Fuera de la jaula, la jaula

I have briefly introduced three PDE techniques for scaling up individual move- ment kernels to population distributions. These three methods are Patlak’s ap- proach (Patlak, 1953) and two recently reviewed methods, the Hyperbolic Scal- ing and Moment Closure methods (Hillen and Painter, 2013). There are two approaches to evaluate the accuracy of using the PDE methods to approximate patterns arising from the movement kernel. One is to compare the transient distri- butions derived from the PDEs in Equations (2.3), (2.9), (2.13) to the movement kernel, while the other is to compare the steady-state distributions in Equations (2.8), (2.12) and (2.18) to the long-term distribution obtained by the Master Equation (Equation 2.21).

By analysing a simple example of a biased random walk, I have demonstrated a comparison between the movement kernel and transient distributions derived by the PDE methods. Applying Patlak’s approach to analyse this example results in a very poor approximation with an overestimated variance, while the Hyperbolic Scaling and Moment Closure methods correctly describe the movement kernel (Figure 2.1).

To understand how PDE techniques performs when estimating the long-term dis- tribution, I have examined three central-place foraging models, characterised by mean velocity functions with different levels of smoothness. When analysing a non-smooth movement kernel, all PDE methods give poor approximations to the long-term distribution (Figures 2.3, 2.6). On the other hand, when considering the model with a differentiable mean velocity function, the accuracy of approx- imations improves and the Moment Closure methods performs better than the others in the range studied (Figures 2.8d, 2.9).

In addition, I have investigated some simple examples of movement in heteroge- neous environments and shown similar results to those observed in central-place foraging models. That is, the PDE methods provide poor approximations if the movement kernel is non-smooth and perform well for smooth models (Figure 2.10). For smooth movement kernels, the Moment Closure method outperforms others (Figure 2.11).

Resource selection analysis by

continuous-time movement

models

Resource selection analysis (RSA) is a fundamental tool for understanding mecha- nisms behind abundance and distributions. It has been strengthened by methods such as step selection analysis (SSA) and integrated step selection analysis (iSSA) because the incorporation of movements makes it straightforward to define the availability of a resource unit by mobility. However, both SSA and iSSA rely on a discrete-time movement model, subject to fixed observation intervals. There- fore, RSA can be improved even further by integrating resource selection into a continuous-time movement modelling framework rather than a discrete-time framework. Moreover, SSA and iSSA compare a used ‘step’, defined by two suc- cessive observations, to some available steps starting with the same source point as the used step. This means SSA and iSSA consider selection at the scale of steps and hence depend on the assumption of a correspondence between decision making and observation scales. By considering movements in continuous time, it is straightforward to deal with irregularly collected data and allow changes in movement decisions. This chapter introduces such a modelling framework on the assumption that animals would be attracted to the place with the best resource quality.

§ 3.1 Modelling framework

Using the OU process as a building block as in Blackwell et al. (2016), I construct a switching OU process, composed of a set of OU processes, each of which repre- sents a random walk towards a target place, or an attraction centre, different to others. That is, as the destination of movement switches because of the change of resources, the OU process used is switched to another OU process which models movement attracted to the new target place. In this way, I model movements in response to the change of resources over time.

Assuming an animal is moving in a 2-dimensional space and its location at time t is x(t), the OU process gives the probability of the animal’s location in time τ as follows:

x(t + τ )|x(t) ∼ M V N (µ(t) + eBτ(x(t) − µ(t)), Λ − eBτΛeB0τ), (3.1)

where M V N refers to “multi-variate normal” distribution and here the 2-dimensional version is considered. The attraction centre at time t is µ(t). The tendency to- wards the attraction centre and uncertainty of the movement are controlled by a 2 × 2 matrix B and the 2 × 2 covariance matrix Λ respectively. Here, I assume B = −bI and Λ = vI with b, v > 0 and I the 2 × 2 identity matrix. The 0s off the diagonal of Λ indicates no correlation between the two coordinates. The parameter b governs the strength of the drift towards the central point, while the parameter v determines the range of strolling around the centre (Blackwell, 1997). Equation (3.1) is a continuous-time analogue of a movement kernel such as those in Chapter 2 but time τ here can be any value rather than being fixed. Here, I assume that the attraction centre µ(t) in Equation (3.1) is decided by using a weighting function to assess the attractiveness of locations in space. A resource selection function (RSF) is commonly used for this purpose as it reflects the probability of an animal using a resource unit in space. It is usually formulated by an exponential function such as (Boyce et al., 2002)

where x is a resource unit, which can be an area in space. The factors influencing movement decisions are incorporated in the vector of predictor covariates, z(x) = (z1(x), · · · , zk(x)), with coefficients β1, · · · , βk representing the extent to which these factors affect movements. Possible drivers of movement can be categorical such as vegetation types, or continuous such as distance to human constructions, etc. (Manly et al., 2002).

I also assume that the animal has complete knowledge of the environment and determines its movement centre µ(t) at time t by selecting the most attractive place in space. That is, the target place is decided by comparing the RSF in Equation (3.2) at all potential destinations and given by (cf. Avgar et al. (2017), Bastille-Rousseau et al. (2017))

µ(t) = µi where w(z(µi)) = max

j∈Ω w(z(µj)). (3.3)

Here, the attraction centre, µi, is the centre of a resource unit, which may be a food patch of any shape. The notation Ω stands for the finite collection of all resource units considered. If the attraction centre determined by Equation (3.3) is not unique because of the equal attractiveness of more than one resource units, then further steps are required to make a decision. For example, randomly select one of the most attractive places. Alternatively, it may be feasible to exclude some minor factors from the RSF rather than considering all possible factors. However, in real life, µi will almost always be unique if the RSF involves continuous covariates.

Note that in this modelling framework, reassessment of the movement centre can occur at any given time. Therefore, to include points where the reassessment might happen in the inference procedure, a Poisson process is used to simulate such points between observations (Blackwell et al., 2016). This is similar to a velocity jump process, which is also a continuous-time movement model and assumes the change of velocity has a Poisson distribution in time (Othmer et al., 1988). While the change of velocity in a velocity jump process relies on a turning kernel, the change of attraction centre in a switching OU process is determined by a RSF. Although it is possible to develop a velocity jump process which describe movement biased towards an attraction centre, it is straightforward to use OU

processes as they intrinsically describe this bias.