Spherical First Passage Time: A tool to investigate area-restricted search in three-dimensional movements

A better understanding of animal movements is of crucial importance for investigating numerous ecological issues. Developments in bio-logging technologies largely contributed to the observation and recording of animal displacements. Recently, several devices were developed to track animals in a three-dimensional space. However, given the larger number of variables, these advances generated new analytical problems and currently, few methods exist to analyse 3-D movements. In this study, we present a new technique, the Spherical First Passage Time (SFPT), to determine the scale of search behaviour in a volume. Building on the development of the First Passage Time (FPT) approach, SFPT measures the time required to cross a sphere along a 3-D path. We used simulations as they provide an opportunity to better understand processes involved in a system. Moreover, they offer the advantage of considerably increasing sample size in cases where empiric data remain scarce. However, in order to be more realistic, simulations were constrained within the physiological and behavioural features inherent to a diving animal, in this case beluga whales. First, we modelled three-dimensional movements as a correlated random walk for which the vertical and horizontal dimensions were considered simultaneously. One restricted search event was included in each simulation. Spatial scales obtained with the SFPT approach were compared to those obtained from the classical FPT analysis over the corresponding horizontal path. Results indicate a significant difference between the two approaches, suggesting that, in most cases, an approach in 2-D misrepresents spatial scale of search behaviour occurring in 3-D. Although we tested the SFPT with the example of a diving marine mammal, we argue that this method is applicable for all animals moving in a three-dimensional space.