Supplement 1
Diversity indices
Shannon-Wiener function (H’):
:
whereS = number of species
pi= ni/N = proportion of total sample belonging to ith species
Any base of logarithms can be used for this index. Hutcheson (1970) provides a method for testing significant differences between samples using the statistic:
whereVar H’ = (N = total number of individuals)
n =
(N1and N2 being the total number of individuals in samples 1 and 2 respectively)
N1= eH’ is another way to express the Shannon-Wiener index in units of number of species (number of equally common species required to produce the value of H’).Peet (1974) recommends N1 as the best heterogeneity measure that is sensitive to the abundances of the rare species in the community.
Brillouin index (HB):
whereN = total number of individuals in the population
ni= number of individuals belonging to species i
Simpson’s index (D)
or when the population is finite
wherepi= ni/N = proportion of species i inthe community
ni= number of individuals belonging to species i
N = total number of individuals in the sample =
S = number of species in the sample
Simpson’s index is usually expressed as 1 – D (Simpson’s index of diversity) or1/D interpreted as the number of equally common species required to generate the observed heterogeneity in the sample.
McIntosh’s measure of diversity (DU)
N = total number of individuals in the sample =
U = is an index that measures the Euclidean distance of the community from the origin if it is considered as a point in a S-dimensional space
Inverse Berger-Parker index (1/d)
d-1 = N /Nmax
where Nmax = the number of individuals in the most abundant species.
Anincrease in the value of this index accompaniesdecrease in diversity and anincrease in dominance.
The Shannon-Wiener index measures the uncertainty in predicting species diversity based on the next individual collected. It assumes that individuals are randomly collected from a large population in which the total number of species is known (Pielou, 1975). The Shannon-Wiener measure increases with the number of species in the community and in theory can reach very high values. In practice, for biological communities falls between 1.5 and 3.5 and rarely surpasses 5.0 (Krebs, 1999; Magurran 1988). For most community samples for which the randomness of the sample cannot be guaranteed or if the community is completely censured with every individual accounted for, the Brillouin index is more appropriate than the Shannon-Wiener index (Magurran, 1988). The Brillouin index like the Shannon-Wiener is sensitive to the abundances of the rare species in the community but although the Shannon-Wiener gives the same values when the number of species and their proportional abundances remain constant, this is not the case for the Brillouin index. Simpson’s index expresses the probability of any two individuals drawn at random from a large community belonging to different species.Simpson’s index is less sensitive in species richness and once species richness exceeds a certain number (~ 10 species) the underlying species abundance is determining the index scores. McIntosh’s diversity index accounts for the spatial heterogeneity as distances between species are taken into account too. The Berger-Parker index is a simple dominance measure independent of number of species in the sample S but influenced by the sample size.
Evenness indices
Evenness measures attempt to quantify the unequal representation of a community (containing few dominant species and many relatively uncommon species) against a hypothetical community in which all species are equally uncommon. The most common approach in constructing evenness indices is to scale a heterogeneity measure relatively to its maximal value. Unfortunately many of the indices of evenness based on this approach are not independent of species richness. According to Smith and Wilson (1996) and Pielou (1969) we have chosen the following indices:
Simpson’s index (E1/D)
whereD = Simpson’s index
S = number of species in the sample
It ranges from 0 to 1
Brillouin’s index (E)
whereHB = Brillouin’s index of diversity
HBmax =
[N/S] = the integer of N/S
r = N – S[N/S]
McIntosh’s measure of evenness (EU)
Based on the McIntosh U index and obtained by the formula:
whereN = total number of individuals in the sample
S = number of species in the sample
Camargo’s index (E’)
whereD = Simpson’s index
pi = proportion of species i in the sample
S = number of species in total sample
Smith and Wilson’s index (Evar)
whereD = Simpson’s index
ni = number of individuals of species i in the sample
S = number of species in total sample
Species abundance models
Geometric series
In a geometric series the abundance of species ranked from most to least abundant will be: ni = NCkk(1 - k)i-1 (May, 1975), where
ni= the number of individuals in the ith species,
N = the total number of individuals,
S = the total number of species and
Ck = [1 – (1 – k)S]-1 is a constant which ensures that =N.
An estimate of k is obtained by iteratively attempting to balance the equation:
where Nmin /N = the proportional abundance of the rarest species (Waite, 2000).
Log series
The log series distribution is described by the formula:
S = aln(1+N/a),
where N = the total number of individuals and S = the total number of species, and a is the log series diversity index.
Fitting the log series distribution requires the estimates of two parameters xand α. An estimate of x is obtained by iteratively attempting to balance the equation:
where
N = the total number of individuals and S = the total number of species.
Once a value for xhas been obtained the α is estimated from the relationship: α= Ν(1 - x)/x. The constant α is used as an index of community diversity whose variance was given by the formula:
Var(α) = (Anscombe, 1950)
Lognormal series:
The lognormal distribution is given by:
where S0 = the number of species in the modal octave
S(R)= the number of species in the Rth octave
α = a measure of the spread of the diastribution
e = thebase of the natural logartithm
Empirical studies have shown that usually α 0.2 (Preston 1962; Whittaker 1977; May 1981). If the lognormal distribution fits well to the data it is possible to use the fitted curve to estimate the theoretical total number of species (S*) present in the community:
or as α 0.2
Although the lognormal distribution is an attractive model for species - abundance relationships in practice is very difficult to fit to ecological data (Hughes, 1986). The number of species recorded in the sample depends on the diversity of the community being sampled and the size of the sample. Some rare species with low abundance appear only when large samples are taken. Thus in order to estimate the total number of species in the community a truncated lognormal distribution is fitted to the data. Among the various methods proposed to this effect we have chosen the method described by Pielou (1975) based on the maximunm likelihood method devised by Cohen (1959).
Broken stick model
The broken stick model (Mc Arthur, 1957; May, 1975) is expressed as:
whereN = the total number of individuals
S = the total number of species
S(n) = the number of species in the abundance class of n individuals
References
Anscombe F.J., 1950: Sampling theory of the negative binomial and logarithmic series distributions. Biometrika, 37, 358-382.
Cohen A.C.Jr., 1959: Simplified estimators for the normal distribution when samples are singly censored or truncated.Technometrics, 1, 217-237.
Hughes R.G., 1986: Theories and models of species abundance. American Naturalist, 128, 879-899.
Hutcheson K., 1970: A test for comparing diversities based on the Shannon formula. Journal of Theoretical Biology, 29, 151-154.
KREBS, C.J. (1999) Ecological Methodology, Addison Wesley Longman, Menlo Park, C.A., 620 pp.
MAGURRAN, A.E. (1988) Ecological Diversity and Its Measurement. Croom Helm, London.
May R.M., 1975: Patterns of species abundance and diversity. In Ecology and Evolution of Communities (eds M.L. Cody and J.M. Diamond), Harvard University Press, Cambridge, MA, pp. 81-120.
May R.M., 1981: Patterns in multi-species communities. In Theoretical Ecology: Principles and Applications (ed. R.M. May), Blackwell, Oxford, pp. 197-227.
MACARTHUR, R.H. (1957) On the relative abundance of species. Proceedings of the NationalAcademy of Sciences, 43, 293-295
Peet R.K., 1974: The measurement of species diversity. Annual Review of Ecology and Systematics, 5, 285-307.
Pielou E.C., 1969: The measurement of diversity in different types of biological collections. Journal of Theoretical Biology 13, 131-144.
Pielou E.C., 1975: “Ecological diversity”, Wiley, New York.
Preston F.W., 1962: Canonical distribution of commonness and rarity. Ecology, 43, 185-215, 410-432.
Smith B. & Wilson J.B., 1996: A consumer’s guide to evenness indices. Oikos 76, 70-82.
Waite S, 2000: Statistical Ecology in Practice. Prentice Hall.
Whittaker R.H., 1977: Evolution and measurement of species diversity. Taxon 21, 231-251.
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