Application of the quadratic entropy indices for diversity studies of drosophilid assemblages

Most diversity indices widely used in ecology (e.g. the Shannon-Wiener index and the Gini-Simpson index) do not reflect taxonomic or other differences among species. Quadratic entropy, proposed by C.R. Rao (1982), takes these differences into account. The aim of this paper is to demonstrate the application of this diversity index in investigations of drosophilid species diversity. Quadratic entropy and Simpson index values are also compared. The elements of the difference matrix in the quadratic form are taxonomic differences (distances) among the species. The difference is defined by the position of the lowest-ranking taxon containing both species. According to the results, the Gini-Simpson index and the taxonomic quadratic entropy are positively correlated. The observed differences are due to the fact that assemblages with greater taxonomic distances produce greater quadratic entropy. Drosophilid species can be assigned to specific life habit classes, more specifically to resource types. After postulating differences between the types, we used the quadratic entropy measure to analyse resource diversity. These kinds of examinations approximate to an operative association of species diversity and environmental diversity.     

Measuring diversity: the importance of species similarity

Realistic measures of biodiversity should reflect not only the relative abundances of species, but also the differences between them. We present a natural family of diversity measures taking both factors into account. This is not just another addition to the already long list of diversity indices. Instead, a single formula subsumes many of the most popular indices, including Shannon's, Simpson's, species richness, and Rao's quadratic entropy. These popular indices can then be used and understood in a unified way, and the relationships between them are made plain. The new measures are, moreover, effective numbers, so that percentage changes and ratio comparisons of diversity value are meaningful. We advocate the use of diversity profiles, which provide a faithful graphical representation of the shape of a community; they show how the perceived diversity changes as the emphasis shifts from rare to common species. Communities can usefully be compared by comparing their diversity profiles. We show by example that this is a far more subtle method than any relying on a single statistic. Some ecologists view diversity indices with suspicion, questioning whether they are biologically meaningful. By dropping the naive assumption that distinct species have nothing in common, working with effective numbers, and using diversity profiles, we arrive at a system of diversity measurement that should lay much of this suspicion to rest.     

Quadratic diversity: Its maximization can reduce the richness of species

In recent decades numerous diversity indices have been introduced. Among them the quadratic entropy index Q expresses the mean difference between two individuals chosen from the community at random. Differing from diversity indices habitually employed, Q does not satisfy a property postulated earlier for those measures. Namely, the uniform distribution of species does not necessarily yield the maximal index value. Q is based on the difference matrix of species. For a given matrix one can seek for the vector yielding the maximum quadratic entropy. This task leads to a quadratic programming problem. Using the appropriate program of a program package, we determined the maximum vector for a genetic difference matrix of crane species, as published in the literature. We discovered that some components (frequencies) in the maximum vector are equal to zero. That is, by maximizing the quadratic diversity some species can be eliminated. We discuss briefly the possible implications of this observation. Moreover, even if all elements in the maximum vector are positive, the elements can differ.     

New multidimensional functional diversity indices for a multifaceted framework in functional ecology

Functional diversity is increasingly identified as an important driver of ecosystem functioning. Various indices have been proposed to measure the functional diversity of a community, but there is still no consensus on which are most suitable. Indeed, none of the existing indices meets all the criteria required for general use. The main criteria are that they must be designed to deal with several traits, take into account abundances, and measure all the facets of functional diversity. Here we propose three indices to quantify each facet of functional diversity for a community with species distributed in a multidimensional functional space: functional richness (volume of the functional space occupied by the community), functional evenness (regularity of the distribution of abundance in this volume), and functional divergence (divergence in the distribution of abundance in this volume). Functional richness is estimated using the existing convex hull volume index. The new functional evenness index is based on the minimum spanning tree which links all the species in the multidimensional functional space. Then this new index quantifies the regularity with which species abundances are distributed along the spanning tree. Functional divergence is measured using a novel index which quantifies how species diverge in their distances (weighted by their abundance) from the center of gravity in the functional space. We show that none of the indices meets all the criteria required for a functional diversity index, but instead we show that the set of three complementary indices meets these criteria. Through simulations of artificial data sets, we demonstrate that functional divergence and functional evenness are independent of species richness and that the three functional diversity indices are independent of each other. Overall, our study suggests that decomposition of functional diversity into its three primary components provides a meaningful framework for its quantification and for the classification of existing functional diversity indices. This decomposition has the potential to shed light on the role of biodiversity on ecosystem functioning and on the influence of biotic and abiotic filters on the structure of species communities. Finally, we propose a general framework for applying these three functional diversity indices.     

The distribution of a generalized diversity index due to Good

Good (1953, 1982) proposed a generalized diversity index which includes as special cases both Shannon's and Simpson's indices. This index can be further generalized as described in Baczkowski et al. (1997, 1998). In this paper the first four moments of this generalized index are derived for both a general species abundance distribution and the case with all species abundances equal, the equiprobable case. This allows the skewness and kurtosis of the index to be determined and thus gives information about the distribution of the index.     

Do biodiversity indices behave as expected from traits of constituent species in simulated scenarios?

It is believed that diversity of plant communities has a positive effect on their productivity. The benefits of diversity are described by “biodiversity indices”, comparing yield of mixtures with yields of monocultures of constituent species. These indices are supposed to capture also the main mechanisms leading to increased yield. We have constructed a spatially explicit individual based model, simulating even-aged stand development, and compared the behaviour of selected biodiversity indices (overyielding, selectivity and complementarity) with expectations based on life history traits of constituent species. The results are based on comparisons of two species mixtures with corresponding monocultures. We designed three scenarios of changes in the two species life history differentiation, and compared the behaviour of the indices with expectation based on it. In the first scenario, selectivity was driven by increasing size inequality of the two species, mostly in accordance with expectations. The second scenario presents increasing shade tolerance of the smaller species that increased complementarity, again mostly as expected. In the last scenario, shortening of length of stress tolerance of the weaker species surprisingly increased values of the biodiversity indices. For each setting, we varied sowing density and spatial pattern of the constituent species. The behaviour of the indices was influenced by both factors, but the effect of density was more pronounced. In particular, at high sowing densities, the most important interactions happened in the very early stages of mixture development, and the behaviour of the indices was often counter-intuitive.     

Additive partitioning of Rao's quadratic diversity: a hierarchical approach

Ecologists have long recognized three different components of species diversity: alpha or within-community diversity (α), beta or between-community diversity (β) and gamma or total species diversity in a region (γ). In this framework, β-diversity has been traditionally linked to the other diversity components through a multiplicative model so that it can be expressed as the ratio between γ-diversity and average α-diversity in a set of plots. Yet, more recently, ecologists are starting to partition diversity using the lesser known approach that α- and β-diversities sum to give the γ-diversity. This additive diversity partitioning is based on the decomposition of concave diversity measures for which the total diversity in a pooled set of communities exceeds (or equals) the average diversity within communities. In this paper, first, I shortly revise additive diversity partitioning for traditional diversity measures that are computed from species relative abundances. Next, I show that, under some specific circumstances, the same model can be extended to Rao's quadratic entropy, a measure that combines species relative abundances and pairwise interspecies differences. Finally, in the framework of taxonomic diversity, Rao's quadratic entropy has another decomposition: the sum over the Simpson indices at all the taxonomic levels. Thus, I show that, combining both partitioning models, the contribution of each level in the taxonomic hierarchy to the α- β- and γ-diversity components of Rao's quadratic entropy is made explicit. The proposed diversity decomposition is illustrated with a worked example on data from a plant community on ultramafic soils in Tuscany (central Italy).     

The apportionment of quadratic entropy: a useful alternative for partitioning diversity in ecological data

Many methods that study the diversity within hierarchically structured populations have been developed in genetics. Among them, the analysis of molecular variance (AMOVA) (Excoffier et al., 1992) has the advantage of including evolutionary distances between individuals. AMOVA is a special case of a far more general statistical scheme produced by Rao (1982a; 1986) and called the apportionment of quadratic entropy (APQE). It links diversity and dissimilarity and allows the decomposition of diversity according to a given hierarchy. We apply this framework to ecological data showing that APQE may be very useful for studying diversity at various spatial scales. Moreover, the quadratic entropy has a critical advantage over usual diversity indices because it takes into account differences between species. Finally, the differences that can be incorporated in APQE may be either taxonomic or functional (biological traits), which may be of critical interest for ecologists.     

Spatio-temporal patterns in the diversity of demersal fish communities off the south coast of South Africa

The diversity of ecological communities has been the focus of many studies. Because biodiversity provides several indicators used in an Ecosystem Approach to Fisheries (EAF) to track changes in fish communities, we investigated the spatial and temporal patterns in the diversity of some demersal fish communities subjected to varying fishing pressure. Depth and catch rate were the most important predictors in explaining changes in diversity followed by longitude and survey year. Diversity, as measured by the various indices except for taxonomic distinctness (∆*), initially declined with increasing depth to about a depth of 80 m, then increased to about 150 m after which it declined. Taxonomic distinctness index (∆*) showed an increase in the taxonomic heterogeneity of the demersal community below the 300-m isobath. Diversity remained relatively constant with increase in longitude to around 24°E (which has the lowest diversity) after which it increased. The assessment of the temporal trend in diversity indicates that survey year has a significant effect on all diversity indices except for ∆*. Diversity increased and dominance declined with time. This may be result of a decline in the abundance of dominant species or an increase in the abundance less dominant species, or a combination of both effects. Multivariate analysis of the set of diversity indices showed three groups of indices: those reflecting species richness (S, Margalef’s d), those measuring mainly taxonomic relatedness (∆*), and those balancing the richness and evenness components of diversity (J′, H′, λ, ∆, Hill’s N1, and Hill’s N2). The relationship between evenness, catch rate, and size was also investigated. Size classes with highest evenness were found to have lowest catch rate and vice versa. This highlights the need to consider the size and trophic level of species when linking diversity to the functioning of ecosystems.     

Indices of Grassland Biodiversity in the Chihuahuan Desert Ecoregion Derived from Remote Sensing

Abstract: We used a relatively simple and direct remote-sensing approach to determine biodiversity values in arid ecosystems and thus identify potential conservation sites. We developed indices based on regression models between grass, shrub, litter, exposed-soil groundcover components, and Landsat thematic mapper satellite imagery reflectance values over a reference site in the northern Chihuahuan Desert in New Mexico. This site supports low-disturbance desert grasslands that have been excluded from livestock grazing for 55 years and moderate-disturbance grasslands that have been under a continuous grazing regime for over 100 years. Greater richness and abundance of noninvasive and nonruderal plant species were associated with the low-disturbance grasslands that had lower shrub abundance, increased litter and grass cover, and lower exposed soil. Using the thematic mapper indices, we computed an additive grassland biodiversity index such that, as exposed soil and shrub values go down, litter and grass values go up, as does the biodiversity index. When the biodiversity index was applied to the reference-site landscape, grasslands previously identified for their high conservation value were detected. As a further test, we applied the indices to a site in Chihuahua, Mexico, that supports similar grasslands but for which there are few other data on condition and conservation values. The soil, grass, and shrub indices were moderately effective in describing the range of variation at the Mexico site, but the litter equation was not. Still, higher biodiversity value in terms of nonruderal plant diversity tended to correspond to higher grass cover and lower soil exposure and a higher overall biodiversity index. Some localized calibration with geologic substrate may be required along with an assessment of the temporal constraints, but generally the index shows promise for quickly and efficiently detecting desert grasslands of high biodiversity conservation value.     

The use of replicated plot,line and point sampling for estimating species abundance and ecological diversity

The paper deals with the problem of estimating diversity indexes for an ecological community. First the species abundances are unbiasedly and consistently estimated using designs based on n random and independent selections of plots, points or lines over the study area. The problem of sampling elusive populations is also considered. Finally, the diversity index estimates are obtained as functions of the abundance estimates. The resulting estimators turn out to be asymptotically (n large) unbiased, even if a considerable bias may occur for a small n. Accordingly, the method of jackknifing is made use of in order to reduce bias.     

A nearly neutral model of biodiversity

S. P. Hubbell's unified neutral theory of biodiversity has stimulated much new thinking about biodiversity. However, empirical support for the neutral theory is limited, and several observations are inconsistent with the predictions of the theory, including positive correlations between traits associated with competitive ability and species abundance and correlations between species diversity and ecosystem functioning. The neutral theory can be extended to explain these observations by allowing species to differ slightly in their competitive ability (fitness). Here, we show that even slight differences in fecundity can greatly reduce the time to extinction of competitors even when the community size is large and dispersal is spatially limited. In this case, species richness is dramatically reduced, and a markedly different species abundance distribution is predicted than under pure neutrality. In the nearly neutral model, species co-occur in the same community not because of, but in spite of, ecological differences. The more competitive species with higher fecundity tend to have higher abundance both in the metacommunity and in local communities. The nearly neutral perspective provides a theoretical framework that unites the sampling model of the neutral theory with theory of biodiversity affecting ecosystem function.     

Effects of landscape design of forest reserves on Saproxylic beetle diversity

Increasing the density of natural reserves in the forest landscape may provide conservation benefits for biodiversity within and beyond reserve borders. We used 2 French data sets on saproxylic beetles and landscape cover of forest reserves (LCFR) to test this hypothesis: national standardized data derived from 252 assessment plots in managed and reserve stands in 9 lowland and 5 highland forests and data from the lowland Rambouillet forest, a forested landscape where a pioneer conservation policy led to creation of a dense network of reserves. Abundance of rare and common saproxylic species and total saproxylic species richness were higher in forest reserves than in adjacent managed stands only in highland forests. In the lowland regional case study, as LCFR increased total species richness and common species abundance in reserves increased. In this case study, when there were two or more reserve patches, rare species abundance inside reserves was higher and common species richness in managed stands was higher than when there was a single large reserve. Spillover and habitat amount affected ecological processes underlying these landscape reserve effects. When LCFR positively affected species richness and abundance in reserves or managed stands, >12‐20% reserve cover led to the highest species diversity and abundance. This result is consistent with the target of 17% forested land area in reserves set at the Nagoya biodiversity summit in 2010. Therefore, to preserve biodiversity we recommend at least doubling the current proportion of forest reserves in European forested landscapes.     

Nonparametric estimation of Shannon’s index of diversity when there are unseen species in sample

A biological community usually has a large number of species with relatively small abundances. When a random sample of individuals is selected and each individual is classified according to species identity, some rare species may not be discovered. This paper is concerned with the estimation of Shannons index of diversity when the number of species and the species abundances are unknown. The traditional estimator that ignores the missing species underestimates when there is a non-negligible number of unseen species. We provide a different approach based on unequal probability sampling theory because species have different probabilities of being discovered in the sample. No parametric forms are assumed for the species abundances. The proposed estimation procedure combines the Horvitz–Thompson (1952) adjustment for missing species and the concept of sample coverage, which is used to properly estimate the relative abundances of species discovered in the sample. Simulation results show that the proposed estimator works well under various abundance models even when a relatively large fraction of the species is missing. Three real data sets, two from biology and the other one from numismatics, are given for illustration.     

Biomasses,catch rates and abundances of demersal fishes,particularly predators of prawns,in a tropical bay in the Gulf of Carpentaria,Australia

The demersal fish fauna of Albatross Bay, in the eastern Gulf of Carpentaria, northern Australia, was sampled on seven cruises from August 1986 to November 1988, using a random stratified trawl survey. Four depth zones between 7 and 45 m were sampled during both day and night. The mean biomass of fish from all seven cruises was 297 kg ha^–1 for days trawls and 128 kg ha^–1 for night trawls. The overall mean catch rates were 922 kg h^–1 for day trawls and 412 kg h^–1 for night trawls. There were marked differences between cruises in both the biomass and catch rate. Approx 890 000 fish of 237 species were collected. Of these, 25 species comprised 82% of the total biomass and 74% of the overall catch rate. The dominant families were Leiognathidae, Haemulidae and Clupeidae, with Sciaenidae and Dasyatidae important at night.Leiognathus bindus was the most abundant species. Twenty-five species occurred in more than 50% of trawls, withCaranx bucculentus the most frequently caught (96% of all trawls). Thirty four species were predators on prawns; their absolute mean biomass was 50 kg ha^–1 during the day and 39 kg ha^–1 at night. The corresponding catch rates were 171 and 125 kg h^–1. Multiple-regression analyses were used to discriminate the effects of diel, seasonal, depth and cruise patterns. Of the 31 most abundant species, 15 showed diel patterns of abundance; 11 species showed seasonal patterns of abundance; 23 species had differential depth distribution; and 13 species showed significant cruise-to-cruise variation in abundance. Cruise variations in abundance were tested against salinity, temperature, tidal exchange, plankton biomass and prawn abundances as well as periods (and lags) of total rainfall prior to sampling. Only total rainfall showed any significant correlation. Total rainfall over a period of 6 wk immediately prior to sampling showed significant positive correlations with the abundances of five species, with overall daytime catch rates, and with the suite of 34 prawn predators. Rainfall and river runoff into Albatross Bay were significantly correlated. In Albatross Bay, the complex of factors affecting fish abundances and the magnitude of between-cruise differences indicate that such tropical communities may be unpredictable and are not seasonally constant. The high catch rates in Albatross Bay relative to similar tropical areas elsewhere are discussed and attributed to the light exploitation of the Albatross Bay stocks. Other than a prawn fishery, there is no commercial trawling in Albatross Bay. Hence, the only fishing mortality is a result of by-catch from prawn trawling. The annual total of such fish by-catch is probably less than 10% of the estimated standing stock of 93 000 tonnes.     

Efficacy of extracting indices from large‐scale acoustic recordings to monitor biodiversity

Passive acoustic monitoring could be a powerful way to assess biodiversity across large spatial and temporal scales. However, extracting meaningful information from recordings can be prohibitively time consuming. Acoustic indices (i.e., a mathematical summary of acoustic energy) offer a relatively rapid method for processing acoustic data and are increasingly used to characterize biological communities. We examined the relationship between acoustic indices and the diversity and abundance of biological sounds in recordings. We reviewed the acoustic‐index literature and found that over 60 indices have been applied to a range of objectives with varying success. We used 36 of the most indicative indices to develop a predictive model of the diversity of animal sounds in recordings. Acoustic data were collected at 43 sites in temperate terrestrial and tropical marine habitats across the continental United States. For terrestrial recordings, random‐forest models with a suite of acoustic indices as covariates predicted Shannon diversity, richness, and total number of biological sounds with high accuracy (R² ≥ 0.94, mean squared error [MSE] ≤170.2). Among the indices assessed, roughness, acoustic activity, and acoustic richness contributed most to the predictive ability of models. Performance of index models was negatively affected by insect, weather, and anthropogenic sounds. For marine recordings, random‐forest models poorly predicted Shannon diversity, richness, and total number of biological sounds (R² ≤ 0.40, MSE ≥ 195). Our results suggest that using a combination of relevant acoustic indices in a flexible model can accurately predict the diversity of biological sounds in temperate terrestrial acoustic recordings. Thus, acoustic approaches could be an important contribution to biodiversity monitoring in some habitats.     

Linking Indices for Biodiversity Monitoring to Extinction Risk Theory

Biodiversity indices often combine data from different species when used in monitoring programs. Heuristic properties can suggest preferred indices, but we lack objective ways to discriminate between indices with similar heuristics. Biodiversity indices can be evaluated by determining how well they reflect management objectives that a monitoring program aims to support. For example, the Convention on Biological Diversity requires reporting about extinction rates, so simple indices that reflect extinction risk would be valuable. We developed 3 biodiversity indices that are based on simple models of population viability that relate extinction risk to abundance. We based the first index on the geometric mean abundance of species and the second on a more general power mean. In a third index, we integrated the geometric mean abundance and trend. These indices require the same data as previous indices, but they also relate directly to extinction risk. Field data for butterflies and woodland plants and experimental studies of protozoan communities show that the indices correlate with local extinction rates. Applying the index based on the geometric mean to global data on changes in avian abundance suggested that the average extinction probability of birds has increased approximately 1% from 1970 to 2009. Conectando Índices para el Monitoreo de la Biodiversidad con la Teoría de Riesgo de Extinción     

RIVPACS models for predicting the expected macroinvertebrate fauna and assessing the ecological quality of rivers

The European Union Water Framework Directive recognises the need for and value of biological monitoring. This paper reviews the modelling approach known as River Invertebrate Prediction and Classification System (RIVPACS for assessing the ecological quality of river sites using macroinvertebrate sampling. The RIVPACS philosophy is to develop statistical relationships between the fauna and the environmental characteristics of a large set of high quality reference sites which can be used to predict the macroinvertebrate fauna to be expected at any site in the absence of pollution or other environmental stress. The observed fauna at new test sites can then be compared with their site-specific expected fauna to derive indices of ecological quality. All methodological decisions in any such model development have implications for the reliability, precision and robustness of any resulting indices for assessing the ecological quality and ecological grade (‘status’) of individual river stretches. The choice of reference sites and environmental predictor variables, the site classification and discrimination methods, the estimation of the expected fauna, and indices for comparing the agreement, or lack of it, between the observed and expected fauna, are all discussed. The indices are assessed on the reference sites and on a separate test set of 340 sites, which are subject to a wide range of types and degrees of impairment.     

Ecological assessment of lakeshore wetland rehabilitation on eastern route of South-to-North Water Transfer Project

To evaluate the ecological effects of lakeshore wetland rehabilitation on the eastern route of the South-to-North Water Transfer Project, species composition, coverage, height, and biomass of wetland communities at 22 sites of the study area on the shore of Nansi Lake in April and May 2007 were investigated. The wetlands under investigation were divided into platform fields, transition zones, and shallow water zones according to differences in elevations, water levels, and human activities. The species richness index, Shannon-Wiener index, Simpson index, and Pielou Evenness index were adopted to delineate and discuss the ecological effects of lakeshore wetland rehabilitation in 22 quadrates. Results showed that the species richness of the wetland areas after 2 years’ rehabilitation amounted to 47 of 24 families, higher than 25 of 20 families in areas without rehabilitation. The biodiversity index and abundance index of rehabilitated areas were also higher than those of platform fields and fish ponds where there was no rehabilitation. In addition, the Shannon-Wiener index, Simpson index, and community evenness index of platform fields in rehabilitated wetland areas were 1.619, 0.745, and 0.860, respectively, higher than those of the platform fields before rehabilitating. The results suggested that the constructed lakeshore wetland played an important role in protecting the diversity of species.