Effective numbers in the partitioning of biological diversity.

Admissible measures of diversity allow specification of the number of types (species, alleles, etc.) that are "effectively" involved in producing the diversity (the "diversity effective number", also referred to as "true diversity") of a community or population. In metacommunities, effective numbers additionally serve in partitioning the total diversity (symbolized by γ) into one component summarizing the diversity within communities (symbolized by α) and an independent component summarizing the differences between communities (symbolized by β). There is growing consensus that the β-component should be treated in terms of an effective number of "distinct" communities in the metacommunity. Yet, the notion of distinctness is shown in the present paper to remain conceptually ambiguous at least with respect to the diversity within the "distinct" communities. To overcome this ambiguity and to provide the means for designing further desirable effective numbers, a new approach is taken that involves a generalized concept of effective number. The approach relies on first specifying the distributional characteristics of partitioning diversity among communities (among which are differentiation, where the same types tend to occur in the same communities, and apportionment, where different types tend to occur in different communities), then developing the indices which measure these characteristics, and finally inferring the effective numbers from these indices.

MAJOR RESULTS: (1) The β-component reflects apportionment characteristics of metacommunity structure and is quantified by the "apportionment effective number" of communities (number of effectively monomorphic communities). Since differentiation between communities arises only as a side effect of apportionment, the common interpretation of the β-component in terms of differentiation is unwarranted. (2) Multiplicative as well as additive methods of partitioning the total type diversity (γ) involve apportionment effective numbers of communities that are based on different apportionment indices. (3) "Differentiation effective numbers" of communities exist but do not conform with the classical concept of partitioning total type diversity into components within and between communities. (4) Differentiation characteristics are measured as effective numbers of distinct types (rather than communities) from the dual perspective, in which the roles of type and community membership are exchanged. This is relevant e.g. in studies of endemism and competitive exclusion. (5) For Shannon-Wiener diversity, all of the differentiation and apportionment effective numbers are equal, with the exception of those representing additive partitioning. (6) Under either perspective, that is dual or non-dual, measures of compositional differentiation (as originally suggested for the assessment of β-diversity) do not figure in the partitioning of total diversity into components, since they do not build on the intrinsic concept of diversity.