Diffusion-enhanced spatio-temporal models for estimating spatial expansion

Spatial expansion is a common feature of many ecological processes, including biological invasions, range shifts, population recovery, and the spread of pathogens and human activities. Although such expansion often arises from diffusive processes, there remain few general statistical methods to quantify these dynamics. Here, we present a diffusion-enhanced spatio-temporal model (DESTM) that approximates diffusive process using Gaussian Markov random fields (GMRF). The model works by incorporating the statistical interaction between spatial diffusion and temporal lags into a joint precision structure of the latent field. To demonstrate, we simulate data from a diffusion movement process using a continuous-time Markov chain (CTMC), and then approximate this mechanistic process using the DESTM. We show that separable spatio-temporal models can lead to large bias in estimates of the local density. In contrast, the DESTM did not appear to bias inferences, and reverts to the separable model when diffusion is not apparent. Finally, we apply the model to two large-scale examples of spatial expansion: (1) the historical spread of the Japanese distant-water longline fleet across the Pacific, representing anthropogenic spatial expansion in the ocean, and (2) the recovery and geographic expansion of Bald Eagle populations across North America. Estimates from both case studies, supported by model comparisons and ecological plausibility, suggest that the DESTM provides a better description of the underlying expansion dynamics, although improvements in forecast performance were more apparent for the Japanese longline fleet. We highlight that the model provides a fast and efficient framework for modeling spatial expansion dynamics, with the potential to represent increasingly complex ecological dynamics through future extensions. We propose that future research explore this framework as a quantitative tool to explain how spatial expansion occurs.