Non-stationary Extensions of the Diffusion-Based Gaussian Matérn Field for Ecological Applications

Abstract

The use of statistical methods informed by partial differential equations (PDEs) and in particular reaction–diffusion PDEs such as ecological diffusion equations (EDEs) has been studied and used to model spatiotemporal processes. In this paper, we consider a stochastic extension of the EDE (SEDE) and discuss its interpretation and main differences from the deterministic EDE. We then leverage a non-stationary extension of the diffusion-based Gaussian Matérn field and show that this extension has SEDE-like behavior. The elucidated connection enables us to find a finite element approximated solution for SEDEs by means of the stochastic partial differential equation (SPDE) Bayesian method. For illustration, we analyze the evolution of white-nose syndrome (WNS) in the continental USA, comparing two models: stationary SEDE and a non-stationary pseudo-SEDE. Our results demonstrate the importance of non-stationarity in wildlife disease modeling and identify spatial explanatory variables for the non-stationarity in the WNS process. Finally, a simulation study is conducted to assess the deviance information criterion for differentiating from the two models, as well as the identifiability of the model parameters.Supplementary materials accompanying this paper appear online.