Transcript Chapter 4

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
Table 4.1. Which plot is more diverse?
plot 1
plot 2
species 1
10
28
species 2
10
1
species 3
10
1
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
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:
S
pi  x i /  x i
i 1
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"

a
Da =   pi 
 i

S
1
1-a
Can think of a as
the weight given to
dominance of a
species.
Diversity (Da)
18
16
14
12
10
8
6
4
2
0
Perfect Equitability
Strong Inequitability
Species richness
Shannon index
Simpson's index
-4
-2
0
2
4
parameter a
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).
D0 = species richness
S
D0 =
p
i
When a = 0, Da is simply species richness.
0
i
D2 and Simpson's index

2
D2 =   pi 
 i

S
-1
=
1
S
2
p
 i
i
Simpson’s (1949) original index (1/D2) is a measure of dominance
rather than diversity
The complement of Simpson's index of dominance is
S
Diversity  1   p
2
i
i
and is a measure of diversity. It is the likelihood that two randomly
chosen 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 use
limits to define D1 as a approaches 1 then
D1 = lim a  1  Da 
S


-1
D1 = log  -  pi log pi
 i

The logarithmic form of D1 is the Shannon-Wiener index (H’), which
measures the “information content” of a sample unit:
S
H '  log( D1) =   pi log pi
i
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?
Plot 1
Plot 2
Sp A
99
50
Sp B
1
50
pA
0.99
0.50
pB
0.01
0.50
2
information content
 H '    pi log pi
i 1
For plot 1
H'  1 099
.  log(0.99)  0.01  log(0.01)  0.024
For plot 2
H'  1 0.5  log(0.5)  0.5  log(0.5)  0.301
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
Underlying
gradient model
Direct gradient
Indirect gradient
No specific gradient
Data type
Quantitative
HC (Whittaker's half changes)
G (gleasons)
Minchin’s R
 (total gradient length)
Axis length in DCA
Presence-absence
D Dissimilarity  (half changes)
w (Whittaker's beta, /-1) =
T (beta turnover)
Minchin’s R
Axis length in DCA
1.0
R
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
Position on Gradient
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.
The amount of change, , is the integral of the rate of change:
b
=
 R(x) dx
a
where a and b refer to the ends of an ecological gradient x.
% Similarity
100
80
60
40
20
0
0
1
2
3
4
5
6
7
Separation along gradient, half
changes
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%.
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.
N
dY i ( x )
R( x ) =  |
|
dx
i=1
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
n 1
G  2 [ IA  PS ( i, i  1)]
i 1
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).
Minchin measured beta diversity using the mean range of the
species’ physiological response function:
Minchin's R 
L
n
r / n
i 1
i
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.
Oksanen and Tonteri (1995) proposed the following measure of total
gradient length:
b
 ab    A ( x )  dx
a
where A is the absolute compositional turnover (rate of change) of the
community between points a and b on gradient x.
Half changes are related to the rate of change by:
HC(a,b)
R(a,b) =
x (a,b)
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):
log( IA) - log( PS ( a,b))
HC( a,b) =
log 2
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).
Beta turnover
measures the amount of change as the "number of communities."
T
| g + l|
=
2
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 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
Sc
w =
1
S
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.
As a rule of thumb:
w < 1 is low
1 < w < 5 is medium
w > 5 is high
Half changes corresponding to the average dissimilarity
among sample units:
D
log(1  D )
=
log( 0.5)
This can be rewritten as
Beta diversity, half changes
7
6
5
4
3
2
1
0
D
1  D  0.5
0
0.2
0.4
0.6
0.8
1
Dissimilarity, proportion
Figure 4.4. Conversion of average dissimilarity, D
measured with a proportion coefficient, to beta
diversity measured in half changes (D).
First-order jackknife estimator
(Heltshe & Forrester 1983, Palmer 1990)
r1( n-1)
Jack 1 = S +
n
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:
2
r1( 2n - 3)
r2( n - 2 )
Jack 2 = S +
n
n( n - 1)
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"
H
J =
log S
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 evenness
components based on the equation
H  = ln(S) + ln(E)
where
E = eH'/S
e is the base of the natural logarithms.
120
0.4
80
Species
Distance
0.2
40
0
Average Distance
Average Number of Species
0.6
0.0
0
40
80
Number of Subplots
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.
Species – Area equations
Arrhenius (1921):
b
S = cA
where
S is the number of species,
A is the area of the sample, and
c and b are fitted coefficients.
In log form:
log S = log c + b log A
Gleason (1922) proposed a similar equation:
S = c + b log A
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).
On-frame
Mountains
Piedmont
Coastal plain
Off-frame
urban/industrial
QA plots
N
Diversity measure
alpha beta
gamma
30
13
19
17
16.4
12.5
11.9
9.2
6.5
4.3
5.5
4.6
107
54
66
64
53
22.7
7.0
158