The results discussed in Chapter 10 have inspired a renewed interest in fundamental questions relating to random searches. We have seen that Lévy flights have scalefree properties such that there is no unique characteristic scale in the random walk flight length (or step length) distribution p(ℓ). In contrast, Wiener noise, unlike Lévy processes, has a well-defined characteristic scale because all moments are finite. Are the high search efficiencies of Lévy flight foraging due to the multiple scales or, equivalently, to the scale-free properties? How many scales would be sufficient to guarantee high encounter rates and search efficiencies? Perhaps scalefree properties are not needed after all, and a few scales would be sufficient. In this chapter, we review search models that contain free parameters embedding characteristic scales.

Correlated random walks with a single scale

The most natural and obvious choice for the fewest number of characteristic scales is one. Correlated random walks (CRWs) appeared the study of ecology when short- and medium-scaled animal movement data were analyzed. CRWs have a single characteristic scale – a correlation length or time that can be quantified via sinuosity. Experiments with ants, beetles, and butterflies were performed in 15 to 20 square meter arenas as well as in their natural environments (and usually lasted fewer than 45 minutes). From these studies, ecologists promptly became aware of the necessity of adding directional persistence to pure random walks to reproduce realistic animal movements [21, 42, 173].