Using likelihood to test for Levy flight search patterns and for general power-law distributions in nature

1. Ecologists are obtaining ever-increasing amounts of data concerning animal movement. A movement strategy that has been concluded for a broad variety of animals is that of Levy flights, which are random walks whose step lengths come from probability distributions with heavy power-law tails. 2. The exponent that parameterizes the power-law tail, denoted mu, has repeatedly been found to be within the Levy range of 1 < mu <= 3. Here, we use Monte Carlo simulations to show that the methods used to infer the value of mu are inaccurate. 3. The widely used method of simply logarithmically transforming a standard histogram of movement lengths has been shown elsewhere to be problematic. Here, we further demonstrate how poor it is, and show that it actually biases estimates of mu towards the Levy range of 1 < mu <= 3, and can bias estimates towards the value of mu = 2. Thus, previous reports of animals undergoing Levy flights, or of mu being close to the reported optimal value of mu = 2, may simply be a consequence of the bias generated by this method. 4. A technique that has been recently recommended is to logarithmically bin the data and then normalize the resulting histogram. We show that this technique also produces biased results, and suffers from similar problems as those just outlined, although to a lesser extent. 5. The proposed solution is to use likelihood. We find that calculating the maximum likelihood estimate of mu gives the most accurate results (having also tested the rank/frequency method). Likelihood has the further advantages of being the easiest method to implement, and of yielding accurate confidence intervals. Results are applicable to power-law distributions in general, and so are not restricted to inference of Levy flights. 6. We also re-analyse a data set of grey seal movements that was originally reported to demonstrate Levy flight behaviour. Using Akaike weights, we test four models, and find no evidence for Levy flights. Overall, our results suggest that Levy flights might not be as common as previously thought.