A moving–resting process with an embedded Brownian motion for animal movements

Animal movements are of great importance in studying home ranges, migration routes, resource selection, and social interactions. The Global Positioning System provides relatively continuous animal tracking over time and long distances. Nevertheless, the continuous trajectory of an animal’s movement is usually only observed at discrete time points. Brownian bridge models have been used to model movement of an animal between two observed locations within a reasonably short time interval. Assuming that animals are in perpetual motion, these models ignore inactivity such as resting or sleeping. Using the latest developments in applied probability, we propose a moving–resting process model where an animal is assumed to alternate between a moving state, during which it moves in a Brownian motion, and a resting state, during which it does not move. Theoretical properties of the process are studied as a first step towards more realistic models for animal movements. Analytic expressions are derived for the distribution of one increment and two consecutive increments, and are validated with simulations. The induced bridge model conditioning on the starting and end points is used to compute an animal’s probability of occurrence in an observation area during the time of observation, which has wide applications in wildlife behavior research.