We address the general question of what is the best statistical strategy to adapt in order to search efficiently for randomly located objects ('target sites'). It is often assumed in foraging theory that the flight lengths of a forager have a characteristic scale: from this assumption gaussian, Rayleigh and other classical distributions with well-defined variances have arisen. However, such theories cannot explain the long-tailed power-law distributions of flight lengths or flight times that are observed experimentally. Here we study how the search efficiency depends on the probability distribution of flight lengths taken by a forager that can detect target sites only in its limited vicinity. We show that, when the target sites are sparse and can be visited any number of times, an inverse square power-law distribution of flight lengths, corresponding to Lévy flight motion, is an optimal strategy. We test the theory by analysing experimental foraging data on selected insect, mammal and bird species, and find that they are consistent with the predicted inverse square power-law distributions.

The study of animal foraging behaviour is of practical ecological importance 1 , and exemplifies the wider scientific problem of optimizing search strategies 2 .Lévy flights are random walks whose step lengths come from probability distributions with heavy power-law tails 3, 4 , such that clusters of short steps are connected by rare long steps. flight durations (time intervals between landing on the ocean) were then calculated as consecutive hours for which a bird remained dry, to a resolution of 1 h. It was assumed that birds landed on the water solely to feed, and that flight durations were thus indicative of distances between prey.Time series for 19 separate foraging trips 7 were pooled to give a total of 363 3 flights. The resulting log-log histogram of flight durations gave a straight line with a slope of approximately 2, and is reproduced in Supplementary Fig. 1 from the original raw data. The crux of the conclusion that the albatrosses were performing Lévy flights was that the slope of 2 implied the probability density function (pdf) of flight durations t (in hours), was 7, 10for t ≥ 1 h (leaving out the normalization constant). This is consistent with the Lévy flight definition that the tail of the pdf is of the power-law form t −µ , where 1 < µ ≤ 3 (though technically this is a Lévy walk 4,7,22 We first analyze a newer, larger, and higher resolution data set of albatross flight durations to test for Lévy flights. In 2004, 20 wandering albatrosses on BirdIsland were each fitted with a salt-water logger and a GPS device. The GPS data were too infrequent (at most one location h −1 ) to give distances between landings, but were needed to estimate each bird's departure time from Bird Island, in order to calculate the duration of the initial flight before first landing on the water (we calculated return flights similarly). The resulting data set of flight records was 4 pooled, as in ref. 7, yielding a total of 1416 flights to a resolution of 10 s (Fig. 1).The flights ≥ 1 h are clearly inconsistent with coming from the power law t −2 ascertained 7 for the 1992 data. Furthermore, data from a power law of any exponent (not just 2) would yield a straight line 23 , and this is clearly not the case.In fact, the flight durations t (in h) are consistent with coming from the shifted gamma distribution given by the pdfwhere y = t − 1/120 accounts for the assumed 30 s period before the bird searches for new food sources (see Methods), s = 0.31 is the shape parameter, r = 0.41 h −1 is the rate parameter, and Γ(·) is the gamma function. Equation (2) is valid for flights >30 s; for shorter flights we have f (t) = 0. The exponential term of (2) dominates for large t, implying Poisson behaviour, such that for long enough flights the birds essentially encounter prey randomly with a constant low probability.A Brownian random walker's displacement increases as t H where H = 1/2.If H > 1/2, we have "superdiffusion" as originally inferred in Fig. 2a The gamma distribution (2) has µ = 1 − s = 0.69. This is such a slow powerlaw ...

Recent advances in spatial ecology have improved our understanding of the role of large-scale animal movements. However, an unsolved problem concerns the inherent stochasticity involved in many animal search displacements and its possible adaptive value. When animals have no information about where targets (i.e., resource patches, mates, etc.) are located, different random search strategies may provide different chances to find them. Assuming random-walk models as a necessary tool to understand how animals face such environmental uncertainty, we analyze the statistical differences between two random-walk models commonly used to fit animal movement data, the Lévy walks and the correlated random walks, and we quantify their efficiencies (i.e., the number of targets found in relation to total displacement) within a random search context. Correlated random-walk properties (i.e., scale-finite correlations) may be interpreted as the by-product of locally scanning mechanisms. Lévy walks, instead, have fundamental properties (i.e., super-diffusivity and scale invariance) that allow a higher efficiency in random search scenarios. Specific biological mechanisms related to how animals punctuate their movement with sudden reorientations in a random search would be sufficient to sustain Lévy walk properties. Furthermore, we investigate a new model (the Lévy-modulated correlated random walk) that combines the properties of correlated and Lévy walks. This model shows that Lévy walk properties are robust to any behavioral mechanism providing short-range correlations in the walk. We propose that some animals may have evolved the ability of performing Lévy walks as adaptive strategies in order to face search uncertainties.

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Do the movements of animals, including humans, follow patterns that can be described quantitatively by simple laws of motion? If so, then why? These questions have attracted the attention of scientists in many disciplines, and stimulated debates ranging from ecological matters to queries such as 'how can there be free will if one follows a law of motion?' This is the first book on this rapidly evolving subject, introducing random searches and foraging in a way that can be understood by readers without a previous background on the subject. It reviews theory as well as experiment, addresses open problems and perspectives, and discusses applications ranging from the colonization of Madagascar by Austronesians to the diffusion of genetically modified crops. The book will interest physicists working in the field of anomalous diffusion and movement ecology as well as ecologists already familiar with the concepts and methods of statistical physics.

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha-->2, and the cases of x(0)-->0 and x(0)-->L. For alpha>2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

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We study the role of dynamical constraints in the general problem of finding the best statistical strategy for random searching when the targets can be detected only in the limited vicinity of the searcher. We find that the optimal search strategy depends strongly on the delay time tau during which a previously visited site becomes unavailable. We also find that the optimal search strategy is always described for large step lengths l by a power-law distribution P(l) approximately l(-mu), with 1