From: McCune, B'

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CHAPTER 4 Species Diversity
Tables, Figures, and Equations
From McCune, B. J. B. Grace. 2002. Analysis
of Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http//www.pcord.com
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Alpha diversity diversity in individual sample
units Beta diversity amount of compositional
variation in a sample (a collection of sample
units) Gamma diversity overall diversity in a
collection of sample units, often
"landscape-level" diversity
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Proportionate diversity measures For an
observed abundance xi, (numbers, biomass, cover,
etc.) of species i in a sample unit, let pi
proportion of individuals belonging to species i

a constant that can be assigned and alters the
property of the measure S number of species Da
diversity measure based on the constant a. The
units are "effective number of species"
Can think of a as the weight given to dominance
of a species.
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Figure 4.1. Influence of equitability on Hill's
(1973a) generalized diversity index. Diversity
is shown as a function of the parameter a for two
cases a sample unit with strong inequitability
in abundance and a sample unit with perfect
equitability in abundance (all species present
have equal abundance see Table 4.1).
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D0 species richness
When a 0, Da is simply species richness.
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D2 and Simpson's index
Simpsons (1949) original index (1/D2) is a
measure of dominance rather than diversity
The complement of Simpson's index of dominance is
and is a measure of diversity. It is the
likelihood that two randomly chosen individuals
will be different species.
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D1 and Shannon-Wiener index If a 1 then D1 is
a nonsense equation because the exponent is 1/0.
But if we use limits to define D1 as a approaches
1 then
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The logarithmic form of D1 is the Shannon-Wiener
index (H), which measures the information
content of a sample unit
The units for D1 are "number of species of equal
abundance The units for H' are the log of the
number of species of equal abundance.
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Box 4.1. How is information related to
uncertainty?
For plot 1
For plot 2
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Table 4.2. Some measures of beta diversity. See
Wilson and Mohler (1983) and Wilson and Shmida
(1984) for other published methods. DCA is
detrended correspondence analysis. A direct
gradient refers to sample units taken along an
explicitly measured environmental or temporal
gradient. Indirect gradients are gradients in
species composition along presumed environmental
gradients
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Figure 4.2. Example of rate of change, R,
measured as proportional dissimilarity in species
composition at different sampling positions along
an environmental gradient. Peaks represent
relatively abrupt change in species composition.
This data set is a series of vegetation plots
over a low mountain range. In more homogeneous
vegetation, the curve and peaks would be lower.
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The amount of change, b, is the integral of the
rate of change
where a and b refer to the ends of an ecological
gradient x.
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Figure 4.3. Hypothetical decline in similarity
in species composition as a function of
separation of sample units along an environmental
gradient, measured in half changes. Sample units
one half change apart have a similarity of 50.
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Wilson and Mohler (1983) introduced "gleasons" as
a unit of species change. This measures the
steepness of species response curves. It is the
sum of the slopes of individual species at each
point along the gradient.
where Y is the abundance of species i at position
x along the gradient. This can be integrated
into an estimate of beta diversity along a whole
gradient with
where PS(a,b) is the percentage similarity of
sample units a and b and IA is the expected
similarity of replicate samples (the similarity
intercept on Fig. 4.3).
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Minchin measured beta diversity using the mean
range of the species physiological response
function
where ri is the range of species i along the
gradient, L is the length of the gradient, and r
and L are measured in the same units.
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Oksanen and Tonteri (1995) proposed the following
measure of total gradient length
where ?A is the absolute compositional turnover
(rate of change) of the community between points
a and b on gradient x.
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Half changes are related to the rate of change
by
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This semi-log plot is the basis for Whittaker's
(1960) method of calculating the number of half
changes along the gradient segment from a to b,
HC(a,b)
where PS(a,b) is the percentage similarity of
sample units a and b IA is the expected
similarity of replicate samples (the y intercept
on the figure just described).
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Beta turnover measures the amount of change as
the "number of communities."
where g the number of species gained, l the
number of species lost the average species
richness in the sample units
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The simplest descriptor of beta diversity and one
that can be applied to any community sample, is
where ? is the landscape-level diversity and is
the average diversity in a sample unit.
Whittaker (1972) stated that a generally
appropriate measure of this is
where ?w is the beta diversity, Sc is the
number of species in the composite sample (the
number of species in the whole data set), and S
is the average species richness in the sample
units.
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As a rule of thumb ?w lt 1 is low 1 lt ?w lt 5
is medium ?w gt 5 is high
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Half changes corresponding to the average
dissimilarity among sample units
This can be rewritten as
Figure 4.4. Conversion of average dissimilarity,
measured with a proportion coefficient, to beta
diversity measured in half changes (?D).
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First-order jackknife estimator (Heltshe
Forrester 1983, Palmer 1990)
where S the observed number of species, r1
the number of species occurring in only one
sample unit, and n the number of sample units.
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The second-order jackknife estimator (Burnham
Overton 1979 Palmer 1991) is
where r2 the number of species occurring in
exactly two sample units.
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Evenness An easy-to-use measure (Pielou 1966,
1969) is "Pielou's J"
where H' is the Shannon-Wiener diversity measure
S is the average species richness. If there
is perfect equitability then log(S) H' and J
1.
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Hayek and Buzas (1997) partitioned H' into
richness and evenness components based on the
equation
where E eH'/S e is the base of the natural
logarithms.
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Figure 4.5. Species-area curve (heavy line) used
to assess sample adequacy, based on repeated
subsampling of a fixed sample (in this case
containing 92 sample units and 122 species).
Dotted lines represent ? 1 standard deviation.
The distance curve (light line) describes the
average Sørensen distance between the subsamples
and the whole sample, as a function of subsample
size.
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Species Area equations Arrhenius (1921)
where S is the number of species, A is the area
of the sample, and c and b are fitted
coefficients. In log form
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Gleason (1922) proposed a similar equation
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Table 4.3. Species diversity of epiphytic
macrolichens in the southeastern United States.
Alpha, beta, and gamma diversity are defined in
the text (table from McCune et al. 1997).