Using spatial gradient analysis to clarify species distributions with application to South African protea

Abstract

Typical ecological gradient analyses consider variation in the response of plants along a gradient of covariate values, but generally constrain themselves to predetermined response curves and ignore spatial autocorrelation. In this paper, we develop a formal spatial gradient analysis. We adopt the mathematical definition of gradients as directional rates of change with regard to a spatial surface. We view both the response and the covariate as spatial surfaces over a region of interest with respective gradient behavior. The gradient analysis we propose enables local comparison of these gradients. At any spatial location, we compare the behavior of the response surface with the behavior of the covariate surface to provide a novel form of sensitivity analysis. More precisely, we first fit a joint hierarchical Bayesian spatial model for a response variable and an environmental covariate. Then, after model fitting, at a given location, for each variable, we can obtain the posterior distribution of the derivative in any direction. We use these distributions to compute spatial sensitivities and angular discrepancies enabling a more detailed picture of the spatial nature of the response–covariate relationship. This methodology is illustrated using species presence probability as a response to elevation for two species of South African protea. We also offer a comparison with sensitivity analysis using geographically weighted regression. We show that the spatial gradient analysis allows for more extensive inference and provides a much richer description of the spatially varying relationships.