Spatial and space–time correlations in ecological models

Space–time autoregressive moving average (STARMA) processes can be used to represent a wide range of theoretical models of ecological variation and statistical models for analyzing ecological data. Many discrete-time, discrete-space ecological processes can be analyzed using STARMA theorems. As an example, one focus is on population genetic models, and using STARMA we obtain not only the usual spatial variance and correlations, but also the space–time correlations. Examples show how this allows one to characterize general space–time population genetic processes in a new and more detailed way. STARMA processes include migration-drift models with general patterns of migration among populations. They also include processes with features that are more realistic for many natural population systems, including various forms of stochastic migration. The space–time correlations are particularly important because they allow us to connect data to theoretical processes, and they can be used for estimating migration rates, model-fitting, testing, and forecasting the future behavior of real systems. The space–time correlations also specify the relationship between spatial correlations at different spatial distances, thus bridging gaps between observations on spatial correlations and inferences about dispersal and selection, and other aspects of the underlying space–time process.