Isolation by distance, trend surface analysis, and spatial autocorrelation

The isolation by distance model is both a population process and a surface model. In this model the surface is, on average, flat in every direction. By contrast, probably most observed genetic surfaces exhibit trends generated by complex long-distance populational processes. When one estimates the parameters of a Malécot-Morton equation for those surfaces, the isolation by distance model does not fit. In the simplest case, in first-order trends (two-dimensional clines) the genetic differentiation increases dramatically, faster per unit distance than it would by isolation by distance alone. When isolation by distance takes place but is hidden through the apparent and complicated relief of a surface, another surface model incorporating trend spatial analysis can bypass the difficulty of estimating the isolation by distance process if approached through the Malécot-Morton equation or through a measure of spatial autocorrelation.