Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao's quadratic index

The diversity of a species assemblage has been studied extensively for many decades in relation to its possible connection with ecosystem functioning and organization. In this view most diversity measures, such as Shannon's entropy, rely upon information theory as a basis for the quantification of diversity. Also, traditional diversity measures are computed using species relative abundances and cannot account for the ecological differences between species. Rao first proposed a diversity index, termed quadratic diversity (Q) that incorporates both species relative abundances and pairwise distances between species. Quadratic diversity is traditionally defined as the expected distance between two randomly selected individuals. In this paper, we show that quadratic diversity can be interpreted as the expected conflict among the species of a given assemblage. From this unusual interpretation, it naturally follows that Rao's Q can be related to the Shannon entropy through a generalized version of the Tsallis parametric entropy.     

A parametric diversity measure combining the relative abundances and taxonomic distinctiveness of species

Most ecological diversity indices summarize the information about the relative abundances of species without reflecting taxonomic differences between species. Nevertheless, in environmental conservation practice, data on species abundances are mostly irrelevant and generally unknown. In such cases, to summarize the conservation value of a given site, so‐called ‘taxonomic diversity’ measures can be used. Such measures are based on taxonomic relations among species and ignore species relative abundances. In this paper, bridging the gap between traditional biodiversity measures and taxonomic diversity measures, I introduce a parametric diversity index that combines species relative abundances with their taxonomic distinctiveness. Due to the parametric nature of the proposed index, the contribution of rare and abundant species to each diversity measure is explicit.     

Strong requirements for weak diversities

Weikard et al . ( Diversity and Distributions , 12 , 215–217) show that the taxonomic diversity measure proposed by Ricotta (2004) violates 'weak species monotonicity'. This condition requires that the addition of a species to a given species set should always increase diversity if abundances change only marginally. They further propose a new taxonomic diversity index that overcomes this drawback. In this paper, some statistical properties of this new diversity index are briefly analysed.     

Quantifying the effects of nutrient addition on the taxonomic distinctness of serpentine vegetation

Traditional diversity indices summarize the information about the relative abundances of species within a community without regard to differences between species. However, intuitively, a community composed of dissimilar taxa is more diverse than a community composed of more similar taxa. Therefore, useful indices of diversity should account for taxonomic relations among species. In this paper, a new parametric diversity index that combines species relative abundances and their taxonomic distinctiveness is used to quantify the way in which soil fertilization affects the diversity of a garigue community on ultramafic soils of Tuscany (central Italy). Results show that, while ultramafic soils generally host plant communities of limited taxonomic diversity with respect to similar communities on other substrates, fertilization significantly enhances the biomass production of species that are not exclusive to ultramafic soils. As a consequence, if diversity is measured combining species relative abundances with their taxonomic distinctiveness, nutrient addition tends to increase the diversity of ultramafic communities.     

Phylogenetic diversity metrics for ecological communities: integrating species richness, abundance and evolutionary history

Phylogenetic information is increasingly being used to understand the assembly of biological communities and ecological processes. However, commonly used metrics of phylogenetic diversity (PD) do not incorporate information on the relative abundances of individuals within a community. In this study, we develop three indices of PD that explicitly consider species abundances. First, we present a metric of phylogenetic-abundance evenness that evaluates the relationship between the abundance and the distribution of terminal branch lengths. Second, we calculate an index of hierarchical imbalance of abundances at the clade level encapsulating the distribution of individuals across the nodes in the phylogeny. Third, we develop an index of abundance-weighted evolutionary distinctiveness and generate an entropic index of phylogenetic diversity that captures both information on evolutionary distances and phylogenetic tree topology, and also serves as a basis to evaluate species conservation value. These metrics offer measures of phylogenetic diversity incorporating different community attributes. We compare these new metrics to existing ones, and use them to explore diversity patterns in a typical California annual grassland plant community at the Jasper Ridge biological preserve.
Ecology Letters (2010) 13: 96–105     

A population genetic comparison of large- and small-bodied sage grouse in colorado using microsatellite and mitochondrial DNA markers

Sage grouse (Centrocercus urophasianus) from southwestern Colorado and southeastern Utah (United States) are 33% smaller than all other sage grouse and have obvious plumage and behavioural differences. Because of these differences, they have been tentatively recog-nized as a separate 'small-bodied' species. We collected genetic evidence to further test this proposal, using mitochondrial sequence data and microsatellite markers to determine whether there was gene flow between the two proposed species. Significant differences in the distribution of alleles between the large- and small-bodied birds were found in both data sets. Analysis of molecular variance (AMOVA) revealed that 65% of the variation in mitochondrial DNA (mtDNA) haplotypes could be explained by the large- vs. small-bodied distinction. Genetic distances and neighbour-joining trees based on allelic frequency data showed a distinct separation between the proposed species, although cladistic analysis of the phylogenetic history of the mitochondrial sequence haplotypes has shown a lack of reciprocal monophyly. These results further support the recognition of the small-bodied sage grouse as a distinct species based on the biological species concept, providing additional genetic evidence to augment the morphological and behavioural data. Furthermore, small-bodied sage grouse had much less genetic variation than large-bodied sage grouse, which may have implications for conservation issues.     

Carabid Beetle and Spider Assemblages along a Forested Urban–Rural Gradient in Southern Finland

To investigate the effects of urbanization on carabid beetles (Carabidae) and ground dwelling spiders (Araneae) a study was completed along a 20km urban–rural forest gradient in the Helsinki–Espoo area of southern Finland. To study changes in assemblage structure, abundance and species richness, these taxa were collected in the year 2000 using pitfall traps, which had been placed in four forest sites within each of the urban, suburban and rural zones. We expected to find changes in the abundances and species richnesses in the two taxa across the urban–rural gradient, but did not find any. Our second and third hypotheses, stating that generalist species and small-bodied species should gain dominance along the gradient from rural to urban sites, were partly supported as carabid specialists were more characteristic of suburban and rural environments whereas generalists were more likely to be collected from rural areas compared to suburban or urban sites. Furthermore, medium to large-sized carabid individuals were more likely to be collected in the rural sites compared to urban forests. We found no evidence for significant changes in spider abundance or species richness across the urban–rural gradient in relation to body size or habitat specialization. We suggest that urbanization does not have significant effects on the total abundances and species richnesses in these two taxa. However, individual species responded differently to urbanization, and there were significant differences in the specialization and body sizes of carabids across the gradient.     

A ‘fast-food approach’ to the standardization of quadratic diversity

Abstract

Intuitively, a community composed of ecologically dissimilar taxa is more diverse than a community composed of more similar taxa. However, since traditional diversity indices such as Shannon's entropy or Simpson's diversity are computed solely from the relative abundances of a given species assemblage, they cannot account for ecological differences between species. There have been recent developments regarding the use of quadratic entropy, a diversity index that incorporates both species relative abundances and a measure of the pairwise ecological differences between species. In this paper we firstly show that under some specific circumstances quadratic entropy can be additively decomposed into α- β- and γ-diversities, a property that renders it a desirable measure of diversity in the ecological practice. Next, we suggest a quick and simple method for obtaining a standardized version of quadratic entropy that may allow an easier interpretation of the resulting diversity values.     

Diversity measurement combining relative abundances and taxonomic distinctiveness of species

We discuss a diversity measure combining information of relative abundances and taxonomic distinctiveness suggested by Ricotta (2004). We show that Ricotta's measure violates weak species monotonicity, a condition that requires that the addition of a species should always increase a diversity index if abundances change only marginally. We suggest an alternative index satisfying weak species monotonicity and apply it to the 'Zeesserveld' forest reserve in the Netherlands.     

Functional regularity: a neglected aspect of functional diversity

Functional diversity has been identified as a key to understanding ecosystem and community functioning. However, due to the lack of a sound definition its nature and measurement are still poorly understood. In the same way that species diversity can be split into species richness and species evenness, so functional diversity can be split into functional richness (i.e. the amount of functional trait/character/attribute space filled) and functional evenness (i.e. the evenness of abundance distribution in functional trait space). We propose a functional regularity index (FRO) as a measure of functional evenness for situations where species are represented only by a single functional trait value (e.g. mean, median or mode), and species abundances are known. This new index is based on the Bulla O index of species evenness. When dealing with functional types or categorical functional traits, the Bulla O or any other accepted species evenness index may be used directly to measure functional evenness. The advantage of FRO is that it supplies a measure of functional evenness for continuous trait data. The FRO index presented in this paper fulfils all the a priori criteria required. We demonstrate with two example datasets that a range of FRO values may be obtained for both plant and animal communities. Moreover, FRO was strongly related to ecosystem function as seen in photosynthetic biomass in plant communities, and was able to differentiate sampling stations in a lagoon based on the functional traits of fish. Thus, the FRO index is potentially a highly useful tool for measuring functional diversity in a variety of ecological situations.     

An information-theoretical measure of taxonomic diversity

Traditional diversity indices are computed from the abundances of species present and are insensitive to taxonomic differences between species. However, a community in which most species belong to the same genus is intuitively less diverse than another community with a similar number of species distributed more evenly between genera. In this paper, we propose an information-theoretical measure of taxonomic diversity that reflects both the abundances and taxonomic distinctness of the species. Unlike previous measures of taxonomic diversity, such as Rao's quadratic entropy, in this new measure the analyzed taxonomic properties are associated with the single species instead of species pairs.     

Phylogenetic balance and ecological evenness

The frequency distribution of numbers of species in taxonomic groups, where many species belong to a few very diverse higher taxa, is mirrored by that of species in most communities, where many individuals belong to a few very abundant species. Various hypotheses mechanistically link a species' community abundance with the diversity of the higher level taxon (genus, family, order) to which it belongs, but empirical data are equivocal about general trends in the relation between rank-taxon diversity and mean abundance. One reason for this inconclusive result may be the effect of the semisubjective nature of rank-based classification. We assessed the relationship between clade diversity and mean species abundance for two diverse tropical tree communities, using both traditional rank-based analysis and two new phylogenetic analyses (based on the ratio of individuals to taxa at each node in the phylogeny). Both rank-based and phylogenetic analyses using taxonomic ranks above the species level as terminal taxa detected a trend associating common species with species-rich families. In contrast, phylogenetic analyses using species as terminal taxa could not distinguish the observed distribution of species abundances from a random distribution with respect to clade diversity. The difference between these results might be due to (1) the absence of a real phylogeny-wide relationship between clade abundance and diversity, (2) the influence of poor phylogenetic resolution within families in our phylogenies, or (3) insufficient sensitivity of our metrics to subtle tree-wide effects. Further development and application of phylogeny-based methods for testing abundance-diversity relationships is needed.     

Parametric scaling from species to growth-form diversity: an interesting analogy with multifractal functions

We propose a measure of divergence from species to life-form diversity aimed at summarizing the ecological similarity among different plant communities without losing information on traditional taxonomic diversity. First, species and life-form relative abundances within a given plant community are determined. Next, using Rényi's generalized entropy, the diversity profiles of the analyzed community are computed both from species and life-form relative abundances. Finally, the speed of decrease from species to life-form diversity is obtained by combining the outcome of both profiles. Interestingly, the proposed measure shows some formal analogies with multifractal functions developed in statistical physics for the analysis of spatial patterns. As an application for demonstration, a small data set from a plant community sampled in the archaeological site of Paestum (southern Italy) is used.     

Biological diversity: Distinct distributions can lead to the maximization of Rao’s quadratic entropy

Rao’s quadratic entropy (QE) is a diversity index that includes the abundances of categories (e.g. alleles, species) and distances between them. Here we show that, once the distances between categories are fixed, QE can be maximized with a reduced number of categories and by several different distributions of relative abundances of the categories. It is shown that Rao’s coefficient of distance (DISC), based on QE, can equal zero between two maximizing distributions, even if they have no categories in common. The consequences of these findings on the relevance of QE for understanding biological diversity are evaluated in three case studies. The behaviour of QE at its maximum is shown to be strongly dependent on the distance metric. We emphasize that the study of the maximization of a diversity index can bring clarity to what exactly is measured and enhance our understanding of biological diversity.     

Macroecology of live-bearing in fishes: latitudinal and depth range comparisons with egg-laying relatives

The habitat heterogeneity hypothesis states that an increase in habitat heterogeneity leads to an increase in species diversity. We tested this hypothesis for a community of small mammals in the semiarid, sand-shinnery-oak ecosystem of the southwestern United States. We used indices of differentiation diversity to quantify differences between two habitat types (blowouts in a sand-shinnery-oak matrix) in terms of species diversity. The Wilson-Shmida index (β_T) considers species composition only, whereas the Morisita-Horn index (C_mH) also takes species abundances into account. We constructed null models to test the hypothesis that differentiation diversity between habitat types is greater than that produced by stochastic processes. Two models were constructed, one based on the random placement of species and one based on the random placement of individuals. No evidence supported the hypothesis that habitat heterogeneity enhances diversity of a landscape by increasing the number of species in an area. Indeed, paired habitats were more similar than chance alone would dictate in terms of species identities. In contrast, habitat heterogeneity affects diversity by significantly altering the relative proportions of species in contrasting habitat types. Because seeds differentially accumulate at the interface between blowouts and matrix, the high productivity of the edge may actually homogenize habitat types in terms of species richness. Nonetheless, blowouts might best be considered to be microhabitats which enhance or complement the value of the matrix even though the species which use either habitat type are identical.     

Plant species identity and diversity effects on different trophic levels of nematodes in the soil food web

Previous studies on biodiversity and soil food web composition have mentioned plant species identity, as well as plant species diversity as the main factors affecting the abundance and diversity of soil organisms. However, most studies have been carried out under limitations of time, space, or appropriate controls. In order to further examine the relation between plant species diversity and the soil food web, we conducted a three-year semi-field experiment in which eight plant species (4 forb and 4 grass species) were grown in monocultures and mixtures of two, four and eight plant species. In addition there were communities with 16 plant species. We analyzed the abundance and identity of the nematodes in soil and roots, including feeding groups from various trophic levels (primary and secondary consumers, carnivores, and omnivores) in the soil food web.
Plant species diversity and plant identity affected the diversity of nematodes. The effect of plant diversity was attributed to the complementarity in resource quality of the component plant species rather than to an increase in total resource quantity. The nematode diversity varied more between the different plant species than between different levels of plant species diversity, so that plant identity is more important than plant diversity. Nevertheless the nematode diversity in plant mixtures was higher than in any of the plant monocultures, due to the reduced dominance of the most abundant nematode taxa in the mixed plant communities. Plant species identity affected the abundances of the lower trophic consumer levels more than the higher trophic levels of nematodes. Plant species diversity and plant biomass did not affect nematode abundance. Our results, therefore, support the hypothesis that resource quality is more important than resource quantity for the diversity of soil food web components and that plant species identity is more important than plant diversity per se.     

Empirical Relationships between Species Richness, Evenness, and Proportional Diversity

Diversity (or biodiversity) is typically measured by a species count (richness) and sometimes with an evenness index; it may also be measured by a proportional statistic that combines both measures (e.g., Shannon-Weiner index or H'). These diversity measures are hypothesized to be positively and strongly correlated, but this null hypothesis has not been tested empirically. We used the results of Caswell's neutral model to generate null relationships between richness (S), evenness (J'), and proportional diversity (H'). We tested predictions of the null model against empirical relationships describing data in a literature survey and in four individual studies conducted across various scales. Empirical relationships between log S or J' and H' differed from the null model when <10 species were tested and in plants, vertebrates, and fungi. The empirical relationships were similar to the null model when >10 and <100 species were tested and in invertebrates. If >100 species were used to estimate diversity, the relation between log S and H' was negative. The strongest predictive models included log S and J'. A path analysis indicated that log S and J' were always negatively related, that empirical observations could not be explained without including indirect effects, and that differences between the partials may indicate ecological effects, which suggests that S and J' act like diversity components or that diversity should be measured using a compound statistic.     

Vole disturbances and plant diversity in a grassland metacommunity

We studied the disturbance associated with prairie vole burrows and its effects on grassland plant diversity at the patch (1 m²) and metacommunity (>5 ha) scales. We expected vole burrows to increase patch-scale plant species diversity by locally reducing competition for resources or creating niche opportunities that increase the presence of fugitive species. At the metacommunity scale, we expected burrows to increase resource heterogeneity and have a community composition distinct from the matrix. We measured resource variables and plant community composition in 30 paired plots representing disturbed burrows and undisturbed matrix patches in a cool-season grassland. Vole disturbance affected the mean values of nine resource variables measured and contributed more to resource heterogeneity in the metacommunity than matrix plots. Disturbance increased local plant species richness, metacommunity evenness, and the presence and abundance of fugitive species. To learn more about the contribution of burrow and matrix habitats to metacommunity diversity, we compared community similarity among burrow and matrix plots. Using Sorenson’s similarity index, which considers only presence–absence data, we found no difference in community similarity among burrows and matrix plots. Using a proportional similarity index, which considers both presence–absence and relative abundance data, we found low community similarity among burrows. Burrows appeared to shift the identity of dominant species away from the species dominant in the matrix. They also allowed subordinate species to persist in higher abundances. The patterns we observed are consistent with several diversity-maintaining mechanisms, including a successional mosaic and alternative successional trajectories. We also found evidence that prairie voles may be ecosystem engineers.     

棉田节肢动物群落的数量与能流量多样性特征分析

根据香农－维纳信息多样性指数的定义,提出了能流量多样性指数的计算公式,并以此分析了棉田生态系统中节肢动物群落特征。结果表明：害虫与天敌群落的能流量多样性指数比数量多样性指数变化更为敏感,更能反映害虫与天敌的群落特征。作者认为,这是一个比较好的多样性测定指标。