Structural aspects of evenness and equalness

In ecology and evolution, measurement of uniformity or non-uniformity of variation aids in understanding adaptation and system stability. Despite the intricacy of problems addressed, a single concept and method of analysis prevails in the assessment of uniformity, namely evenness. Relevant entities of analysis, however, range from individuals, their genetic type or phenotype, pairs of these, to assemblages at various levels of organization. Entities, in turn, may be represented by frequencies/abundances, area occupied, pair differences, or diversity (of assemblages). The evenness concept views inequality in representation in terms of dominance of single entities (unevenness), but it does not cover the omnipresent phenomenon of differentness or heterogeneity as it appears in intermediate representations (unequalness). Here the concepts of evenness and equalness are expanded for application to more kinds of entities and their representations and related to common notions of equability and regularity. Beyond frequency distributions, a coherent approach to variability in similarity or difference of pairs of entities (pair differences) enables inclusion of facets of structure. Commensurability among all notions is then reached by specification of appropriate indicators and indices. Although evenness and equalness both address ideas of uniformity, they can lead to divergent and partially unexpected but consistent conclusions in their various applications. These arise especially for dominance and heterogeneity in representation as distinct deviations from ideal uniformity. Among such conclusions are: • the evaluation of complex functional traits for uniformity characteristics yields partially inconsistent results when based on the evenness concept, while this inconsistency does not arise under the equalness concept; • equalness indices consistently quantify regularity (branching symmetry) in the topology of phylogenies, while assessment of the uniformity of branch lengths and related metric characteristics of phylogenies is largely independent of topological characteristics; • for clines considering difference gradients for complex characters, analysis of uniformity at different levels shows that large degrees of uniformity can be realized only when the entity representations are difference increments; • the evenness variant of (non-)uniformity shows little sensitivity to specifically structural characteristics. • Evenness and the recent concept of equalness describe two basic variants of uniformity. • The variants differ in that evenness focuses on dominance and equalness on heterogeneity. • In the assessment of structural uniformity, equalness rather than evenness is the relevant variant. • In phylogenies, topological and metric uniformity must be treated separately. • Clinal uniformity is revealed by using difference increments along transects.