From: McCune, B. & J. B. Grace. 2002. Analysis ofEcological Communities. MjM Software Design,Gleneden Beach, Oregon http://www.pcord.com
Tables, Figures, and Equations
CHAPTER 4
Species Diversity
Alpha diversity: diversity in individual sample units
Beta diversity: amount of compositional variation in asample (a collection of sample units)
Gamma diversity: overall diversity in a collection ofsample units, often "landscape-level" diversity
Proportionate diversity measures
For an observed abundance xi, (numbers, biomass, cover, etc.) of speciesi 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 "effectivenumber of species"
Can think of a asthe weight given todominance of aspecies.
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 withstrong inequitability in abundance and a sample unit with perfect equitability inabundance (all species present have equal abundance; see Table 4.1).
D0 = species richness
When a = 0, Da is simply species richness.
D2 and Simpson's index
Simpson’s (1949) original index (1/D2) is a measure of dominancerather than diversity
The complement of Simpson's index of dominance is
and is a measure of diversity. It is the likelihood that two randomlychosen individuals will be different species.
D1 and Shannon-Wiener index
If a = 1 then D1 is a nonsense equation because the exponent is 1/0. But if we uselimits to define D1 as a approaches 1 then
The logarithmic form of D1 is the Shannon-Wiener index (H’), whichmeasures 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.
Box 4.1. How is information related to uncertainty?
For plot 1
For plot 2
Table 4.2. Some measures of beta diversity. See Wilson and Mohler (1983) andWilson and Shmida (1984) for other published methods. “DCA” is detrendedcorrespondence analysis. A direct gradient refers to sample units taken along anexplicitly measured environmental or temporal gradient. Indirect gradients aregradients in species composition along presumed environmental gradients
Figure 4.2. Example of rate of change, R, measured as proportionaldissimilarity in species composition at different sampling positions along anenvironmental gradient. Peaks represent relatively abrupt change in speciescomposition. This data set is a series of vegetation plots over a low mountainrange. In more homogeneous vegetation, the curve and peaks would be lower.
The amount of change, , is the integral of the rate of change:
where a and b refer to the ends of an ecological gradient x.
Figure 4.3. Hypothetical decline in similarity in species composition as afunction of separation of sample units along an environmental gradient,measured in half changes. Sample units one half change apart have asimilarity of 50%.
Wilson and Mohler (1983) introduced "gleasons" as a unit ofspecies change. This measures the steepness of species responsecurves. It is the sum of the slopes of individual species at eachpoint along the gradient.
where Y is the abundance of species i at position x along thegradient. This can be integrated into an estimate of betadiversity along a whole gradient with
where PS(a,b) is the percentage similarity of sample units a and band IA is the expected similarity of replicate samples (the similarityintercept on Fig. 4.3).
Minchin measured beta diversity using the mean range of thespecies’ physiological response function:
where ri is the range of species i along the gradient, L is the length of thegradient, and r and L are measured in the same units.
Oksanen and Tonteri (1995) proposed the following measure of totalgradient length:
where A is the absolute compositional turnover (rate of change) of thecommunity between points a and b on gradient x.
Half changes are related to the rate of change by:
This semi-log plot is the basis for Whittaker's (1960) method of calculatingthe 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 yintercept on the figure just described).
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:
The simplest descriptor of beta diversity and one that can be applied toany community sample, is
where is the landscape-level diversity and is the average diversity in asample unit. Whittaker (1972) stated that a generally appropriatemeasure of this is
where
w is the beta diversity,
Sc is the number of species in the composite sample (the number ofspecies in the whole data set), and
S is the average species richness in the sample units.
As a rule of thumb:
w < 1 is low
1 < w < 5 is medium
w > 5 is high
Half changes corresponding to the average dissimilarityamong sample units:
Figure 4.4. Conversion of average dissimilarity,measured with a proportion coefficient, to betadiversity measured in half changes (D).
This can be rewritten as
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.
The second-order jackknife estimator (Burnham & Overton 1979; Palmer 1991)is:
where r2 = the number of species occurring in exactly two sample units.
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.
Hayek and Buzas (1997) partitioned H' into richness and evennesscomponents based on the equation
where
E = eH'/S
e is the base of the natural logarithms.
Figure 4.5. Species-area curve (heavy line) used to assess sample adequacy, based onrepeated subsampling of a fixed sample (in this case containing 92 sample units and 122species). Dotted lines represent 1 standard deviation. The distance curve (light line)describes the average Sørensen distance between the subsamples and the whole sample, asa function of subsample size.
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:
Gleason (1922) proposed a similar equation:
Table 4.3. Species diversity of epiphytic macrolichens in thesoutheastern United States. Alpha, beta, and gamma diversityare defined in the text (table from McCune et al. 1997).