Estimating scale-dependent covariate responses using two-dimensional diffusion derived from the stochastic partial differential equation method

Species distribution models (SDMs) are widely used to standardize spatially unbalanced data, project climate impacts and identify habitat for conservation. SDMs typically estimate the impact of local environmental conditions by estimating a dome-shaped or non-parametric 'environmental response function'. However, ecological responses often integrate across local habitat conditions, such that species density depends on habitat at the location of sampling but also at nearby locations. To address this, we extend methods from the stochastic partial differential equation (SPDE) method that is widely used in INLA, which approximates spatial correlations based on local diffusion over a finite-element mesh (FEM). We specifically introduce the sparse inverse-diffusion operator on a FEM and apply this operator to covariates to efficiently calculate a spatially weighted average of local habitat that is then passed through pointwise basis expansion to predict species densities. We show that this operator has several useful properties, that is conservation of mass, efficient scaling of computational time with spatial resolution, and invariance to linear (scale and offset) transformations of covariates. We test this covariate-diffusion method using a simulation experiment and show that it can correctly recover a non-local environmental response while collapsing to a local (pointwise) response when warranted. We apply it to monitoring data for 25 bottom-associated fishes in the eastern Bering Sea and 20 bird species in the western United States. This application confirms that non-local responses in the eastern Bering Sea case study are parsimonious for 26 species-maturity combinations, while 18 collapse to the pointwise method. Estimates suggest that some species-maturity combinations avoid proximity to the continental slope, beyond what is predicted by local bathymetry in isolation. By contrast, in four of the 20 bird species the diffused human population density covariate is more parsimonious than the original covariate. The covariate-diffusion method introduced here constitutes a fast and efficient approach to modelling non-local covariate effects. This flexible method may be useful in cases when covariates influence nearby population densities, for instance due to movement of the sampled species or its important biological or physical drivers.