Understanding animal habitat preferences and movement patterns is key to effective conservation. We develop new types of spatial and movement models (see also Movement Models).

Shown in the mural is a linear Stochastic Partial differential equation (SPDE). We use this equation for some of the spatial modelling approaches. The solution of this equation is used in spatial and spatio-temporal models to account for dependence inherent in data.

This leads particularly flexible modelling that is is still computationally feasible. For example, we can use this approach to model directly on the surface of the earth rather than having to project onto a flat surface first, avoiding the distortions that come with any type of projection.

The tricky part is solving this equation for *x* in the continuous spatial domain which is computationally expensive.
A good approximation of the (distribution of the) solutions to this continuous space SPDE is obtained by using the finite element method,
taking the advantage of the connection between Gaussian Markov random fields (GMRFs) of graphs and SPDEs in continuous space.
The SPDE approach provides us with the computational advantage of having sparse precision matrices resulting from the Markovian property of GMRFs.

Some of the research conducted at CREEM includes developing methods to assess potential impacts of wind farm construction in offshore areas. This led to the development of the MRSea R Package which allows the fitting of spatially adaptive one and two dimensional smooths in regions of complex topography (see our spline page) and for correlated model residuals.

More recently (2017), the Scottish Government commissioned the work of Mackenzie, Scott-Hayward, Paxton and Burt to develop spatially explicit power analysis methods and software (resulting in the R package MRSeaPower) to allow a wide range of users to carry out spatially explicit power analyses. This allows the used to see what the chance of detecting a particular change in animal numbers is, given the current survey regime and distribution of animals. For example, you might wish to find out the power to detect a 25% decline in the population.

This image shows that if you want 80% power to detect a change for this particular survey/animal, then you would also need to accept that 8.6% of the time you will detect a change when there isn't one.

Humans are excellent at visually comparing images and identifying differences between them. However, they are pretty bad at quantifying those differences. In ecology, comparing maps is often required, for instance when comparing changes in species distributions over time. When comparing ecological maps, you need to account for interdependencies between cells (abundance of animals in one grid cell may be related to the abundance of animals in the next grid cell). We used a methodology originally developed in computer science to assess the quality of JPEG image compression, and enhanced it for application to ecological problems. We enhanced the Structural Similarity Index (SSIM) to include uncertainty from the underlying maps being compared, and account for edge effects.

An advantage of the comparison algorithm is that different levels of summary metrics can be used. We used a case study of sperm whales in the Mediterranean Sea to compare groups and singleton assemblages and show patterns in spatial structure that could not be identified visually. A key finding was that groups (likely females) and single animals (likely males) interacted in a mutually exclusive way, in an area where there were high abundances of both assemblages. The open-access paper Novel application of a quantitative spatial comparison tool to species distribution data is published in Ecological Indicators, and the supplementary information includes self-sufficient R code and data.

Sperm whale (*Physeter microcephalus*).

Map comparison between the predicted probability of occurrences of group & singleton sperm whales.

Efficiently modelling non-stationarity in spatial ecological models (EMNS) is a feasibility project funded by the Engineering and Physical Sciences Research Council,
facilitated by SECURE, University of Glasgow. I am the Principal Investigator of the project, which is a collaboration between some fantastic colleagues in Norway, the Netherlands, South Africa, and UK, including Dr Janine Illian and Dr Sophie Smout. The project aims to characterise distributions of species from movement data in a concise analytical framework, and use non stationary random fields. We are using black eagle (*Aquila verreauxii*) and grey seal (*Halichoerus grypus*) data as case studies for the project, and fitting models
using Integrated Nested Laplace Approximations (INLA) and Stochastic Partial Differential Equations (SPDE) approach in R-INLA. A poster outlining some of the project's aims can be found here.

Black eagle (*Auila verreauxii*). Photo credit: Mario Moreno.

CREEM Software Downloads: MRSea Package

Research by Dr. Lindesay Scott-Hayward

Research by Dr. Monique Mackenzie