Biological fitness and the Price Equation in class-structured populations

Price׳s extended covariance selection mathematics is applied to class-structured populations with additional assumptions, to derive the 'genetic Price Equation with class structure'. Each individual belongs to a class, and there may be overlapping generations; the equation is genetic because the trait is restricted to an arbitrary weighted sum of allele frequencies. Two special cases are then considered, a demography-like case corresponding to Fisher's fundamental theorem of natural selection, and a sex-ratio-like case corresponding to Fisher's sex ratio argument: these differ in whether it is natural to assume that the per-capita or the total reproductive values of each class are maintained from the parental to the descendant population. These cases also match the two existing attempts to eliminate from the effects of natural selection those passive changes in allele frequencies that are caused by the class structure, and suggest improvements in one of them. In each case a more specialised Price Equation, and a 'fundamental theorem of natural selection', are proved, which hold out of class-structure equilibrium, showing that passive changes can be eliminated in more than one way, and hinting at the possibility of a more general formulation. Previous class-structured Price Equations and a 'fundamental theorem' are linked to these results. The power of Price's formal approach is vividly illustrated by this lucid conspectus of otherwise self-standing theories with confusing interconnections.