Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns

Abstract

For many years, the dominant conceptual framework for describing non-oriented animal movement patterns has been the correlated random walk (CRW) model in which an individual's trajectory through space is represented by a sequence of distinct, independent randomly oriented ‘moves’. It has long been recognized that the transformation of an animal's continuous movement path into a broken line is necessarily arbitrary and that probability distributions of move lengths and turning angles are model artefacts. Continuous-time analogues of CRWs that overcome this inherent shortcoming have appeared in the literature and are gaining prominence. In these models, velocities evolve as a Markovian process and have exponential autocorrelation. Integration of the velocity process gives the position process. Here, through a simple scaling argument and through an exact analytical analysis, it is shown that autocorrelation inevitably leads to Lévy walk (LW) movement patterns on timescales less than the autocorrelation timescale. This is significant because over recent years there has been an accumulation of evidence from a variety of experimental and theoretical studies that many organisms have movement patterns that can be approximated by LWs, and there is now intense debate about the relative merits of CRWs and LWs as representations of non-orientated animal movement patterns.

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

“…Pooling together movement tracks of non-identical individuals (of the same species) can create the appearance of a Lévy walk [48]. In some cases, a correlated random walk can be mistaken for a Lévy walk as it results in a similar pattern [54]. Also, animals of different taxa often employ more than one movement mode [38,39]; if each of those modes is a Brownian walk, their mixture results in a composite Brownian walk that can have the appearance of a Lévy walk [16,28].…”

Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks

“…Pooling together movement tracks of non-identical individuals (of the same species) can create the appearance of a Levy flight [125]. In some cases, a correlated random walk can be mistaken for a Lévy walk as it results in a similar pattern [142]. Also, animals of different taxa often employ more than one movement mode [96,103]; if each of those modes is a Brownian walk, their mixture results in a composite Brownian walk that can have the appearance of a Levy flight [39,69].…”

On correlations between certain random variables associated with first passage Brownian motion

“…Consider a one-dimensional Brownian motion with negative drift, This area variable, whose probability density ) , ( 0 x A P  and associated properties have been analysed in some detail (see [1,2] and references therein), has recently been found to arise in the study of intermittent burst processes in solar physics [3], non-orientated animal movement patterns [4], DNA breathing dynamics [5] and evolutionary dynamics in the context of colonising new territories [6]. The latter invokes certain discrete lattice models, in turn making a natural link to earlier work on compact directed percolation [7], the dynamics of avalanche processes [8] and queueing phenomena such as traffic jams [9].…”