Lévy or Not? Analysing Positional Data from Animal Movement Paths

The Lévy walk hypothesis asserts that the optimal search strategy for a forager under specific conditions is to make successive movement steps that have uniformly random directions and lengths drawn from a probability distribution that is heavy-tailed. This idea has generated a huge amount of interest, with numerous studies providing empirical evidence in support of the hypothesis and others criticising some of the methods employed in these. The most common method for identifying Lévy walk behaviour in movement data is to fit a set of candidate distributions to the observed step lengths using maximum likelihood methods. Commonly used candidate distributions are the exponential distribution and the power-law (Pareto) distribution, both on an infinite and a finite (truncated) range. Data sets for which the relative fit of a power-law distribution is better than that of an exponential are typically classified as Lévy walks. However, the movement pattern of the Lévy walk is similar to that of an animal that switches between two behavioural modes in a composite correlated random walk (CCRW) movement process. Recent studies have shown that standard approaches can misidentify the CCRW process as a Levy walk. This misidentification can be due to the methods used to sample and process the data, a failure to assess the absolute fit of the candidate distributions, or the lack of a strong alternative model. In this chapter, we simulate a CCRW process and show that including a composite exponential distribution in the set of candidate M.J. Plank Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand e-mail: [email protected] M. Auger-Méthé Department of Biological Sciences, University of Alberta, Edmonton, Canada e-mail: [email protected] E.A. Codling ( ) Departments of Mathematical Sciences and Biological Sciences, University of Essex, Colchester, UK e-mail: [email protected] M.A. Lewis et al. (eds.), Dispersal, Individual Movement and Spatial Ecology, Lecture Notes in Mathematics 2071, DOI 10.1007/978-3-642-35497-7 2, © Springer-Verlag Berlin Heidelberg 2013 33 34 M.J. Plank et al. distributions can alleviate the problem of misidentification. However, in some cases sampling and processing of the CCRW data can cause a power-law distribution to have a better fit than a composite exponential. In such cases, the absolute goodnessof-fit of the power-law distribution is typically poor, indicating that none of the candidate distributions are a good model for the data. We discuss the relevance of these results for the analysis of empirical movement data.