## Indices Based on Diversity

Diversity is perhaps the most used concept in assessing pollution, assuming that diversity and environmental disturbances have an inverse relation. We may consider three main categories of diversity measures:

1. Indices that measure the species richness, which are essentially measurements of the number of species in a defined sampling unit.

2. Models of the abundance of species, as the K-dominance curves or the lognormal model, which describe the distribution of their abundance, going from those that represent situations in which there is a high uniformity, to those that characterize cases in which the abundance of each species is very unequal.

3. Indices based on the proportional abundance of different species, which intend to account for enrichment and uniformity in a simple expression. This category of indices can also be divided into those based on (a) statistics, (b) information theory, and (c) species dominance.

Other types of measures, such as the average taxonomic diversity and distinctness, which account for taxonomical, numerical, ecological, genetical, and filogenetical aspects of diversity, have also been used to evaluate biodiversity in the marine environment.

Shannon-Wiener index. This index was elaborated by C. E. Shannon and W. Weaver in 1949 and is based on the information theory. It assumes that individuals are sampled at random, out of an 'indefinitely large' community, and that all the species are represented in the sample and can be estimated according to the algorithm:

where pi is the proportion of individuals belonging to species i in the sample. The real value of pi is unknown, but it is estimated through the ratio N/N, where Ni is the number of individuals of the species i and N the total number of individuals.

The units for the index depend on the log used: for log2 the unit is bits/individual; for loge 'natural bels' and 'nat'; and for log10 'decimal digits' and 'decits'.

Values usually range from 0 to 5, and maximal values above 5 bits/individual are very rare. Since diversity is a logarithmic measurement, its relatively asymptotic character decreases the index sensitivity in the range ofvalues close to the upper limit. Low values are considered an indication of pollution, but there is an obvious lack of objectivity when trying to establish significant thresholds regarding pollution effects. In fact, detractors of this index base their criticisms on its lack of sensitivity when it comes to detecting the initial stages of pollution. This index is extensively treated in Shannon-Wiener Index.

Pielou evenness index. This index, introduced by Pielou in 1969, is a measure of how evenly distributed abundance is among the species that exist in a community:

where H9max is the maximum possible value of Shannon diversity.

The values ofthis index may vary from 0 to 1, where 1 represents a community with perfect evenness, and decreases to 0 as the relative abundances of the species diverge from evenness.

Margalef index. The Margalef index, elaborated by R. Margalef in 1958, quantifies diversity by relating specific richness to the total number ofindividuals:

where 5 is the number of species and N the total number ofindividuals.

No reference values were proposed or established, which has always been a practical difficulty.

This index is extensively treated in Margalefs Index.

Berger-Parker index. This index, elaborated by W. H. Berger and F. L. Parker in 1970, expresses the proportional importance ofthe most abundant species, and may be computed using the following algorithm:

where nmax is the number of individuals of the one most abundant species and N is the total number of individuals. The index values may vary from 0 to 1 and, contrary to other diversity indices, higher values correspond to a lower diversity. This index is extensively treated in Berger-Parker Index.

Simpson index. This index, elaborated by E. H. Simpson in 1949, accounts for the probability that whatever two individuals randomly sampled from an infinitely large community could belong to the same species:

where pi is the proportion of individuals from species i in the community. To calculate the index for a finite community, the following algorithm can be used:

where n is the number of individuals of species i and N is the total number of individuals.

Like the Berger-Parker index, the Simpson's index may vary from 0 to 1; it has no dimensions, and, in the same way, higher values correspond to lower diversity. This index is treated extensively in Simpson Index.

Hulbert index. This index elaborated by S. H. Hulbert in 1971 is formulated in the following equation:

where N is total number of individuals in a sample and N' is the number ofindividuals ofthe ith species.

The validity of this index depends on the assumption that the individuals of each species are randomly distributed, which is not always the case.

Fisher's a index. This index was introduced by Fisher in 1943 - it is formulated as

a where 5 is the number of taxa, n is the number of individuals, and a is the Fisher's a, which is the shape parameter, fitted by maximum likelihood, under the assumption that the species abundance distribution follows a log series. This has been shown to be the case for some ecological data sets, but can by no means be universally assumed, which restricts the index use to genuine (integral) counts.

Rarefaction curves. Rarefaction curves are plots of the number of individuals on the x-axis against those of the number of species on the y-axis. The more diverse the community the steeper and more elevated the rarefaction curve.

Ranked species abundance (dominance) curves. It consists of ranking the species (or higher taxa) in decreasing order of their importance in terms of abundance or biomass. The ranked abundances, expressed as percentages of the total

abundance of all species, are plotted against the relevant species rank.

K-dominance curves. The K-dominance curve is an index introduced by P. J. Lambshead in 1983 and it is the representation of the accumulated percentage of abundance versus the logarithm ofthe sequence ofspecies ranked in a decreasing order. The slope ofthe straight line obtained allows the valuation of the pollution grade. The higher the slope, the higher the diversity. K-dominance curves are extensively treated in k-Dominance Curves.

Average taxonomic diversity and distinctness measures. Several measures integrating information usually provided by species richness and other diversity indices were proposed. Such measures are based on the different species abundances (denoted by xi, the number of individuals of species i in the sample) and on the taxonomic distance (wj through the classification tree between every pair of individuals (the first from species i and the second from species j ). The measures proposed by R. M. Warwick and K. R. Clarke in 1995 are provided later. In the equations, xi represents the abundance of the ith of s species observed, n (=Pjx) is the total number of individuals in the sample, and xj is the 'distinctness weight' given to the path length linking species i and j in the taxonomy.

1. Taxonomic diversity (A). It consists of the average taxonomic distance apart from every pair of individuals in the sample or the expected path length between any two individuals chosen at random:

where the double summation is over all pairs of species i and j (ij = 1,2,..., S; i < j), and n = ^ ixi, the total number of individuals in the sample.

2. Average taxonomic distinctness. To remove the dominating effect of the species abundances distribution, it was proposed that the average taxonomic diversity index be divided by the Simpson index to give the average taxo-nomic distinctness index:

When quantitative data are not available and samples provided simple lists of species (presence/absence data), the average taxonomic distinctness takes the following form:

where s is the observed number of species in the sample and the double summation ranges over all pairs i and j of the species (i < j).

3. Total taxonomic distinctness (TTD). The TTD was proposed by Warwick and Clarke in 2001 as a useful measure of total taxonomic breadth of an assemblage, as a modification of species richness, which allows for the species interrelatedness:

This measure is the average taxonomic distance from species i to every other species, summed over all species, i = 1,2, ..., s.

4. Variation in taxonomic distinctness (STTD). This measure was proposed in order to account for the 'evenness' of the different taxa distribution across the hierarchical taxo-nomic tree:

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