Estimating scale-dependent covariate responses using two-dimensional diffusion derived from the SPDE 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 applying a pointwise basis expansion, thereby estimating a dome-shaped or non-parametric “environmental response function”. However, ecological responses integrate across local habitat conditions, such that the 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, i.e., conservation of mass, linear computational time with spatial resolution, and a uniform stationary distribution, where the latter ensures that estimated responses are invariant 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 25 species-maturity combinations, while 20 collapse to the null 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 only 2 of the 20 bird species is the diffused human population density covariate 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.