Decompositions of Price's formula in an inhomogeneous population structure

The central tool for the study of allele frequency change due to selection is the remarkably simple but powerful formula of Price [Nature 227 (1970) 520]. Here, I provide what might be called a structural analysis of this formula. The formula essentially accumulates the average allele frequency change over many instances of a fitness-determining interaction, but there are different ways of organizing this average and these lead to quite different computational algorithms. I present three of these: an analysis by population state, an analysis by recipient and an analysis by actor. A comparison of these can lead to a heightened understanding of the different factors behind selective allele frequency change. In particular, I pay attention to the effects of structural inhomogeneity on reproductive value (RV) and emphasize that Price's formula measures RV-weighted allele frequency change. I examine in detail a simple example as a crucial way of cementing the different theoretical pathways. My aim was to produce a simple transparent presentation and therefore I work with a simple population structure and have omitted a number of technical details that are found elsewhere.