Net displacement and temporal scaling: Model fitting, interpretation and implementation

Abstract

Abstract Net displacement is an integral component of numerous ecological processes and is critically dependent on the tortuosity of a movement trajectory and hence on the temporal scale of observation. Numerous attempts have been made to quantitatively describe net displacement while accommodating tortuosity, typically evoking a power law, but scale‐dependency in tortuosity limits the utility of approaches based on power law relationships that must assume scale‐invariant tortuosity. We describe a phenomenological model of net displacement that permits both scale‐variant and scale‐invariant movement. Movement trajectories are divided into pairs of relocations specifying start‐ and end‐points, and net displacements between points are calculated across a vector of time intervals. A bootstrap is implemented to create new datasets that are independent both across and within time intervals, and the model is fitted to the bootstrapped dataset using log–log regression. We apply this model to simulated trajectories and both fine‐grain and coarse‐grain trajectories obtained from an Aldabra giant tortoise Aldabrachelys gigantea, African elephants Loxodonta africana, black‐backed jackals Canis mesomelas and Northern elephant seals Mirounga angustirostris. The model was able to quantify the characteristics of net displacement from simulated movement trajectories corresponding to both scale‐variant (e.g. correlated random walks) and scale‐invariant (e.g. random walk) movement models. Furthermore, the model produced identical outputs across time vectors corresponding to different intervals and absolute ranges of time for scale‐invariant models. The model characterized the tortoise as generally exhibiting long scale‐invariant steps, which was corroborated by visual comparison of model outputs to observed trajectories. Elephants, jackals and seals exhibited movement parameters consistent with their known movement behaviours (nomadism, territoriality and widely ranging searching). We describe how the model may be used to compare movements within and between species, for example by partitioning movement into scale‐variant and scale‐invariant components, and by calculating a unitless net displacement scaled to the basal movement capacities of an animal. We also identify several useful derived quantities and realistic parameter ranges and discuss how the model may be implemented in a variety of ecological studies.