On the notion of dispersion: from dispersion to diversity

Dispersion and diversity are two variants of the notion of biological variation that are often not properly distinguished even though they address intrinsically disparate aspects. Dispersion focuses on the assessment of (variable) differences among individuals or types, while diversity focuses on the assessment of numbers of distinct types. Here, ‘type’ is a proxy for biological entities such as species, genotype and phenotype culminating in individuals at the highest level of resolution. The present paper bridges the apparent gap between dispersion and diversity by showing how numbers of types and individuals can be treated as a dispersion characteristic (the number-characteristic). Drawing from the general concept of effective quantities, it is shown that the contribution of difference arrays to dispersion measures can always be summarized in terms of an effective difference. If the number-characteristic is realized in addition, numbers of types (of individuals) can be determined that are effectively distinct at specified levels of difference. Whereas measures of diversity that replace abundances of types by differences of each type from other types can also be considered as numbers of effectively different types, they do not explicitly involve levels of distinctness among types. By contrast, dispersion effective numbers of types gain their sole validity from explicit reference to the degree to which types effectively differ from each other. In particular, dispersion effective numbers obey the diversity criterion only at levels of distinctness equal to or greater than the effective difference in that they do not exceed the actual number of types and become equal to that number only for uniform type distributions. The corresponding ‘proper dispersion effective number’ thus sets an upper limit to the effective number of types in a community and meets the characteristics of a diversity effective number. Examples are outlined that have broader application in phylogenetics, population genetics and ecology.