Sampling rate and misidentification of Lévy and non‐Lévy movement paths

Abstract

A large number of empirical studies have attributed Lévy search patterns to the foraging movements of animals. Typically, this is done by fitting a power-law distribution with an exponent of 1 < mu < or = 3 to the observed step lengths. Most studies record the animal's location at equally spaced time intervals, which are sometimes significantly longer than the natural time scale of the animal's movements. The collected data thus represent a subsample of the animal's movement. In this paper, the effect of subsampling on the observed properties of both Lévy and non-Lévy simulated movement paths is investigated. We find that the apparent properties of the observed movement path can be sensitive to the sampling rate even though Lévy search patterns are supposedly scale-independent. We demonstrate that, in certain contexts and dependent on the sampling rate used in observation, it is possible to misidentify a non-Lévy movement path as being a Lévy path. We also demonstrate that a Lévy movement path can be misidentified as a non-Lévy path, but this is dependent on the value of mu of the original simulated path, with the greatest uncertainty for mu = 2. We discuss the implications of these results in the context of studies of animal movements and foraging behavior.

Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

“…Moreover, there are concerns as to whether concrete evidence indicating a particular movement pattern can be obtained at all due to the complexity and variability of the behavioural response that pests exhibit to inherently stochastic environmental factors. Standard statistical tools sometimes cannot distinguish between power law and exponential rate of decay in the step size distribution [51]. The observed movement type can depend on technical details of data collection such as the time scale of the study [3] and/or the time resolution at which the data are obtained [32].…”

Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion

“…This composite BM approach-from randomly mixing bouts of scale-specific movement at different scales-is feasibly described as intra-patch movement mixed with less frequent inter-patch movement [24,28]. Plank & Codling [29] extended Benhamou's [24] exemplification of Lévy look-alike processes from composite BM, by exploring a broader range of scale constants and conditions for the scale-specific components under a range of sampling lags on the generated series. They found that a strong directional persistence (correlated random walk) for the coarser-scale process in combination with low frequency of appearance of this component relative to the finer-scaled process led to an LW-like pattern.…”

“…Hence, this field of research is currently in a state of confusion and controversy. Not only has the traditional approach towards testing for log -log linearity based on equation (1.1) (regression analysis) been questioned and more sophisticated statistical methods advocated (in particular, the Akaike model approach), but even these advanced approaches have been cast in doubt for some parameter range of b [29]. Thus, some alternative protocols for distinguishing between these processes from a statistical mechanical perspective, by studying fractal properties of space use, have been proposed [16,34].…”

Predicting oscillatory dynamics in the movement of territorial animals

“…Two such examples are correlated random walks [9 -12] and Lévy walks [13,14]. Whereas the former has a long mathematical history [15], the latter has been proposed more recently [16] and only in the last decade has it been used as a tool to analyse and interpret animal movement data [17][18][19]; however, its use has not been immune to criticisms [20][21][22][23][24][25]. Here, we are interested in exploring the dynamics of a territorial animal in two dimensions when its movement possesses some degree of persistence.…”