Animal space use: distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk


A. O. Gautestad, Centre for Ecological and Evolutionary Synthesis, Dept of Biology, Univ. of Oslo, PO Box 1066 Blindern, NO-0316 Oslo, Norway. E-mail:


How to differentiate between scale-free space use like Lévy walk and a two-level scale-specific process like composite random walk (mixture of intra- and inter-patch habitat movement) is surrounded by controversy. Composite random walk may under some parameter conditions appear Lévy walk-like from the perspective of the path’s distribution of step lengths due to superabundance of very long steps relative to the expectation from a classic (single-level) random walk. However, a more explicit focus on the qualitative differences between studying movement at a high resolution mechanistic (behavioral) level and the more coarse-grained statistical mechanical level may contribute to resolving both this and other issues related to scaling complexity. Specifically, a re-sampling of a composite random walk at larger time lags than the micro-level unit time step for the simulation makes a Lévy-look-alike step length distribution re-shaping towards a Brownian motion-like pattern. Conversely, a true Levy walk maintains its scaling characteristics upon re-sampling. This result illustrates how a confusing pattern at the mechanistic level may be resolved by changing observational scale from the micro level to the coarser statistical mechanical meso- or macro-scale. The instability of the composite random walk pattern under rescaling is a consequence of influence of the central limit theorem. I propose that a coarse-graining test – studying simulated animal paths at a coarsened temporal scale by re-sampling a series – should be routinely performed prior to comparing theoretical results with those patterns generated from GPS data describing animal movement paths. Fixes from terrestrial mammals are often collected at hourly intervals or larger, and such a priori coarse-grained series may thus comply better with the statistical mechanical meso- or macro-level of analysis than the behavioral mechanics observed at finer resolutions typically in the range of seconds and minutes. If fixes of real animals are collected at this high frequency, coarse graining both the simulated and real series is advised in order to bring the analysis into a temporal scale domain where analytical methods from statistical mechanics can be applied.