradient of concentration; this idea describes Brownian motion and is analogous to Fourier's law of heat conduction. A prediction of the theory of Fickian diffusion is that the mean squared displacement of a 'random walker' increases linearly with time, not superlinearly (for example, quadratically) or sublinearly (for example, as the square root). Any process that is inconsistent with Fick's laws is known as anomalous diffusion: superdiffusion leads to superlinear growth of the mean squared displacement with time, whereas subdiffusion leads to sublinear growth.

Much of the interest in Lévy flights is due to their superdiffusive properties. The rare but long steps of Lévy flights follow the simple rule that a step of length l is chosen from a probability density function P(l) l−μ, with 1 < μ ≤ 3. Such an asymptotic power-law function is sometimes termed a 'fat-tailed distribution' because the tail falls off much more gently than for a Gaussian (or normal) distribution — hence it is 'fat'. This property lies at the heart of the interesting and unusual behaviour of Lévy flights.

Why would animals adopt a Lévy-flight foraging strategy? In a Brownian random walk, the walker frequently returns to the same place (Fig. 1). By contrast, Lévy walkers can outperform Brownian walkers by revisiting sites far less often1,3. Biological Lévy flights were first conjectured to be a useful search strategy in 1985, during a NATO Advanced Study Institute conference4. A decade later, an analysis of data obtained from recording devices attached to the legs of wandering albatrosses generated broad interest in biological Lévy flights and prompted many further investigations5.

Figure 1: Comparison of a Lévy flight with a Brownian random walk.
Figure 1

a, Walks of identical total length of 1,000 unit steps are shown drawn to scale. Lévy flights (or walks) have ultra-long steps, which are absent from Brownian walks. b, A close-up of the Brownian walk, in which the walker returns many times to previously visited locations (a phenomenon known as 'oversampling'). By contrast, the Lévy walker occasionally takes long jumps to new territory. This reduction in oversampling is part of the theoretical basis for interest in the Lévy-flight foraging hypothesis, which predicts that Lévy flights offer higher search efficiencies in environments where prey is scarce. Humphries et al.2 show that marine predators often move in patterns that are consistent with this hypothesis. (Plots modified from ref. 10.)

Formulation of the Lévy-flight foraging hypothesis arose from a study1 that showed mathematically that Lévy flights, characterized by an inverse square distribution of step lengths, optimize random searches under specific conditions, when targets (such as prey) are scarce. Scarcity of resources is common, and observations of Lévy flights have been extended to species ranging from dinoflagellates6 to fish7, and even to human movement8. Such findings have been the subject of a News Feature in these pages9. Moreover, random search has blossomed into an interdisciplinary subfield of physics3, as discussed in a new book on the physics of foraging10.

The results reported by Humphries et al.2 give a convincing answer to whether the Lévy-flight foraging hypothesis stands up to empirical scrutiny — one of the more controversial problems in theoretical movement ecology. The authors' data set of more than 107 measurements, compiled from electronic tags on 55 individual fish — sharks, tuna, billfish and a sunfish— is an order of magnitude larger than the last reported data set7. They find strong evidence of Lévy flights, but, as predicted theoretically, these flights are not universal. Lévy flights are expected in places where prey is scarce (such as the open ocean), whereas a Brownian strategy is more likely to occur where prey is abundant (as in marine regions where the mixing of water bodies produces high densities of phytoplankton, zooplankton and organisms higher in the food chain). The observed2 pattern of switching between search modes is not entirely consistent with these expectations. But it is nonetheless plausible, as seen for instance in the data on a blue shark that moved from the prey-rich waters of the western English Channel to the oceanic environment of the Bay of Biscay.

The fact that some organisms perform Lévy flights has deep implications that transcend those for marine ecosystems, and it raises many questions. Did humans disperse from Africa superdiffusively rather than diffusively? Does pollen from genetically modified crops spread superdiffusively? What are the consequences if influenza epidemics spread superdiffusively? In a reaction-diffusion context, superdiffusion leads to significantly increased overall reaction rates, because the reacting species — which may be chemical or biological — meet each other more often. What more can be learned about such interactions? These questions and many more await investigation.

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  1. Gandhimohan M. Viswanathan is at the Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas 57072-970, Brazil.  gandhi.viswanathan@pq.cnpq.br

    • Gandhimohan M. Viswanathan

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