Factorization of joint metacommunity diversity into its marginal components: an alternative to the partitioning of trait diversity

Diversity in metacommunities is traditionally viewed to consist of the diversity within communities ( $$\alpha$$ α ) that is complemented by the differences between communities ( $$\beta$$ β ) so as to result in the total diversity ( $$\gamma$$ γ ) of the metacommunity. This perception of the partitioning of diversity, where $$\beta$$ β is a function of $$\gamma$$ γ and $$\alpha$$ α (usually $$\beta =\gamma /\alpha$$ β = γ / α with all components specified as effective numbers), has several drawbacks, among which are (1)  $$\alpha$$ α is an average that can be taken over communities in many ways, (2) complete differentiation among communities cannot always be uniquely inferred from $$\alpha$$ α and $$\gamma$$ γ , (3) different interpretations of $$\beta$$ β as effective number of communities (e.g., distinct or monomorphic) are possible, depending on the choice of ideal situations to which the respective effective numbers refer, and (4) associations between types (species, genotypes, etc.) and community affiliations of individuals are not explicitly covered by $$\alpha$$ α and $$\gamma$$ γ . Item (4) deserves special regard when quantifying metacommunity diversity. It is argued that this requires consideration of the joint distribution of type-community combinations together with its diversity (joint diversity) and its constituent components: type and community affiliation. The quantification of both components can be affected by their association as realized in the joint distribution. It is shown that under this perception, the joint diversity can be factorized into a leading and an associated component, where the first characterizes the minimum number of communities required to obtain the observed joint diversity given the observed type distribution, and the second specifies the effective number of types represented in the minimally required number of communities. Multiplication of the two yields the joint diversity. Interchanging the roles of community and type, one arrives at the dual factorization with leading minimum number of types and associated effective number of communities. The two dual factorizations are unambiguously defined for all measures of diversity and can be used, for example, to indicate structural characteristics of metacommunities, such as type differentiation among communities and associated type polymorphism. The information gain of the factorization approach is pointed out in comparison with the classical and more recent modified approaches to partitioning total type diversity into diversity within and between communities. The use of factorization in analyses of latent community subdivision is indicated.