The Annals of Applied Statistics

A spatially varying stochastic differential equation model for animal movement

James C. Russell, Ephraim M. Hanks, Murali Haran, and David Hughes

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Animal movement exhibits complex behavior which can be influenced by unobserved environmental conditions. We propose a model which allows for a spatially varying movement rate and spatially varying drift through a semiparametric potential surface and a separate motility surface. These surfaces are embedded in a stochastic differential equation framework which allows for complex animal movement patterns in space. The resulting model is used to analyze the spatially varying behavior of ants to provide insight into the spatial structure of ant movement in the nest.

Article information

Ann. Appl. Stat., Volume 12, Number 2 (2018), 1312-1331.

Received: May 2016
Revised: September 2017
First available in Project Euclid: 28 July 2018

Permanent link to this document

Digital Object Identifier

Animal movement stochastic differential equations potential surface Camponotus pennsylvanicus


Russell, James C.; Hanks, Ephraim M.; Haran, Murali; Hughes, David. A spatially varying stochastic differential equation model for animal movement. Ann. Appl. Stat. 12 (2018), no. 2, 1312--1331. doi:10.1214/17-AOAS1113.

Export citation


  • Albertsen, C. M., Whoriskey, K., Yurkowski, D., Nielsen, A. and Mills, J. (2015). Fast fitting of non-Gaussian state-space models to animal movement data via Template Model Builder. Ecology 96 2598–2604.
  • Altizer, S., Bartel, R. and Han, B. A. (2011). Animal migration and infectious disease risk. Science 331 296–302.
  • Avgar, T., Baker, J. A., Brown, G. S., Hagens, J. S., Kittle, A. M., Mallon, E. E., McGreer, M. T., Mosser, A., Newmaster, S. G., Patterson, B. R. et al. (2015). Space-use behaviour of woodland caribou based on a cognitive movement model. J. Anim. Ecol. 84 1059–1070.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data. CRC Press, Boca Raton.
  • Baylis, A. M., Orben, R. A., Arnould, J. P., Christiansen, F., Hays, G. C. and Staniland, I. J. (2015). Disentangling the cause of a catastrophic population decline in a large marine mammal. Ecology 96 2834–2847.
  • Bestley, S., Jonsen, I. D., Hindell, M. A., Harcourt, R. G. and Gales, N. J. (2015). Taking animal tracking to new depths: Synthesizing horizontal–vertical movement relationships for four marine predators. Ecology 96 417–427.
  • Beyer, H. L., Morales, J. M., Murray, D. and Fortin, M.-J. (2013). The effectiveness of Bayesian state-space models for estimating behavioural states from movement paths. Methods Ecol. Evol. 4 433–441.
  • Brillinger, D. R. (2003). Simulating constrained animal motion using stochastic differential equations. In Probability, Statistics and Their Applications: Papers in Honor of Rabi Bhattacharya. Institute of Mathematical Statistics Lecture Notes—Monograph Series 41 35–48. IMS, Beachwood, OH.
  • Brillinger, D. R. (2007). A potential function approach to the flow of play in soccer. J. Quant. Anal. Sports 3 3.
  • Brillinger, D. (2012). A particle migrating randomly on a sphere. In Selected Works of David Brillinger 73–87. Springer, New York.
  • Brillinger, D. R. and Stewart, B. S. (1998). Elephant-seal movements: Modelling migration. Canad. J. Statist. 26 431–443.
  • Brillinger, D., Preisler, H. and Wisdom, M. (2011). Modelling particles moving in a potential field with pairwise interactions and an application. Braz. J. Probab. Stat. 25 421–436.
  • Brillinger, D. R., Stewart, B. S. and Littnan, C. L. (2008). Three months journeying of a Hawaiian monk seal. In Probability and Statistics: Essays in Honor of David A. Freedman 246–264. IMS, Beachwood, OH.
  • Brillinger, D. R., Preisler, H. K., Ager, A. A., Kie, J. and Stewart, B. S. (2001). Modelling movements of free-ranging animals. Univ. Calif. Berkeley Statistics, Technical Report 610.
  • Brillinger, D. R., Preisler, H. K., Ager, A. A., Kie, J. G. and Stewart, B. S. (2002). Employing stochastic differential equations to model wildlife motion. Bull. Braz. Math. Soc. (N.S.) 33 385–408.
  • Brillinger, D. R., Preisler, H. K., Ager, A. A. and Kie, J. (2012). The use of potential functions in modelling animal movement. In Selected Works of David Brillinger 385–409. Springer, Berlin.
  • Brost, B. M., Hooten, M. B., Hanks, E. M. and Small, R. J. (2015). Animal movement constraints improve resource selection inference in the presence of telemetry error. Ecology 96 2590–2597.
  • Cremer, S., Armitage, S. A. O. and Schmid-Hempel, P. (2007). Social immunity. Curr. Biol. 17 R693–R702.
  • de Boor, C. (1978). A Practical Guide to Splines. Applied Mathematical Sciences 27. Springer, New York–Berlin.
  • Dodge, S., Bohrer, G., Weinzierl, R., Davidson, S. C., Kays, R., Douglas, D., Cruz, S., Han, J., Brandes, D. and Wikelski, M. (2013). The environmental-data automated track annotation (Env-DATA) system: Linking animal tracks with environmental data. Mov. Ecol. 1 3.
  • Eilers, P. H. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statist. Sci. 89–102.
  • Flegal, J., Haran, M. and Jones, G. (2008). Markov chain Monte Carlo: Can we trust the third significant figure? Statist. Sci. 23 250–260.
  • Gardiner, C. (1986). Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer Ser. Synergetics 13 149–168.
  • Gelfand, A. E., Diggle, P., Guttorp, P. and Fuentes, M. (2010). Handbook of Spatial Statistics. CRC Press, Boca Raton.
  • Gibert, J. P., Chelini, M.-C., Rosenthal, M. F. and DeLong, J. P. (2016). Crossing regimes of temperature dependence in animal movement. Glob. Change Biol. 22 1722–1736.
  • Hanks, E. M., Hooten, M. B. and Alldredge, M. W. (2015). Continuous-time discrete-space models for animal movement. Ann. Appl. Stat. 9 145–165.
  • Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. J. Comput. Graph. Statist. 5 299–314.
  • Johnson, D. (2013). crawl: Fit continuous-time correlated random walk models to animal movement data. R package version 1.4-1.
  • Johnson, D., London, J., Lea, M. and Durban, J. (2008). Continuous-time correlated random walk model for animal telemetry data. Ecology 89 1208–1215.
  • Jones, G., Haran, M., Caffo, B. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101 1537–1547.
  • Jonsen, I. (2015). bsam: Bayesian state-space models for animal movement. R package version 0.43.1.
  • Killeen, J., Thurfjell, H., Ciuti, S., Paton, D., Musiani, M. and Boyce, M. S. (2014). Habitat selection during ungulate dispersal and exploratory movement at broad and fine scale with implications for conservation management. Mov. Ecol. 2 15.
  • Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications. Imperial College Press, London.
  • Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
  • Mersch, D. P., Crespi, A. and Keller, L. (2013). Tracking individuals shows spatial fidelity is a key regulator of ant social organization. Science 340 1090–1093.
  • Northrup, J. M., Anderson, C. R. and Wittemyer, G. (2015). Quantifying spatial habitat loss from hydrocarbon development through assessing habitat selection patterns of mule deer. Glob. Change Biol. 21 3961–3970.
  • Preisler, H. K., Ager, A. A. and Wisdom, M. J. (2013). Analyzing animal movement patterns using potential functions. Ecosphere 4 art32.
  • Preisler, H. K. and Akers, R. P. (1995). Autoregressive-type models for the analysis of bark beetle tracks. Biometrics 259–267.
  • Preisler, H. K., Ager, A. A., Johnson, B. K. and Kie, J. G. (2004). Modeling animal movements using stochastic differential equations. Environmetrics 15 643–657.
  • Quevillon, L. E., Hanks, E. M., Bansal, S. and Hughes, D. P. (2015). Social, spatial, and temporal organization in a complex insect society. Scientific Reports.
  • Rode, K. D., Wilson, R. R., Regehr, E. V., Martin, M. S., Douglas, D. C. and Olson, J. (2015). Increased land use by Chukchi Sea polar bears in relation to changing sea ice conditions. PLoS ONE 10 e0142213.
  • Russell, J. C., Hanks, E. M., Haran, M. and Hughes, D. (2018). Supplement to “A spatially varying stochastic differential equation model for animal movement.” DOI:10.1214/17-AOAS1113SUPP.
  • Scharf, H. R., Hooten, M. B., Fosdick, B. K., Johnson, D. S., London, J. M. and Durban, J. W. (2016). Dynamic social networks based on movement. Ann. Appl. Stat. 10 2182–2202.
  • Thiebault, A. and Tremblay, Y. (2013). Splitting animal trajectories into fine-scale behaviorally consistent movement units: Breaking points relate to external stimuli in a foraging seabird. Behav. Ecol. Sociobiol. 67 1013–1026.
  • Toledo, S., Kishon, O., Orchan, Y., Bartan, Y., Sapir, N., Vortman, Y. and Nathan, R. (2014). Lightweight low-cost wildlife tracking tags using integrated transceivers. In Education and Research Conference (EDERC), 2014 6th European Embedded Design 287–291.
  • Watkins, K. S. and Rose, K. A. (2013). Evaluating the performance of individual-based animal movement models in novel environments. Ecological Modelling 250 214–234.
  • Wikle, C. K. and Hooten, M. B. (2010). A general science-based framework for dynamical spatio-temporal models. Test 19 417–451.

Supplemental materials

  • Supplement to “A spatially varying stochastic differential equation model for animal movement”. We provide additional information including prior distribution specification, full conditional distributions, analysis of the discretization error, an application to simulated data and an application to the spread of pathogens.