Partitioning of diversity: the "within communities" component
It is routinely understood that the total diversity within a metacommunity (γ-diversity) can be partitioned into one component summarizing the diversity within communities (α-diversity) and a second component representing the contribution of diversity (or differences) between communities (β-diversity). The underlying thought is that merging differentiated communities should raise the total diversity above the average level of diversity within the communities. The crucial point in this partitioning criterion is set by the notion of "diversity within communities" (DWC) and its relation to the total diversity. The common approach to summarizing DWC is in terms of averages. Yet there are many different ways to average diversity, and not all of these averages stay below the total diversity for every measure of diversity, corrupting the partitioning criterion. This raises the question of whether conceptual properties of diversity measures exist, the fulfillment of which implies that all measures of DWC obey the partitioning criterion. It is shown that the straightforward generalization of the plain counting of types (richness) leads to a generic diversity measure that has the desired properties and, together with its effective numbers, fulfills the partitioning criterion for virtually all of the relevant diversity measures in use. It turns out that the classical focus on DWC (α) and its complement (β as derived from α and γ) in the partitioning of total diversity captures only the apportionment perspective of the distribution of trait diversity over communities (which implies monomorphism within communities at the extreme). The other perspective, differentiation, cannot be assessed appropriately unless an additional level of diversity is introduced that accounts for differences between communities (such as the joint "type-community diversity"). Indices of apportionment IA (among which is GST and specially normalized versions of β) and differentiation ID are inferred, and it is demonstrated that conclusions derived from IA depend considerably on the measure of diversity to which it is applied, and that in most cases an assessment of the distribution of diversity over communities requires additional computation of ID.