The Langevin diffusion as a continuous‐time model of animal movement and habitat selection

Abstract

The utilization distribution of an animal describes the relative probability of space use. It is natural to think of it as the long‐term consequence of the animal's short‐term movement decisions: it is the accumulation of small displacements which, over time, gives rise to global patterns of space use. However, many estimation methods for the utilization distribution either assume the independence of observed locations and ignore the underlying movement (e.g. kernel density estimation), or are based on simple Brownian motion movement rules (e.g. Brownian bridges). We introduce a new continuous‐time model of animal movement, based on the Langevin diffusion. This stochastic process has an explicit stationary distribution, conceptually analogous to the idea of the utilization distribution, and thus provides an intuitive framework to integrate movement and space use. We model the stationary (utilization) distribution with a resource selection function to link the movement to spatial covariates, and allow inference about habitat preferences of animals. Standard approximation techniques can be used to derive the pseudo‐likelihood of the Langevin diffusion movement model, and to estimate habitat preference and movement parameters from tracking data. We investigate the performance of the method on simulated data, and discuss its sensitivity to the time scale of the sampling. We present an example of its application to tracking data of Steller sea lions Eumetopias jubatus. Due to its continuous‐time formulation, this method can be applied to irregular telemetry data. The movement model is specified using a habitat‐dependent utilization distribution, and it provides a rigorous framework to estimate long‐term habitat selection from correlated movement data. The Langevin movement model can be written as a linear model, which allows for very fast inference. Standard tools such as residuals can be used for model checking.

A ‘How to’ guide for interpreting parameters in habitat‐selection analyses

“…We also note that alternative modelling frameworks exist with parameters that directly describe relative intensities of use at both fine and coarse scales (e.g. Michelot, Gloaguen, et al, 2019). These new analytical developments hold exciting promises to bridge the micro scale of animal movement be- (Signer et al, 2017), and we expect this approach to become more flexible in the near future, allowing users to forecast not only steady-state utilization distributions but also transient movement patterns such as migration and dispersal.…”

A ‘How-to’ Guide for Interpreting Parameters in Habitat-Selection Analyses

“…(13) by u(s , t) and then integrating over G with respect to s ) for its steady state [70,71], or in some cases, by translating the fitted model into a partial differential equation model with analytical steady-state distribution [72]. We also note that alternative modeling frameworks exist with parameters that directly describe long-term relative risk [73][74][75], but these methods are more computationally challenging to implement, and therefore, less likely to be widely used in applied settings. The amt package has a basic capacity to simulate the utilization distribution based on a parameterized integrated step-selection function [69], and we expect this approach to become more flexible in the near future.…”

“…(13) by u(s , t) and then integrating over G with respect to s ) for its steady state (Potts et al, 2014a(Potts et al, , 2014b, or in some cases, translating the fitted model into a partial differential equation model with analytical steady-state distribution (Potts & Schlägel, 2020). We also note that alternative modeling frameworks exist with parameters that directly describe relative intensities of use at both fine and coarse scales (e.g., Michelot et al, 2019bMichelot et al, , 2019aMichelot, Blackwell, Chamaillé-Jammes, & Matthiopoulos, 2020). These new analytical developments hold exciting promises to bridge the micro scale of animal movement behavior with the macro scale of animal spatial distribution but are more computationally challenging to implement.…”

An integrated path for spatial capture–recapture and animal movement modeling

“…surface could also be based on the standard parametric form for RSFs (Manly et al, 2007) using an approximate Langevin diffusion movement model (Michelot et al, 2019b):…”