December 2023 Continuous-time modelling of behavioural responses in animal movement
Théo Michelot, Richard Glennie, Len Thomas, Nicola Quick, Catriona M. Harris
Author Affiliations +
Ann. Appl. Stat. 17(4): 3570-3588 (December 2023). DOI: 10.1214/23-AOAS1776


There is great interest in ecology to understand how wild animals are affected by anthropogenic disturbances, such as sounds. For example, behavioural response studies are an important approach to quantify the impact of naval activity on marine mammals. Controlled exposure experiments are undertaken where the behaviour of animals is quantified before, during, and after exposure to a controlled sound source, often using telemetry tags (e.g., accelerometers or satellite trackers). Statistical modelling is required to formally compare patterns before and after exposure, to quantify deviations from baseline behaviour. We propose varying-coefficient stochastic differential equations (SDEs) as a flexible framework to model such data with two components: (1) time-varying baseline dynamics, modelled with nonparametric or random effects of time-varying covariates, and (2) a nonparametric response model, which captures deviations from baseline. SDEs are specified in continuous time, which makes it straightforward to analyse data collected at irregular time intervals, a common situation for animal tracking studies. We describe how the model can be embedded into a state-space modelling framework to account for measurement error. We present inferential methods for model fitting, model checking, and uncertainty quantification (including on the response model). We apply this approach to two behavioural response study data sets on beaked whales: a satellite track and high-resolution depth data. Our results suggest that the whales’ horizontal movement and vertical diving behaviour changed after exposure to the sound source, and future work should evaluate the severity and possible consequences of these responses. These two very different examples showcase the versatility of varying-coefficient SDEs to measure changes in behaviour, and we discuss implications of disturbances for the whales’ energetic balance.

Funding Statement

TM, RG, CH, and LT were funded by the U.S. Office of Naval Research, Grant N000141812807.


We are very grateful to Rob Schick, Will Cioffi, Alan Gelfand, Josh Hewitt, Stacy DeRuiter, and Brandon Southall for discussions about the data and models. The data from four of the five DTags were collected as part of the SOCAL-BRS project, primarily funded by the U.S. Navy’s Chief of Naval Operations Environmental Readiness Division and subsequently by the U.S. Navy’s Living Marine Resources Program. Additional support for environmental sampling and logistics was also provided by the Office of Naval Research, Marine Mammal Program. All research activities for that study were authorized and conducted under U.S. National Marine Fisheries Service permit 14534, Channel Islands National Marine Sanctuary permit 2010-004, U.S. Department of Defense Bureau of Medicine and Surgery authorization, a federal consistency determination by the California Coastal Commission, and numerous institutional animal care and use committee authorizations. The data from the satellite tag and one of the DTags were collected as part of the Atlantic BRS project under National Marine Fisheries Service scientific research permit numbers 17086 and 20605 to Robin W. Baird. The tagging protocol was approved by the Institutional Animal Care and Use Committee at Cascadia Research Collective. This work was supported by the U.S. Fleet Forces Command through the Naval Facilities Engineering Command Atlantic under Contract No. N62470-15-D-8006, Task Order 50, Issued to HDR, Inc. We thank all members of the field teams involved in both the SOCAL and Atlantic BRS projects.


Download Citation

Théo Michelot. Richard Glennie. Len Thomas. Nicola Quick. Catriona M. Harris. "Continuous-time modelling of behavioural responses in animal movement." Ann. Appl. Stat. 17 (4) 3570 - 3588, December 2023.


Received: 1 December 2022; Revised: 1 May 2023; Published: December 2023
First available in Project Euclid: 30 October 2023

MathSciNet: MR4661711
Digital Object Identifier: 10.1214/23-AOAS1776

Keywords: beaked whale , behavioural response study , diffusion process , Stochastic differential equation

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.17 • No. 4 • December 2023
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