Transcript Concepts of diversity - Chezshaw.demon.co.uk

Concepts of diversity
Peter Shaw
USR
At the moment you probably have a good idea of what
ecological diversity is. By the end of the lecture you will
know more but probably feel more confused! This is because
diversity is not a unitary concept but has several strands;
different emphases give rise to different answers. (And as
an analytical tool for actual practical decisions, I think that
diversity generates more heat than light.)
•Diversity is a deceptive concept. Magurran (1991) likens
the concept of diversity to an optical illusion; the more it is
examined, the less clear it becomes.
What I want to cover:
Types of diversity
Spercies abundance curves
Diversity indices
The usefulness thereof
• Ecologists recognise that the diversity of an ecological
system has 2 facets:
– species number (=richness)
– evenness of distribution
Both these systems have 30 individuals and
3 species. Which is more diverse?
Sp. A Sp. B Sp. C
10
10
10
And how about this?
Sp. A Sp. B Sp. C
1
1
28
A B C
10 2 3
D E F
1 1 1
Figure 2.1 The difference between species richness and
diversity. The diagram shows two samples, both containing
the same number of species and individuals (16 individuals
from 4 species), but with a clearly different balance of
species composition.
Habitat 1 (low diversity)
Habitat 2 (high diversity)
Types of diversity
The word has multiple meanings, multiple facets. There is a primary
distinction into 3 levels, corresponding to different measurement scales:
α, ß and γ
Alpha diversity is the diversity of organisms within a selected habitat or
sample, and is quantified by indices and by rank abundance models.
Beta diversity is an index of the rate of increase of alpha diversity as new
habitats are sampled, so is a measure of the turnover of species along a
spatial gradient.
Finally, gamma diversity is the full diversity (species richness) of an
entire sampled landscape or gradient.
More on α, ß and γ diversity
Scale: These definitions of alpha, beta and gamma diversity are
scale dependent, so that a patch size that a mammal ecologist would
consider to be one habitat (measuring alpha diversity) would be a
mosaic of micro-habitats which a microbial ecologist might count as
containing gamma diversity.
Scalar/vector: α and
γ diversity are scalar quantities, ie may be
represented by a single number. ß diversity is a vector: it must have
a directional component as well as a magnitude.
Species richness
This is what people often (not always) mean when they talk of
biodiversity. The value of species richness is, in principle, easy: you list
all the species in the habitat and count them. IMHO this is as good, if
not better, than any of the more complex indices as a tool to assess the
‘value’ of a habitat. BUT this has flaws:
1: This index is usually criticised as being too sensitive to the occurrence
of rare species.
2: Are you sure of your list? The more samples you take the more
species you find – this links to rank abundance curves (coming next..).
And that’s assuming your taxonomic basis is solid. There are two
unrelated tools available to deal with the problem that you can’t take
infinite samples: Bootstrapping and Rarefaction.
Bootstrapping
Here you (or rather your PC) constructs a curve showing how
the rate of species appearance declines with increasing sample
number. A function is then fitted and its asymptote estimated.
In effect this tells you how many species you WOULD have
found if you had taken infinite samples – the estimated species
richness of the site.
This comes with one health warning: GIGO!
Number
spp
Estimated asymptote
Number samples
Rarefaction
Here you want to compare species richness in two populations from
which you took unequal numbers of observations For example you might
use this to estimate how many spp would have been found if you
sampled 1000 organisms each time. You take Q samples, totalling N
organisms from Q species (species 1 has S1 individuals, Sp2 has S2 etc).
You want to estimate how many spp would have been found in a sample
of R organisms:
Where N
Expected
number of
species in
sample of K
organisms
N-Si
i=Q
Σ
i=1
1-
R
N
R
R
N!
(N-R!)*R!
Number of ways
of picking N
items from a list
of R when order
does not matter
Rank Abundance curves
Since the idea of diversity indices is to encapsulate the relative
abundances of different species, a logical extension of this is simply to
examine the relative abundances visually! The standard tool here is a
rank abundance curve, where the Y axis is log-transformed density and
the X axis is rank order of density (1st, 2nd, 3rd, …87th)
The actual pattern described by these lines can be shoe-horned
into different mathematical models, each making different
assumptions about how the community is arranged. The
actual patterns seen in the field range over most of the
available models and never quite fit any of them!
Broken stick: a space/resource is
broken into random segments, each
length corresponding to a species’
resource allocation (forest birds).
Log-normal (forest plants)
Geometric: each successive sp occupies a
constant proportion of the remaining
space/resource (sub-alpine plants)
Log series alpha
Diversity indices
• Today I only intend to cover 2 indices (my favourites!). These
are: The Simpson index, The Shannon index
– Be aware that there are many more. Hill (1973) noted that
one function could generate infinitely many diversity indices,
each credible. Here was have a total of N individuals from S
species: we define the proportion of the ith species as pi.
1/(1-n)
i=S
Calculate:
Σ
i=1
Pin
When n = 0 this is S – species
richness. N=1 turns out to be the
Shannon index (tho the maths are
rather tricky..). N=2 gives us the
Simpson index. N=3,.. are
unused but perfectly good indices.
Higher values of N downweight
rare species.
The Simpson Index D
• This is an intuitively simple, appealing index. It is the
probability that two consecutive samples drawn
from the same population will be different species.
• It involves sampling individuals from a population one at a
time (like pulling balls out of a hat).
• What is the probability of sampling the same species twice
in 2 consecutive samples? Call this p.
– If there is only 1 species, p = 1.0
– If all samples find different species, p = 0.0
• The probability of sampling species i = pi.
• Hence p(sampling species 1 twice) = pi * pi.
• Hence p(sampling any species twice)
– = p(sp1) + p(sp2)… +p(spN)
• Hence the simplest version of this index = i pi * pi
• This has the counter-intuitive property that 0 = infinite
diversity, 1 = no diversity
• Hence the usual formulation is:
• Simpson’s diversity D
= 1 - i pi * pi
Applying the Simpson index to the
communities listed previously:
Sp. A Sp. B Sp. C
10
10
10
p = 1/3 p=1/3 p = 1/3
D = 1-(1/9 + 1/9 + 1/9)
= 0.667
Sp. A
Sp. B
Sp. C
1
1
28
p=1/30 p = 1/30 p = 28/30
D = 1-(1/900 + 1/900 + 784/900)
= 14/900 = 0.0156
Applied to the demo communities
given previously, this tells what was
already obvious: the first community
had a higher diversity (measured as
evenness of species distribution)
Figure 2.3: The spreadsheet showing the calculation of Simpson’s
index for two observations within the Wimbledon common dataset.
(This sheet shows the cell entries used. Circled cells should be copied
and pasted up to their greyed-out boundaries).
A
1
2
Achillea millefolia
4 Arrhenatherium elatius
5 Festuca rubra
6 Calluna vulgaris
7 Deschampsia flexuosa
8 Heracleum sphondylium
B
raw data
heath 1
C
spoil 1 *
3
G H
heath
F
pi
spoil 1
I
pi2
spoil 1
*
heath
=F4*F4
=F5*F5
0
10 *
=B3/B$12
=C3/C$12
*
0
70 *
=B4/B$12
=C4/C$12
*
=E3*E3
=E4*E4
=B4*B4
80
0*
etc
..
*
etc
..
5
0*
..
..
*
..
..
25
0*
..
..
*
..
..
0
0*
..
..
*
..
..
0
5*
..
..
*
..
..
0
1*
*
..
..
*
*
..
..
*
*
..
..
=1-H14
=1-I14
Trifolium repens
10 Vicia sativa
9
11
12 sum:
13
14
D E
*
=SUM(B3:B10)
=SUM(C
3:C10)
*
etc
..
Simpson's diversity:
=F3*F3
The Shannon (or ShannonWiener) index
• Very often mis-named as the Shannon-Weaver
index, this widely used index comes out of some
very obscure mathematics in information theory. It
dates back to work by Claude Shannon in the Bell
telephone company labs in the 1940s.
• It is the most commonly used, most mis-used and
least understood (!) diversity index.
To calculate:
• H = -i[pi*log(pi)]
• Note that you have the choice of
logarithm base. Really you
should use base 2,when H defines
the information content of the
dataset in bits.
• To do this use
• log2(x) = log10(x) / log10(2)
• An oddity is that the index varies with number of
species, as well as their evenness.
• The maximum possible score for a community with N
species is log(N) – this occurs when all species are
equally frequent.
• Because of this (and log-base problems, and the
difficulty of working out what the score actually means)
I prefer to convert H into an evenness index; H as a
proportion of what it could be if all species were equally
frequent. This is E, the evenness.
• E = H / log(N).
• E = 1 implies total evenness, E  0 implies the opposite.
• This index is independent of log base and of species
number.
Applying the Shannon index to the
communities listed previously:
Sp. A Sp. B Sp. C
10
10
10
p = 1/3 p=1/3 p = 1/3
H10 = -3*(log10(1/3)*1/3)
= -3*(log10(1/3)*1/3)
= 0.477
E = H / log(3) = 1,
ie this community is as
even as it could be
Sp. A
Sp. B
Sp. C
1
1
28
p=1/30 p = 1/30 p = 28/30
H10 = 1/30*log10(1/30)
+ 1/30*log10(1/30)
+ 28/30*log10(28/30)
= 0.126
E = H / log(3) = 0.265
Figure 2.5: The spreadsheet showing the calculation of Shannon’s index for two
observations within the Wimbledon common dataset. This sheet shows the cell entries
used. Circled cells should be copied and pasted up to their greyed-out boundaries. Note that
the value 8 used in the equitability calculation is the number of species in the dataset. This
could also be obtained by counting the number of names in column B.
A
C
1
B
raw data
2
Heath 1
3 Achillea millefolia
Arrhenatherium
4 elatius
5 Festuca rubra
6 Calluna vulgaris
Deschampsia
7 flexuosa
Heracleum
8 sphondylium
9 Trifolium repens
10
11 Vicia sativa
12
13
14
15
16
sum:
E
pi
F
G
H
pi * log10pi
I
J
Spoil 1*
Heath 1
Spoil 1
*
Spoil 1
*
0
10 *
=B3/C$12
=C3/D$12
*
Heath 1
= IF(E3 = 0, 0,
E3*LOG10(E3))
= IF(F3 = 0, 0,
F3*LOG10(F3))
*
0
70 *
=B4/C$12
=C4/D$12
*
80
5
0*
0*
=B5/C$12
=C5/D$12
= IF(F4 = 0, 0,
F4*LOG10(F4))
= IF(F5 = 0, 0,
F5*LOG10(F5))
etc..
..
*
*
= IF(E4 = 0, 0,
E4*LOG10(E4))
= IF(E5 = 0, 0,
E5*LOG10(E5)')
etc..
..
*
*
25
0*
..
..
*
..
..
*
0
5
1
..
..
..
..
..
..
..
..
..
..
*
*
**
..
..
*
*
**
=SUM(H3:H10)
=SUM(I3:I10)
=0-H12
=0-I12
=H14/LOG10(2)
=H14/LOG10(8)
=I14/LOG10(2)
=I14/LOG10(8)
0
0
0
D
=SUM(B3: =SUM(
B10)
C3:C10)
*
*
**
=SUM(E3:E
10)
=SUM(F3:F10)
Shannon diversity
base 10:
Shannon diversity
base 2:
Equitability
*
Other diversity indices
The Berger-Parker Index is the simplest and most easily understood
diversity index, since it only calculates the proportion of the commonest
species in a sample:
d = Nmax/N
The Brillouin index
It is calculated as:
HB =
ln(N!) - Σln(ni!)
N
Where HB = the Brillouin index, N = total number of individuals in the
sample, ni = number of individual of species i, N! means the factorial of
N = 1 * 2 * 3 * 4... * N, ln(x) = natural logarithm of x (or logarithm base
e)
Measures of Beta diversity
I have said a lot about alpha diversity – since this is easy! Beta
diversity is more messy since alpha diversity is mixed with
spatial information. This is handled in several ways which can
be squeezed under the heading of Ordination: this includes
direct gradient analysis (plotting species along a pre-measured
axis), mosaic diagrams and indirect ordinations (later in the
course).
One particular form of ordination, Bray-Curtis ordination, was
invented to handle beta diversity along perceived spatial
gradients.
Figure 4.2. Vegetation zonation away from a stream edge in
a floodplain forest (redrawn from Hughes & Cass 1997).
Open Typha
water swamp
Onoclea
zone
Shrub zone
0
25
Distance from stream edge, m
Closed
tree
canopy
50
Hardwood
canopy
75
2000
Lodgepole
pine
Red fir
Subalpine
conifer
Jeffrey
pine
pinyon
Wet meadows
3000
Alpine dwarf scrub
Mixed conifers
Ponserosa pine
1000
Montane
hardwood
hardwoods
Elevation, m
4000
Chaparral
Montane
hardwood
Valley-foothill hardwood
Annual grassland
Moist….…………………Dry
Figure 4.3 A mosaic diagram, in
this case showing the distribuition
of vegetation types in relation to
elevation and moisture in the
Sequoia national park. This is an
example of a direct ordination,
laying out communities in relation
to two well-understood axes of
variation. Redrawn from Vankat
(1982) with kind permission of the
California botanical society.
Does diversity matter?
I want to avoid any attempt at analysing the ethical question ‘Does
Biodiversity Matter?’ and remain focussed on assessing whether
there are any sound ecological reasons for using diversity measures
beyond giving you one more statistic in your thesis :=)
The diversity – stability hypothesis. (questionable, probably a
long-lived myth)
The diversity – functionality hypothesis: diverse ecosystems
function ore effectively as measured by rates of energy capture,
nutrient cycling etc. This comes out of John Lawton’s Ecotrons,
mini-ecosystems where species assemblage is controlled.
Diversity driving conservation decisions: Nice idea but in
practice diversity indices tend to be lowest for really important
habitats like lowland heaths or the mountain gorilla habitat!!