Elsevier

Spatial Statistics

Volume 50, August 2022, 100637
Spatial Statistics

A flexible movement model for partially migrating species

https://doi.org/10.1016/j.spasta.2022.100637Get rights and content

Abstract

We propose a flexible model for a partially migrating species, which we demonstrate using yearly paths for golden eagles (Aquila chrysaetos). Our model relies on a smoothly time-varying potential surface defined by a number of attractors. We compare our proposed approach using varying coefficients to a latent-state model, which we define differently for migrating, dispersing, and local individuals. While latent-state models are more common in the existing animal movement literature, varying coefficient models have various benefits including the ability to fit a wide range of movement strategies without the need for major model adjustments. We compare simulations from the models for three individuals to illustrate the ability of our model to better describe movement behavior for specific movement strategies. We also demonstrate the flexibility of our model by fitting several individuals whose movement behavior is less stereotypical.

Introduction

Movement behavior within species is often highly variable across individuals and years. While some animal populations follow similar migratory trajectories or travel in groups, many display partial migration, meaning seasonal migration is observed only in a fraction of individuals in the population (Chapman et al., 2011). Non-migratory strategies include residential (i.e., sedentary), nomadic, and dispersal behaviors (Mueller and Fagan, 2008). Current inference frameworks for partially migrating species require researchers to first define explicit movement strategies exhibited by the species (e.g., Fullman et al., 2021, Poessel et al., 2016). Researchers then classify individual paths as one of the defined strategies using methods based on spatially explicit measures or model selection (Cagnacci et al., 2016). Classification is often followed by descriptive statistics for each movement strategy or interpretation of statistical models formulated for each movement strategy (Fullman et al., 2021).

Golden eagles (Aquila chrysaetos) display partial migration (Poessel et al., 2016). Understanding the yearly movement strategies of golden eagles is important for conservation and management of the species. Golden eagles’ high mobility, for example, carries individuals across political boundaries, forcing management efforts for the same individuals to be shared by multiple governing bodies (Brown et al., 2017).

Morales et al. (2010) argue for the importance of understanding the links between movement and population dynamics. Population-level inference using a hierarchical structure depends on individual-level models (e.g., Hooten et al., 2016), so it is essential to develop individual-level models that describe behavior well. The task of developing realistic individual-level models becomes more difficult the more heterogeneous the population.

In Fig. 1, we display year-long paths for three individuals that used three movement strategies, which could be described as residence, migration, and dispersal. We define residence as attraction to a single location throughout the year. We define migration as a path where the individual spends a portion or all of the summer season in a single location and a portion or all of winter in a more southern location. We define dispersal as a path where the individual is attracted to one location for a period starting in the beginning of the year and switches to a new location for the remainder of the year.

Partitioning groups of golden eagles based on movement strategy can be a challenging task due to the presence of ‘less-stereotyped’ or ‘mixed’ cases (Cagnacci et al., 2016). Some authors have suggested movement strategies in partially migrating populations would be better described as existing on a continuum, which would better accommodate those less-stereotyped cases (Ball et al., 2001, Cagnacci et al., 2016). Thus there is a need for flexible models that are capable of fitting multiple movement strategies, without predefining those strategies.

Varying coefficient models that allow behavior to transition smoothly in time have recently received attention in the animal movement literature for being a more flexible and realistic alternative to the latent-state model (Michelot et al., 2020, Russell et al., 2018, Russell et al., 2017). In this work, we describe a single varying coefficient model which utilizes a stochastic differential equation (SDE) framework similar to that of Eisenhauer and Hanks (2020). We fit the varying coefficient model for a variety of movement paths including those displaying residential, migratory, and dispersal behavior. The advantage of this varying coefficient framework is that the same model can easily be used to provide insight into movement behavior that fits one of these three categories, as well as behavior that does not clearly fit into only one of these categories. Our proposed model can produce realistic simulated paths for a range of movement strategies.

We compare our approach to a latent-state model within the same SDE framework, and we show that our varying coefficient model better describes movement behavior. Latent-state models are commonly used in animal movement modeling (Pirotta et al., 2018, Patterson et al., 2017), and there exist popular R packages that can be used to easily fit these types of models for animal telemetry data (Michelot et al., 2016, McClintock and Michelot, 2018). The latent-state models are not as flexible as our varying coefficient model and need to be specified differently depending on the movement strategy. We defined different sets of states for residential, dispersal, and migratory movement strategies.

In Section 2, we describe the golden eagle data and motivate the selection of the subset we focus on in this paper. In Section 3, we describe the SDE model framework which is common to all models we consider in this paper. In Section 4, we present our varying coefficient model. In Sections 5 Resident example, 6 Dispersal example, 7 Migrant example, we fit the varying coefficient and alternative models to three paths we selected to illustrate three stereotypic movement strategies: residence, dispersal, and migration. We then illustrate how the varying coefficient model can be used to fit a wider range of movement behaviors in Section 8. Lastly, we summarize the results and suggest areas for future work in Section 9.

Section snippets

Golden eagle telemetry data

We obtained satellite telemetry data for 68 individuals, each of which was tracked for at least 1 year in the western United States. Tagging of eagles and collection of data was funded by the National Raptor Program of the U.S. Fish and Wildlife Service (USFWS), and we accessed the data through collaboration with the USFWS and one of its contractors, Eagle Environmental, Inc. Movement paths for all individuals, based on hourly GPS locations accurate to within 19 m, are shown in Fig. 2. Most of

A stochastic differential equation model framework for animal movement

We considered a flexible SDE model framework following Russell et al. (2018) and Hanks et al. (2017). We adopt the notation of Eisenhauer and Hanks (2020). The continuous time model for an animal’s position rt at time t is drt=vtdtdvt=β(vtμ(rt))dt+σIdwt where vt is the velocity of the animal at time t, β is the coefficient of friction (Nelson, 1967) which controls autocorrelation in movement, μ(rt) is the mean drift in the direction of movement, σ is a scalar that controls the variance in the

Flexible model for partially migratory species

The varying coefficient model we considered fixes the number of attractors m=8 and allows kit for i=1,2,,m to change smoothly over time. We chose to use m=8 as an overestimate of the number of attractors, and we used shrinkage methods to effectively select a subset of the attractors (Marra and Wood, 2011). It is clear that at least two attractors are needed for a migrant or disperser model, and the additional attractors might capture stopover sites or other irregular behavior. We chose the

Resident example

We define a resident as an individual attracted to the same location throughout the year. In this section, we compare our varying coefficient model from Section 4 to a single-state model formulated specifically for the residential movement strategy. For this comparison, we chose a path consisting of a single year of data for one individual. We visually determined that this path, shown in Fig. 4A, displayed a residential movement strategy.

Dispersal example

We define dispersal as an individual that switches from being attracted to one location to being attracted to a second location at some point in the year and remains attracted to the second location for the rest of the year. The path we analyzed as a path displaying dispersal was collected in the year 2018. We compare our varying coefficient model described in Section 4 to a latent-state model formulated specifically for dispersal.

Migrant example

We defined a migrant as an individual who switches seasonally from being attracted to a southern location to a northern location and back to the original southern location throughout the year. The path we analyzed was collected from an eagle in the year 2012. We compare our varying coefficient model described in Section 4 to a latent-state model formulated specifically for migration.

Fitting boundary individuals

An important benefit of the flexible movement model described in Section 4 is its ability to fit a wide range of movement behaviors, including those that do not clearly fit the migrant, resident, or dispersal stereotypes. There are many such individuals in the golden eagle dataset since they are a partially migrating species. Three examples of less-stereotyped paths are shown in Fig. 7 along with simulations from the varying coefficient model fit to each path and the varying coefficients.

Discussion and future work

We have described a flexible model using varying coefficients for fitting individual movement paths for a partially migrating species. We compared our varying coefficient model described in Section 4 to latent-state models within the same SDE model framework for three individual golden eagles. For these three individuals displaying migration, residence, and dispersal, simulations from our varying coefficient model more closely resembled the true paths. We also illustrated the ability of our

Funding information

This material is based upon work supported by the NSF under Award No. NSF DMS-2015273. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the NSF.

References (27)

  • MarraG. et al.

    Practical variable selection for generalized additive models

    Comput. Statist. Data Anal.

    (2011)
  • BallJ.P. et al.

    Partial migration by large ungulates: characteristics of seasonal moose alces alces ranges in northern Sweden

    Wildlife Biology

    (2001)
  • BrillingerD.R. et al.

    The use of potential functions in modelling animal movement

  • BrownJ.L. et al.

    Patterns of spatial distribution of golden eagles across North America: how do they fit into existing landscape-scale mapping systems?

    J. Raptor Res.

    (2017)
  • CagnacciF. et al.

    How many routes lead to migration? Comparison of methods to assess and characterize migratory movements

    J. Anim. Ecol.

    (2016)
  • ChapmanB.B. et al.

    The ecology and evolution of partial migration

    Oikos

    (2011)
  • EisenhauerE. et al.

    A lattice and random intermediate point sampling design for animal movement

    Environmetrics

    (2020)
  • FullmanT.J. et al.

    Variation in winter site fidelity within and among individuals influences movement behavior in a partially migratory ungulate

    PLoS One

    (2021)
  • HanksE.M. et al.

    Reflected stochastic differential equation models for constrained animal movement

    J. Agric. Biol. Environ. Stat.

    (2017)
  • HootenM.B. et al.

    Hierarchical animal movement models for population-level inference

    Environmetrics

    (2016)
  • KloedenP.E. et al.

    Stochastic differential equations

  • McClintockB.T. et al.

    momentuHMM: R package for generalized hidden Markov models of animal movement

    Methods Ecol. Evol.

    (2018)
  • MichelotT. et al.

    Varying-coefficient stochastic differential equations with applications in ecology

    (2020)
  • Cited by (0)

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