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Optimizing the success of random searches

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Abstract

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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References

  1. Tsallis,C. Lévy distributions. Phys. World 10, 42–45 (1997).

    CAS  Google Scholar 

  2. Schlesinger,M. F., Zaslavsky,G. M. & Frisch,U. (eds) Lévy Flights and Related Topics in Physics (Springer, Berlin, 1995).

    Google Scholar 

  3. Levandowsky,M., Klafter,J. & White,B. S. Swimming behavior and chemosensory responses in the protistan microzooplankton as a function of the hydrodynamic regime. Bull. Mar. Sci. 43, 758–763 (1988).

    Google Scholar 

  4. Schuster,F. L. & Levandowsky,M. Chemosensory responses of Acanthamoeba castellani: Visual analysis of random movement and responses to chemical signals. J. Eukaryotic Microbiol. 43, 150–158 (1996).

    CAS  Google Scholar 

  5. Cole,B. J. Fractal time in animal behaviour: The movement activity of Drosophila. Anim. Behav. 50, 1317–1324 (1995).

    Google Scholar 

  6. Viswanathan,G. M. et al. Lévy flight search patterns of wandering albatrosses. Nature 381, 413–415 (1996).

    ADS  CAS  Google Scholar 

  7. Shlesinger,M. F. & Klafter,J. in On Growth and Form (eds Stanley, H. E. & Ostrowsky, N.) 279–283 (Nijhoff, Dordrecht, 1986).

    Google Scholar 

  8. Berkolaiko,G., Havlin,S., Larralde,H. & Weiss,G. H. Expected number of distinct sites visited by N discrete Lévy flights on a one-dimensional lattice. Phys. Rev. E 53, 5774–5778 (1996).

    ADS  CAS  Google Scholar 

  9. Berkolaiko,G. & Havlin,S. Territory covered by N Lévy flights on d-dimensional lattices. Phys. Rev. E 55, 1395–1400 (1997).

    ADS  CAS  Google Scholar 

  10. Larralde,H., Trunfio,P., Havlin,S., Stanley,H. E. & Weiss,G. H. Territory covered by N diffusing particles. Nature 355, 423–426 (1992).

    ADS  CAS  Google Scholar 

  11. Larralde,H., Trunfio,P., Havlin,S., Stanley,H. E. & Weiss,G. H. Number of distinct sites visited by N random walkers. Phys. Rev. A 4, 7128–7138 (1992).

    ADS  Google Scholar 

  12. Szu, H. in Dynamic Patterns in Complex Systems (eds Kelso, J. A. S., Mandell, A. J. & Shlesinger, M. F.) 121–136 (World Scientific, Singapore, 1988).

    Google Scholar 

  13. Mantegna,R. N. & Stanley,H. E. Stochastic process with ultra-slow convergence to a gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73, 2946–2949 (1994).

    ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  14. Shlesinger,M. F. & Klafter,J. Comment on “Accelerated diffusion in Josephson junctions and related chaotic systems”. Phys. Rev. Lett. 54, 2551 (1985).

    ADS  CAS  PubMed  Google Scholar 

  15. Heinrich,B. Resource heterogeneity and patterns of movement in foraging bumble-bees. Oecologia 40, 235–245 (1979).

    ADS  PubMed  Google Scholar 

  16. Focardi,S., Marcellini,P. & Montanaro,P. Do ungulates exhibit a food density threshold—a field-study of optimal foraging and movement patterns. J. Anim. Ecol. 65, 606–620 (1996).

    Google Scholar 

  17. Berg,H. C. Random Walks in Biology (Princeton Univ. Press, Princeton, 1983).

    Google Scholar 

  18. Kot,M., Lewis,M. & van der Driessche,P. Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042 (1996).

    Google Scholar 

  19. Schulman,L. S. Time's Arrows and Quantum Measurement (Cambridge Univ. Press, Cambridge, 1997).

    Google Scholar 

  20. Sugihara,G. & May,R. Applications of fractals in ecology. Trends Ecol. Evol. 5, 79–86 (1990).

    CAS  PubMed  Google Scholar 

Download references

Acknowledgements

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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Authors and Affiliations

  1. Center for Polymer Studies and Department of Physics, Boston University, Boston, 02215, Massachusetts, USA

    G. M. Viswanathan, Sergey V. Buldyrev, Shlomo Havlin & H. Eugene Stanley

  2. International Center for Complex Systems and Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal-RN, 59072-970, Brazil

    G. M. Viswanathan

  3. Departamento de Física, Universidade Federal de Alagoas, Maceió-AL, 57072-970, Brazil

    G. M. Viswanathan

  4. Gonda-Goldschmied Center and Department of Physics, Bar Ilan University, Ramat Gan, Israel

    Shlomo Havlin

  5. Lyman Laboratory of Physics, Harvard University, Cambridge, 02138, Massachusetts, USA

    M. G. E. da Luz & E. P. Raposo

  6. Departamento de Física, Universidade Federal do Paraná, Curitiba-PR, 81531-970, Brazil

    M. G. E. da Luz

  7. Departamento de Física, Laboratório de Física Teórica e Computacional, Universidade Federal de Pernambuco, Recife-PE, 50670-901, Brazil

    E. P. Raposo

Authors
  1. G. M. Viswanathan
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  2. Sergey V. Buldyrev
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  3. Shlomo Havlin
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  4. M. G. E. da Luz
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  5. E. P. Raposo
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  6. H. Eugene Stanley
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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). https://doi.org/10.1038/44831

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