Retaining the double dimension of species diversity: Application of partially ordered set theory and Hasse diagrams

The measurement of species diversity has been a central task of community ecology from the mid 20th century onward. The conventional method of designing a diversity index is to combine values for species richness and assemblage evenness into a single composite score. The literature abounds with such indices. Each index weights richness and evenness in a different fashion. The conventional approach has repeatedly been criticized since there is an infinite number of potential indices which have a minimum value when S (species richness) = 1 and a maximum value when S = N (number of individuals). We argue that partial order theory is a sound mathematical fundament and demonstrate that it is an attractive alternative for comparing and ranking biological diversity without the necessity of combining values for species richness and evenness into an ambiguous diversity index. The general principle of partial ordering is simple: one particular assemblage is regarded as more diverse than another when both its species richness and its evenness are higher. Assemblages are not comparable with each other when one has a higher value for species richness and a lower value for evenness. Hasse diagrams can graphically represent partially ordered communities. Linear extensions and rank-frequency distributions reveal the potential of partial order theory as a means to support decisions when assemblage ranking is desired.