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Optimizing the success of random searches


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 distributions1,2 of flight lengths or flight times3,4,5,6 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.

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Figure 1: Foraging strategy.
Figure 2: a, b, The product of the mean free path λ and the foraging efficiency η against the Lévy parameter μ in one dimension for different values of λ, found from equations (2) and (3) (rv = 1) for the case of non-destructive foraging (a) and from simulations (b).
Figure 3: Foraging by bumble-bees and deer.

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We thank V. Afanasyev, N. Dokholyan, I. P. Fittipaldi, P. Ch. Ivanov, U. Laino, L. S. Lucena, E. G. Murphy, P. A. Prince, M. F. Shlesinger, B. D. Stosic and P. Trunfio for discussions, and CNPq, NSF and NIH for financial support.

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Correspondence to G. M. Viswanathan.

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Viswanathan, G., Buldyrev, S., Havlin, S. et al. Optimizing the success of random searches. Nature 401, 911–914 (1999).

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