Théo Michelot
email: tmichelot1@sheffield.ac.uk

Marie-Pierre Étienne

Pierre Gloaguen

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

Keywords: animal movement, continuous time, resource selection, step selection, Langevin diffusion, potential function, utilisation distribution

1. The utilisation distribution describes the relative probability of use of a spatial unit by an animal. 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, most utilisation distribution models either ignore the underlying movement, assuming the independence of observed locations, or are based on simplistic Brownian motion movement rules. 2. 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 utilisation distribution, and thus provides an intuitive framework to integrate movement and space use. We model the stationary (utilisation) distribution with a resource selection function to link the movement to spatial covariates, and allow inference into habitat selection. 3. 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 Stellar sea lions (Eumetopias jubatus). 4. Due to its continuous-time formulation, this method can be applied to irregular telemetry data. It provides a rigorous framework to estimate long-term habitat selection from correlated movement data.

Introduction

A crucial concept in animal ecology is the utilisation distribution, "the probability density function that gives the probability of finding an animal at a particular location" [START_REF] Anderson | The home range: A new nonparametric estimation technique: Ecological archives e063-001[END_REF]. In recent decades, improvements in tracking technologies have produced large amounts of animal location data, at a high spatio-temporal resolution. Statistical methods have been developed to estimate the UD from telemetry observations, and to link animal movement to habitat preferences and space use [START_REF] Hooten | Animal movement: statistical models for telemetry data[END_REF]. Most popular approaches to estimate the utilisation distribution from tracking data are non-parametric, and use empirical histograms [START_REF] Nielson | Estimating resource selection with count data[END_REF] or kernel density estimation [START_REF] Anderson | The home range: A new nonparametric estimation technique: Ecological archives e063-001[END_REF][START_REF] Worton | Kernel methods for estimating the utilization distribution in home-range studies[END_REF] For example, the stationary distribution is uniform over the study region for Brownian motion movement models [START_REF] Skellam | Random dispersal in theoretical populations[END_REF], and it is a normal distribution for Ornstein-Uhlenbeck based movement models [START_REF] Blackwell | Random diffusion models for animal movement[END_REF].

In this work, we describe a new continuous-time and space mechanistic movement model. The animal's position is modelled as a diffusion process with a drift towards the gradient of its stationary (utilisation) distribution, bringing together the ideas of [START_REF] Brillinger | Handbook of Spatial Statistics, chapter 26. Handbooks of Statistical Methods[END_REF] and [START_REF] Michelot | Linking resource selection and step selection models for habitat preferences in animals[END_REF]. As in [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], the limiting distribution of the process is the utilisation distribution. The movement model is based on the Langevin diffusion, which has also been used to construct a MCMC algorithm [START_REF] Roberts | Optimal scaling of discrete approximations to Langevin diffusions[END_REF]. As this model belongs to the class of potential-based models, inference can be performed from movement data using different estimation methods for stochastic differential equations (SDEs), such as pseudo-likelihood methods which are simple to implement [START_REF] Gloaguen | Stochastic differential equation based on a multimodal potential to model movement data in ecology[END_REF]. We show here how this parametric model can also be linked to step selection approaches when the utilisation distribution is parameterized as a simple function of environmental covariates. Point estimators and confidence intervals of habitat selection parameters can easily be derived in a classical approximated inference framework.

In Section 2, the proposed movement model is formulated in its general form, and in a specific "covariatedependent" form, using a resource selection function. Section 3 describes a pseudo-likelihood method based on the Euler discretization scheme, for the estimation of habitat selection parameters from telemetry data. In Section 4, we assess the performance of inference methods in simulations, and we discuss conditions under which the model parameters can be recovered. In Section 5, we present the analysis of three trajectories of Stellar Sea Lions (Eumetopias jubatus) with the Langevin movement model, with four environmental covariates as potential drivers of their movement.

Langevin movement model

General formulation

We denote by X t ∈ R d the location of an animal in d-dimensional space at time t ≥ 0, and π : R d → R its utilisation distribution [START_REF] Worton | Kernel methods for estimating the utilization distribution in home-range studies[END_REF]. The utilisation distribution is the probability density function such that, for any area A ⊂ R d ,

P(X t ∈ A) = A π(z)dz (1)
In the following, we will focus on the case d = 2, by far the most common in movement ecology, although the framework could be extended to higher dimensions. We propose to describe the continuous-time location process of the animal (X t ) t≥0 with a Langevin diffusion for the density π, defined as the solution to the stochastic differential equation

dX t = 1 2 ∇ log π(X t )dt + dW t , (2)
where W t is a standard Brownian motion, ∇ is the gradient operator, and with initial condition X 0 = x 0 . Under some easily-satisfied technical conditions (that can be found in [START_REF] Dalalyan | Theoretical guarantees for approximate sampling from smooth and log-concave densities[END_REF], Equation (2) has a unique solution, which is a continuous-time continuous Markov process. It describes the animal's movements as the combination of a drift towards higher values of its utilisation distribution π (informed by the gradient of log π), and a random component given by the Brownian motion. Crucially, the solution is a continuous-time Markov process whose asymptotic stationary distribution is π as defined in Equation (1) [START_REF] Roberts | Exponential convergence of langevin distributions and their discrete approximations[END_REF]. The Langevin diffusion is thus a natural choice for the basis of a continuous-time model of animal movement, with a known steady-state distribution.

In its standard formulation, however, the Langevin diffusion cannot readily be used to model animal movement. Indeed, the speed of the process described above is only determined by the shape of the underlying utilisation distribution π, whereas it should be possible for two animals to move at different speeds on the same long-term distribution of space use. To allow for this flexibility, we introduce an additional parameter γ 2 , and we define the Langevin movement model (with speed), as the solution to dX t = γ 2 2 ∇ log π(X t )dt + γdW t , X 0 = x 0 .

(3)

In the following, γ 2 will be referred to as the speed parameter. The model is parameterized in terms of γ 2 (rather than γ) as it is also the variance parameter of the random Brownian motion component. Figure 1 shows two tracks simulated from the Langevin movement model on an artificial utilisation distribution, for two different values of γ 2 . Although the two tracks explore space at very different speeds, they have the same equilibrium distribution. 1 2 3 4 5 π q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q ∆ = 50 γ 2 = 1 q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ∆ = 1000 and , after ∆ = 50 and ∆ = 1000 time units. Although the process with γ 2 = 1 is much slower to explore space, the properties of the Langevin equation guarantee that both processes have the same stationary distribution π.

γ 2 = 100

Including covariates

We link the utilisation distribution of the animal to spatial covariates with the standard parametric form of resource selection functions (RSF),

π(x|β) = exp J j=1 β j c j (x) Ω exp J j=1 β j c j (z) dz , (4)
where c j (x) is the value of the j-th covariate at location x, Ω ⊂ R d is the study region, and β = (β 1 , . . . , β J ) is a vector of unknown parameters. Larger values of β j indicate stronger selection of the j-th covariate. The denominator in the right-hand side of Equation ( 4) is a normalising constant, and is necessary to ensure that π(x|β) is a probability density function with respect to x. Note that Equation (3) requires log π to be a smooth function, i.e. with continuous first-order partial derivatives. If π is modelled by a resource selection function (Equation ( 4)), then

∇ log π(x|β) = J j=1
β j ∇c j (x).

(5) Therefore, it is supposed here that all covariates c j are differentiable, and that their gradient can be computed at each point x, either analytically, or by numerical approximation. In most real data sets, the covariate functions c j are measured at discrete points in space. There is generally no analytical form for the gradient, and it is necessary to interpolate the covariate fields so that its gradient can be approximated. In Sections 4.2 and 5, bilinear interpolation is considered to obtain continuous covariate functions. As a consequence of this interpolation, the Langevin movement model cannot handle categorical covariates. Indeed, a categorical covariate field cannot be interpolated into a continuous function, and it is generally not possible to derive a measure of its gradient.

Inference

The continuous-time location process of the animal (X t ) t≥0 is observed discretely at times t 0 < t 1 < • • • < t n , and these observations are denoted by (x 0 , x 1 , . . . , x n ). We consider J spatial covariates c 1 , . . . , c J , measured on a grid over the study region. θ denotes the vector of all parameters of the Langevin movement model defined in Section 2, i.e. θ = (β 1 , . . . , β J , γ 2 ). This section describes an inference method to estimate θ, from telemetry and habitat data.

Euler approximation of the likelihood

The likelihood of the observed locations, given θ, can be expressed using the transition density of the process (X t ) t≥0 . The transition density is the probability density function of the random variable X t+∆ given that X t = x t , and we denote it by q ∆ (x|x t , θ). By the Markov property of the solution to Equation (3), and assuming that the first position is deterministic, the likelihood function is

L(θ; x 0:n ) = n-1 i=0 q ∆i (x i+1 |x i , θ), (6)
where x 0:n is shorthand for the set of observations, and

∆ i = t i+1 -t i .
As discussed in [START_REF] Gloaguen | Stochastic differential equation based on a multimodal potential to model movement data in ecology[END_REF], in many practical cases, the density q ∆ is intractable, and the likelihood L(θ; x 0:n ) cannot be evaluated. To circumvent this problem, pseudo-likelihood approaches can be used as approximations. In these approaches, the intractable transition density in Equation ( 6) is replaced by the p.d.f. of a known distribution (usually, Gaussian), with moments given by a discretisation scheme.

The most common pseudo likelihood approach for discretely observed diffusion is the Euler discretization scheme [START_REF] Iacus | Simulation and inference for stochastic differential equations: with R examples[END_REF]. In the Euler discretization (for d = 2), the transition density of the Langevin diffusion is approximated by the following Gaussian density between t i and t i+1 , for i = 0, . . . , n -1,

X i+1 |{X i = x i } = x i + γ 2 ∆ i 2 ∇ log π(x i |β) + ε i+1 , ε i+1 ind ∼ N 0, γ 2 ∆ i I 2 , (7)
where I 2 is the 2 × 2 identity matrix. Under this approximation, the transition density of the process can then be written

q ∆i (x i+1 |x i , θ) = φ x i+1 x i + γ 2 ∆ i 2 ∇ log π(x i |β); γ 2 ∆ i I 2 ,
where φ(•|µ; Σ) is the p.d.f. of the multivariate normal distribution with mean µ and covariance matrix Σ. This expression can be plugged into Equation ( 6) to obtain the likelihood of a track x 0:n . The Euler discretization can also be used to simulate (approximately) from the Langevin movement model, as illustrated in the simulations of Section 4. The quality of the scheme decreases for longer time steps of simulation [START_REF] Kessler | Statistical methods for stochastic differential equations[END_REF], Chapter 1).

Maximum likelihood estimation

The pseudo-likelihood function could be optimised numerically to obtain estimates of all model parameters. However, if π is modelled with the resource selection function of Equation ( 4), the maximum likelihood estimate θ can simply be obtained using standard linear model equations.

Plugging Equation ( 5) into Equation ( 7), we can write a standard linear model, in the following matrix form. Let Y i = (X i+1 -X i )/ √ ∆ i be the (two-dimensional) normalized random increment of the process between t i and t i+1 , and denote

Y =             Y 0,1 . . . Y n-1,1 Y 0,2 . . . Y n-1,2             , D = 1 2             ∂c1(x0) ∂z1 ∂c2(x0) ∂z1
. . . . . . ∂c J (xn-1)

∂z2 ,            
,

where Y i = (Y i,1 , Y i,2
), and ∂/∂z i denotes the partial derivative with respect to the i-th spatial coordinate. Moreover, let T ∆ be the (2n) × (2n) diagonal matrix with i-th and (n + i)-th diagonal terms equal to ∆ i-1 , for i = 1, . . . , n. Then, the Euler approximation of the Langevin movement model can be rewritten as

Y = (T ∆ D)ν + E, (8)
where E is a 2n-vector of N (0, γ 2 ) variables, and ν = γ 2 β. The estimators for ν and γ 2 are derived from standard linear model theory, as

ν = D T 2 ∆ D -1 D T ∆ Y , and γ2 = 1 2n -J Y -Ŷ 2 ,
where Ŷ = (T ∆ D) ν is the predicted value of Y . In Appendix A, we show that an unbiased estimator of β is then given by

β = (2n -J -2) ν (2n -J)γ 2 ,
with the following covariance formula

Cov( βj , βk ) = 2β j β k 2n -J -4 + Υ jk γ 2 1 + 2 2n -J -4 ,
where Υ jk := (D T 2 ∆ D) -1 jk . Using the asymptotic normality of maximum likelihood estimators, we can obtain a confidence interval of any level α ∈ (0, 0.5), that is valid for a large enough n. For each covariate coefficient β j ,

CI α (β j ) = βj + z α/2 × V( βj ); βj -z α/2 × V( βj )
where z α/2 is the quantile of level α/2 of a standard Gaussian distribution, and V( βj ) = Cov( βj , βj ). The detail of the derivation is given in Appendix A.

The results above derive from the use of a linear model to approximate the solution to a SDE.

For the Langevin movement model based on a RSF, as defined in Section 2, the Euler approximation therefore provides explicit estimates and confidence intervals. Note that the Euler estimator is biased due to the approximation made in Equation ( 7) (see Kessler et al., 2012, Chapter 1). Therefore, both the estimate and the confidence interval must be interpreted with caution, as they depend on the quality of the scheme. The potential use of other discretization schemes is discussed in Section 6.

Simulation study

In this section, we assess the performance of the inference method described in Section 3 in two simulation scenarios. In both cases, we simulate movement tracks from the Langevin process, using the Euler discretization given in Equation 7. We simulate covariates and define an artificial utilisation distribution, expressed as a resource selection function, as shown in Equation ( 4). The objective is to recover the habitat selection parameters {β 1 , . . . , β J } and the speed parameter γ 2 .

Scenario 1

We first consider a fully controlled simulation scenario, where the covariate fields are given by smooth analytical functions. In this idealized case, the gradient of the covariate functions, and thus of the utilisation distribution, can be calculated exactly at any point in the plane. The utilisation distribution π is defined as a RSF (Equation (4)) of three covariates c 1 , c 2 and c 3 , given by

c j (z) = α j exp(-(z -a j ) Σ j (z -a j )) × sin ω j 1 (z 1 -a j 1 ) × sin ω j 2 (z 1 -a j 2 ) , j = 1, 2, c 3 (z) = z 2 ,
where alpha j , a j = (a j 1 , a j 2 ), ω j = (ω j 1 , ω j 2 ), and Σ j = diag{σ j 1 , σ j 2 } are known simulation parameters whose values are given in Appendix B. For the simulations, we choose the resource selection parameters β 1 = -1, β 2 = 0.5, and β 3 = -0.05, and the speed parameter γ 2 = 1.

The first two covariates are smooth functions, for which the gradient can easily be derived. The third covariate is the squared distance to the centre of the map, and is used to include a weak force of attraction towards the centre (here, the point (0, 0), somewhat related to the home range of the individual). These three covariates functions are shown in Figure 2.

Inference was performed independently on 600 data sets. Each data set was a trajectory of 300 points, simulated from the Langevin movement model. The tracks were first generated at a fine time resolution (∆ = 0.01), to minimise the effect of the Euler approximation, and they were then thinned to time intervals of 0.5 time units. We estimated all model parameters using the Euler method, presented in Section 3.2. We considered two different settings: (i) the true analytic gradient is used in the estimation, and (ii) the covariates are discretized on a 8 × 8 regular grid, and the gradient is obtained through the interpolation of the covariates. This second setting corresponds to the more realistic case where covariates are only observed on a discrete grid, and the gradient needs to be approximated. The gradient approximations were performed for the covariates c 1 and c 2 using the R package nloptr [START_REF] Ypma | Ecological scale and seasonal heterogeneity in the spatial behaviors of giant pandas[END_REF]. The gradient of the Euclidean distance c 3 is computed exactly in both cases, as it would be in a real analysis.

Boxplots of the parameter estimates in the 600 replications are shown in Figure 3. All parameters were correctly estimated in this benchmark scenario, even when the covariates were discretized to a coarse grid.

Scenario 2

We considered a second simulation scenario, with randomly-generated covariate fields on a discrete grid, more similar to real environmental data. The main objective of this scenario is to investigate the effect of the sampling frequency on the estimation.

To simulate covariates, we used a procedure similar to that described by [START_REF] Avgar | Integrated step selection analysis: bridging the gap between resource selection and animal movement[END_REF]. We defined a spatial grid over [-50, 50] × [-50, 50], with cells of size 1. For each cell, a random uniform value was generated on (0, 1), and the covariate field was obtained with a two-dimensional moving average filter over a circular region of radius ρ. Here, ρ measures the degree of spatial autocorrelation of the simulated covariate. We then normalized the covariate field, to range between 0 and 1. Using this procedure, we simulated two covariates c 1 and c 2 , with the same autocorrelation parameter ρ = 10. Then, we simulated trajectories from the Euler scheme described in Equation ( 7), with π defined as the (normalized) RSF with coefficients (β 1 , β 2 ) = (2, 4) . Plots of the simulated covariates, and of the utilisation distribution used in the simulations, are shown in Figure 4. We simulated 200 trajectories from the Langevin movement model, at a fine temporal resolution of ∆ = 0.01. We then subsampled each trajectory, for different time resolutions ∆ ∈ {0.01, 0.02, 0.05, 0.1, 0.25, 0.5, 1}, to emulate data sets obtained at different observation rates. From each thinned data set, we kept the first 250 locations of each of the 200 trajectories, leading to a total of 50,000 locations.

We fitted the Langevin movement model to each thinned data set, using the estimators given in Section 3.2. We evaluated the gradients of the covariates at each simulated location with a bilinear interpolation method. This procedure was done using the function interp.surface from the R package fields (Douglas [START_REF] Nychka | fields: Tools for spatial data[END_REF]. We then obtained the approximated gradient of the interpolated covariates with the R package nloptr. Point estimates and 95% confidence intervals of the habitat selection parameters β and the speed parameter γ 2 are displayed in Figure 5.

The estimates of the speed parameter γ 2 were very close to the true value, for all simulation experiments. However, there was a lot of variability in the accuracy and precision of habitat selection parameter estimates. The uncertainty on the estimates of the habitat selection parameters decreased as the time interval increased. This is not surprising: all trajectories had the same number of locations, such that those with longer time intervals explored a larger proportion of the study region. Tracks with longer time intervals therefore covered a larger range of covariate values. Similarly to standard linear model analyses, the uncertainty on the coefficients is larger when the observed range of explanatory variables in Equation ( 8) is narrow. From ∆ = 0.05 to ∆ = 1, the estimates of β 1 and β 2 both decreased as the time interval increases, leading to an underestimation of the parameters for longer time intervals. This is a common problem for the estimation q q q q q q q 0 2 4 6 8 0.01 0.02 0.05 0.10 0.25 0.50 1.00 Time interval β 1 q q q q q q q 0 2 4 6 8 0.01 0.02 0.05 0.10 0.25 0.50 1.00 Time interval β 2 q q q q q q q 0.490 of discretely observed diffusion processes, because the consistency of the estimators requires ∆ to tend towards 0 (for more details, see [START_REF] Kessler | Statistical methods for stochastic differential equations[END_REF]. For long time intervals, the habitat selection parameters are underestimated in absolute value, i.e. the strength of the (positive or negative) effect is underestimated. This bias is a side effect of the underestimation of the speed of the process for long time lags between observations. As the time interval increases, the estimated utilisation distribution becomes flatter, to reflect our growing uncertainty about the effect of the covariates on the short-term movement. In the extreme, for very long time intervals, we would have no information about the selection process, and the estimated utilisation distribution would be flat, corresponding to a uniform distribution of space use over the study region. Note that, although the strength of selection was underestimated in the simulations with long time intervals, the sign of the effect -i.e. selection or avoidance -was always estimated correctly (Figure 5).

To investigate the performance of the method for the analysis of data sets collected at irregular time intervals, we ran a similar experiment where the observations were thinned at random. The results were very similar to the simulations with regular intervals, and are presented in Appendix C. These findings confirm that, due to its continuous-time formulation, the Langevin movement model can directly be used on tracking data collected irregularly.

Example analysis

In this section, we fit the Langevin movement model to a data set described by [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], collected on Steller sea lions (Eumetopias jubatus) in Alaska. The data set comprises three trajectories, obtained from three different individuals, for a total of 2672 Argos locations. The time intervals were highly irregular, with percentiles P 0.025 = 6min, P 0.5 = 1.28h, P 0.975 = 17.4h. In addition to the locations, [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF] provided four spatial covariates over the study region, at a resolution of 1km: bathymetry (c 1 ), slope (c 2 ), distance to sites of interest (c 3 ), and distance to continental shelf (c 4 ). The sites of interest were either haul-out or rookery sites. Maps of the covariates are shown in Figure 6, and we refer the readers to [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF] for more detail about the data set.

To correct for the measurement error, and to follow the preprocessing performed by [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], we first fitted a continuous-time correlated random walk (CTCRW) to the tracks, using the R package crawl [START_REF] Johnson | Continuous-time correlated random walk model for animal telemetry data[END_REF][START_REF] Johnson | crawl: an r package for fitting continuous-cime correlated random walk models to animal movement data[END_REF]. The CTCRW is a continuous-time state-space model, that can be used on irregular and noisy telemetry data. The package crawl implements the Kalman filter for this state-space model. We used the code provided by [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF] to fit the CTCRW to each track, and obtained predicted locations for the times of the observations.

We then fitted the Langevin movement model to the filtered tracks, using the inference method of Section 3. Most of the computation time is needed to evaluate the gradient of each covariate at all observed locations, which took about 1.5 min on a 2GHz i5 CPU. Like in the simulation study of Section 4.2, the covariates were interpolated, so that their gradient could be evaluated at each filtered location. The point estimates and 95% confidence intervals, obtained from the equations of Section 3.2, are given in Table 1. The estimated utilisation distribution, and its logarithm (for comparison with [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], are plotted in Figure 7.

Estimate 95% CI

β 1 1.39 • 10 -4 (-3.87 • 10 -7 , 2.79 • 10 -4 ) β 2 0.12 (-0.14, 0.37) β 3 -2.50 • 10 -5 (-3.58 • 10 -5 , -1.41 • 10 -5 ) β 4

3.47 • 10 -6 (2.05 • 10 -7 , 6.73 • 10 -6 )

Table 1: Maximum likelihood estimates and % confidence intervals obtained with the Euler scheme for the sea lion analysis.

The 95% confidence intervals of the parameters for two of the covariates (bathymetry and slope) include zero, i.e. we cannot draw conclusions about their effects on the sea lions' movement. The effect of the distance to sites of interest, β 3 , was estimated to be negative. This indicates that the model captures the attraction of the sea lions towards the sites of interest (rookeries and haul-out sites). On the other hand, the effect of the distance to the shelf, β 4 , was estimated to be positive. Although this seems to indicate that the animals tend to move away from the shelf, it may also be an artifact caused by the strong collinearity between the covariates c 3 and c 4 . derivatives computed using numerical methods. To compare the Euler and the Ozaki scheme, we repeated the simulation study of Section 4.1, using the Ozaki scheme for the estimation. (The results are not shown here.) The theoretical advantages of the Ozaki scheme were counterbalanced by the need of this secondorder interpolation, and the Euler scheme provided more reliable estimates. Therefore, in the context of the Langevin movement model, the Euler scheme is typically more robust to numerical approximations. To avoid any discretization scheme, it would be interesting to consider exact algorithms for diffusion processes, as suggested in [START_REF] Gloaguen | Stochastic differential equation based on a multimodal potential to model movement data in ecology[END_REF]. However, so far, exact methods of inference have been restricted to a small class of diffusions, and could not be readily applied to the cases presented here.

In the case study of Section 5, we used a two-stage approach to deal with the measurement error. We first fitted a state-space model, the continuous-time correlated random walk, to filter the Argos locations. Then, we fitted the Langevin movement model to the filtered tracks. There are several drawbacks to the two-stage approach. Indeed, it is difficult to propagate the uncertainty from the measurement error (although multiple imputation could be used; see e.g. [START_REF] Scharf | Imputation approaches for animal movement modeling[END_REF]. Besides, the two stages are not consistent, because the first stage ignores the environmental effects that are estimated in the second stage. To avoid this issue, the two steps could be integrated into a state-space model that incorporates measurement error directly on top of the Langevin movement process. The state equation of the full model is given by the transition density of the Langevin movement model, or a discretization of it (like the one given in Equation ( 7)). A natural choice for the observation equation would be xi = x i + η i , where xi is the noisy observed location, x i is the true location, and η i ∼ N (0, σ 2 obs I 2 ) models the measurement error. Under the Euler scheme, the approximate transition density is normal, and a Kalman filter can be used to compute the pseudo-likelihood of this hierarchical state-space model. This extension is conceptually straightforward, and should be done in the near future. We now note that

E[ν j νk ] = Cov(ν j , νk ) + E[ν j ]E[ν k ] = γ 2 Υ jk + β j β k γ 4 ,
where Υ jk := (D T 2 ∆ D) -1 jk . Moreover:

V 1 γ2 =
2(2n -J) 2 γ 4 (2n -J -2) 2 (2n -J -4) as γ 2 (2n -J)γ 2 ∼ Inv-χ 2 (2n -J)

E 1 γ4 = V 1 γ2 + E 1 γ2 2 = (2n -J) 2 γ 4 (2n -J -2) 2 2 2n -J -4 + 1 .
Finally, we can rewrite

Cov( βj , βk ) = V 1 γ2 E[ν j ]E[ν k ] + E 1 γ4 Cov(ν j , νk ) = (2n -J) 2 (2n -J -2) 2 2β j β k 2n -J -4 + Υ jk γ 2 1 + 2 2n -J -4
.

B Simulation parameters

The simulation parameters used in Section 4.1 are given in Table A1.

C Simulation study with irregular intervals

We repeated the second simulation scenario (Section 4.2), with irregular time intervals. Starting from the same full simulated data set (∆ = 0.01), we thinned the observations at random, to obtain irregularlysampled locations. We ran two experiments: (1) mean time interval ∆ = 0.05, (2) mean time interval

Figure 1 :

 1 Figure 1: Left: Artificial utilisation distribution π. Right: Trajectories simulated from the Langevin movement model on π, with two different values of the speed parameter γ 2and , after ∆ = 50 and ∆ = 1000 time units. Although the process with γ 2 = 1 is much slower to explore space, the properties of the Langevin equation guarantee that both processes have the same stationary distribution π.

Figure 2 :

 2 Figure 2: Artificial covariates fields for the simulation scenario of Section . .

Figure 3 :

 3 Figure 3: Estimates for model parameters on experiments replications of scenario . The dotted lines show the real values used in the simulations.

Figure 4 :

 4 Figure 4: Simulated covariate fields c1 and c2, and utilisation distribution obtained with β = (2, 4) . used in the second simulation scenario.

Figure 5 :

 5 Figure 5: Estimates of the habitat selection parameters β and of the speed parameter γ 2 for different observation intervals, in the simulation study of Section . . The vertical lines show % confidence intervals, and the red dotted lines are the true values of the parameters. The x axis is on the log scale.

Figure 6 :

 6 Figure 6: Covariate maps for the sea lion analysis.

  bution of β. We first use the covariance matrix of β,Cov( βj , βk ) = E j ]E[ν k ]by independence of the estimators γ and ν.

Table A1 :

 A1 Simulation parameters for Section . .

Acknowledgements

TM was supported by the Centre for Advanced Biological Modelling at the University of Sheffield, funded by the Leverhulme Trust, award number DS-2014-081. We thank Paul Blackwell for discussions during the early developments of the method.

Data accessibility

The Steller sea lion data set used in Section 5 is provided by [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF].

q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q qq q q q q q q q q q q qq qq q q q qq q q q q q q q q q q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q qq q q qq q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q qq q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q 300 400 500 600 700 -2200

-2000 -1800 -1600 Easting (km) Northing (km) 5.0e-06 1.0e-05 1.5e-05 π q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q qq qq q q q qq q q q q q q q q q q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q qq q q qq q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qqq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q 300 400 500 We presented this analysis as a proof of concept for the Langevin movement model, and several directions could be explored further. As for any regression model, standard model selection criteria such as the AIC could be used to choose the best set of covariates. Following [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], we could also estimate the parameters separately for the three seals, to capture inter-individual heterogeneity.

Discussion

This work introduces a new model of animal movement, based on the Langevin diffusion process, that integrates the movement with space use and habitat selection. The movement model follows the idea of potential-based movement models proposed by [START_REF] Preisler | Modeling animal movements using stochastic differential equations[END_REF] and [START_REF] Brillinger | Handbook of Spatial Statistics, chapter 26. Handbooks of Statistical Methods[END_REF], and it is explicitly connected to the animal's utilisation distribution, from stationarity properties of the Langevin diffusion process [START_REF] Roberts | Exponential convergence of langevin distributions and their discrete approximations[END_REF]. If spatial covariates are available, the long-term (utilisation) distribution can be modelled with a resource selection function, embedded in the movement process, to infer habitat preferences. The Langevin movement model therefore describes animal movement in response to spatial covariates, i.e. step selection. Pseudo-likelihood methods can be used to obtain estimates of the habitat selection parameters in a classical linear model framework, from which an estimated utilisation distribution can be computed. The Langevin movement model is formulated in continuous time, and it can deal with location data collected at irregular time intervals, without the need to interpolate them. Similarly, because it models movement in continuous space (unlike the method presented by [START_REF] Wilson | Estimating animal utilization densities using continuous-time markov chain models[END_REF], the interpretation of the results is not tied to a particular space discretisation.

In this paper, we used the Euler discretization scheme to approximate the likelihood of the model. This scheme is the most widely-used method to carry out inference for discretely-observed diffusion processes, when the transition density is not analytically tractable (see [START_REF] Preisler | Modeling animal movements using stochastic differential equations[END_REF][START_REF] Brillinger | Handbook of Spatial Statistics, chapter 26. Handbooks of Statistical Methods[END_REF][START_REF] Russell | A spatially varying stochastic differential equation model for animal movement[END_REF], for applications in ecology). There exist other pseudo-likelihood approaches, and [START_REF] Gloaguen | Stochastic differential equation based on a multimodal potential to model movement data in ecology[END_REF] argued that better inferences could be obtained with more refined schemes. In particular, they found that the Ozaki discretization provided more reliable results in their applications. However, the Ozaki scheme requires the evaluation of the partial derivatives of the drift, i.e. the (partial) second derivatives of log π in the Langevin movement model. If π is modelled with a resource selection function, then this would require the evaluation of the second derivatives of the covariate fields, from Equation (5). In practice, the covariates must be interpolated to a spatially continuous function, and their second

A Estimators

This appendix proves that β and γ2 are unbiased estimators and derived the confidence intervals for the corresponding parameters.

A.1 Unbiased estimators

We defined ν and γ2 as

Following standard linear model properties, ν ∼ N γ 2 β, D T 2 ∆ D -1 . Besides, Cochran's theorem implies that ν and γ2 are independent and that (2n -J)γ 2 /γ 2 ∼ χ 2 (2n -J), where γ2 is an unbiased estimator of γ 2 . Considering β defined as β = ν/γ 2 , we have

From the properties given above, we have

and we obtain

Thus, to obtain an unbiased estimator of β, we define β := (2n -J -2)/(2n -J) β.

A.2 Confidence intervals

As said above, (2n -J)γ 2 /γ 2 ∼ χ 2 (2n -J), so that a confidence interval for γ 2 is given by CI α (γ) = γ 2n -J q 1-α/2,2n-J ; γ 2n -J q α/2,2n-J , where q α/2,2n-J stands for the quantile of order α/2 from a χ 2 distribution with 2n -J degrees of freedom.

It is also straightforward to derive a confidence interval or even a confidence ellipsoid using the distri-∆ = 0.50. The estimates and 95% confidence intervals of the habitat selection parameters β 1 and β 2 are given in Table A2. We compare them to the results of the simulations conducted in Section 4.2 with regular intervals. The estimates obtained in the simulations with regular and irregular time intervals are very similar, and the standard errors are virtually identical.

regular irregular

Mean interval ∆ = 0.05 ∆ = 0.5 ∆ = 0.05 (SD: 0.045) ∆ = 0.5 (SD: 0.49) This confirms that the continuous-time formulation of the Langevin movement model can accommodate irregular time intervals, without the need to interpolate prior to the analysis.