Computing additive ββ-diversity from presence and absence scores: A critique and alternative parameters

Whittaker first proposed to measure the variation in species composition among plots or ββ-diversity as the ratio between regional diversity (γγ-diversity) and average local diversity (αα-diversity). More recently, an alternative way of partitioning diversity for which ββ-diversity is obtained as the difference between γγ-diversity and average αα-diversity has become very popular for linking the structure of species assemblages to ecosystem functioning in a spatially explicit manner. Unfortunately, additive ββ-diversity computed from species presences and absences suffers from the major drawback of being dependent on regional species richness. For instance, if the separation between ββ-diversity and γγ-diversity is incomplete, so that variation in species composition is affected by species richness, then differences in ββ-diversity values among different sets of plots could reflect differences in the species count rather than any fundamental difference in species composition among the plots. Based on the above observation, in this paper I will first propose a basic requirement for ββ-diversity measures that adequately captures our intuitive notion of independence of species richness. Next, I will show that additive ββ-diversity computed from species presence and absence scores can be interpreted within the framework of fuzzy set theory. Finally, based on this unusual “fuzzy” interpretation of additive ββ-diversity, I will introduce two families of parametric ββ-diversity measures whose members have varying sensitivities to the presence of rare and frequent species.