Mean squared displacement and sinuosity of three-dimensional random search movements

Correlated random walks (CRW) have been used for a long time as a null model for animal's random search movement in two dimensions (2D). An increasing number of studies focus on animals' movement in three dimensions (3D), but the key properties of CRW, such as the way the mean squared displacement is related to the path length, are well known only in 1D and 2D. In this paper I derive such properties for 3D CRW, in a consistent way with the expression of these properties in 2D. This should allow 3D CRW to act as a null model when analysing actual 3D movements similarly to what is done in 2D. Properties of Correlated Random Walks (CRW) in two-dimensional (2D) space, which are classically used to model 2D random search movements and thereby to serve as null model for more complex movements, are well known (Codling et al., 2008). This is not yet the case for CRW in three-dimensional (3D) space, whereas an increasing number of studies based on various recording techniques focus on 3D movements (Wilson et al., 2008; Voesenek et al., 2016; Le Bras et al., 2017; de Margerie et al., 2018). In particular, we are still missing mathematical expressions for two key metrics of CRWs in 3D space: the mean squared displacement (MSD), i.e. the expected value of the squared beeline distance from the starting point, and the path sinuosity, which corresponds to the spatial component of the diffusion coefficient (Benhamou, 2006). Here I show how these metrics can be expressed for 3D CRW, consistently with their expressions for 2D CRW.