Minimizing errors in identifying Lévy flight behaviour of organisms

Summary

• 1
  Lévy flights are specialized random walks with fundamental properties such as superdiffusivity and scale invariance that have recently been applied in optimal foraging theory. Lévy flights have movement lengths chosen from a probability distribution with a power-law tail, which theoretically increases the chances of a forager encountering new prey patches and may represent an optimal solution for foraging across complex, natural habitats.
• 2
  An increasing number of studies are detecting Lévy behaviour in diverse organisms such as microbes, insects, birds, and mammals including humans. A principal method for detecting Lévy flight is whether the exponent (µ) of the power-law distribution of movement lengths falls within the range 1 < µ ≤ 3. The exponent can be determined from the histogram of frequency vs. movement (step) lengths, but different plotting methods have been used to derive the Lévy exponent across different studies.
• 3
  Here we investigate using simulations how different plotting methods influence the µ-value and show that the power-law plotting method based on 2^k (logarithmic) binning with normalization prior to log transformation of both axes yields low error (1·4%) in identifying Lévy flights. Furthermore, increasing sample size reduced variation about the recovered values of µ, for example by 83% as sample number increased from n = 50 up to 5000.
• 4
  Simple log transformation of the axes of the histogram of frequency vs. step length underestimated µ by c.40%, whereas two other methods, 2^k (logarithmic) binning without normalization and calculation of a cumulative distribution function for the data, both estimate the regression slope as 1 − µ. Correction of the slope therefore yields an accurate Lévy exponent with estimation errors of 1·4 and 4·5%, respectively.
• 5
  Empirical reanalysis of data in published studies indicates that simple log transformation results in significant errors in estimating µ, which in turn affects reliability of the biological interpretation. The potential for detecting Lévy flight motion when it is not present is minimized by the approach described. We also show that using a large number of steps in movement analysis such as this will also increase the accuracy with which optimal Lévy flight behaviour can be detected.