Modelling Killer Whale Feeding Behaviour Using a Spatially Adaptive Complex Region Spatial Smoother (CReSS) and Generalised Estimating Equations (GEEs)

Abstract

To develop appropriate spatial conservation planning for individual species, it is important to understand their habitat requirements and in particular to identify areas where critical life-history process such as breeding, weaning or feeding take place. The process of defining critical habitat often ignores behavioural aspects of animal distribution, which for highly migratory species like baleen whales whose feeding and breeding grounds are clearly demarcated and widely separated is not a problem. However, for other species like the endangered ‘Eastern North Pacific southern resident’ killer whale stock, critical life-history processes occur in the same waters. This killer whale stock lives in a topographically complex region (many islands) off the west coast of Canada/USA, which makes accurate mapping of densities or behaviours difficult using traditional generalised additive models. We present results on the spatial distribution of southern resident killer whale feeding grounds in 2006, using a binomial, complex region spatial smoothing model within a generalised estimating equation framework, which allows for both complex topography and correlated residuals. The model performs well and suggests a region to the south of San Juan Island as an area with a high probability of feeding, which could not have been as accurately established from a more traditional presence–absence model. We also calculate estimates of precision, which other studies did not include, enabling more informed management decisions for spatial conservation planning. A vignette containing the code along with an R workspace and function file is provided to allow the user to fit the models presented in this paper.

Supplementary materials accompanying this paper appear on-line.

Appendices

Appendix 1: Calculation for Minimum and Maximum Range Parameter

The following was the formula used to find the minimum and maximum values for parameter r:

$$\begin{aligned} \left( r_{min}=\sqrt{\frac{\bar{d}}{-\text {log}(0.1)}}, r_{max}=\sqrt{\frac{\bar{d}}{-\text {log}(0.99)}}\right) \end{aligned}$$

(6)

where $\bar{d}$ is the mean distance between all possible knot locations and all data locations. The values returned represent the value for r that gives a basis function value of 0.1 and 0.99 at the distance given. These give an appropriately local basis (using $r_{min}$) and global basis ($r_{max}$). Figure 5 shows an example basis using the minimum, maximum and middling value for r.

Fig. 5

Graphic showing a single basis (s) with three different values for the range parameter, $r_s$. Small values of $r_s$ lead to local bases and large values to global ones (Color figure online).

Appendix 2: Autocorrelation

See Fig. 6.

Fig. 6

Autocorrelation plot for the model residuals. The grey lines represent the correlation for each individual panel.

Appendix 3: Block-Diagonal Error Matrix

Example of a block-diagonal error matrix for 2 panels and 4 observations in each using an AR(1) model for the correlation structure:

$\left[ \begin{array}{cccccccc} 1 &{} \rho &{} \rho ^2 &{} \rho ^3 &{} 0 &{} 0 &{} 0 &{} 0 \\ \rho &{} 1 &{} \rho &{} \rho ^2 &{} 0 &{} 0 &{} 0 &{} 0 \\ \rho ^2 &{} \rho &{} 1 &{} \rho &{} 0 &{} 0 &{} 0 &{} 0 \\ \rho ^3 &{} \rho ^2 &{} \rho &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} \rho &{} \rho ^2 &{} \rho ^3 \\ 0 &{} 0 &{} 0 &{} 0 &{} \rho &{} 1 &{} \rho &{} \rho ^2 \\ 0 &{} 0 &{} 0 &{} 0 &{} \rho ^2 &{} \rho &{} 1 &{} \rho \\ 0 &{} 0 &{} 0 &{} 0 &{} \rho ^3 &{} \rho ^2 &{} \rho &{} 1\\ \end{array}\right]$

The AR(1) model sees the correlation between residuals decay as the time interval between observations increases.

Appendix 4: Supplementary Material

A vignette containing the code along with an R workspace and function file is provided to allow the user to fit the models presented in this paper.