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Relative-Abundance Patterns
“No other general attribute of ecological communities
besides species richness has commanded more
theoretical and empirical attention than
relative species abundance”
“Commonness, and especially rarity, have long fascinated ecologists…,
and species abundance is of central theoretical and practical
importance in conservation biology…”
“In particular, understanding the causes and consequences of rarity is a
problem of profound significance because most species are uncommon
to rare, and rare species are generally at greater risk to extinction”
Photo of Steve Hubbell from UCLA; quotes from Hubbell (2001, pg. 30)
Relative-Abundance Patterns
Empirical distributions of relative abundance (two graphical representations)
2. Dominance-diversity
(or rank-abundance)
diagram, after
Whittaker (1975)
1. Frequency histogram
(Preston plot)
5
40
Log10 abundance
Frequency (no. of spp.)
45
35
30
25
20
15
10
4.5
4
3.5
3
2.5
2
1.5
1
5
0.5
0
0
1
2
3
4
5
6
7
8
9
10
Log2 abundance
11
12
13
14
15
16
0
50
100
150
200
250
300
350
Rank
Data from BCI 50-ha Forest Dynamics Plot, Panama; 229,069 individual trees of 300
species; most common species nHYBAPR=36,081; several rarest species nRARE=1
Relative-Abundance Patterns
Approaches used to better understand relative abundance distributions
1. Descriptive (inductive):
Fitting curves to empirical
distributions
2. Mechanistic (deductive):
Creating mechanistic
models that predict the
shapes of distributions
A combined approach: create simple, mechanistic models that predict
relative abundance distributions; adjust the model to maximize
goodness-of-fit of predicted distributions to empirical distributions
Preston (1948) advocated log-normal distributions
Log normal fit to the data
Number of species
60
40
20
1
16
256
4096
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)
Preston (1948) advocated log-normal distributions
Number of species
The left-hand portion of the curve may be missing beyond the
“veil line” of a small sample
20
1
16
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)
Preston (1948) advocated log-normal distributions
Number of species
The left-hand portion of the curve may be missing beyond the
“veil line” of a small sample
40
20
1
16
256
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)
Preston (1948) advocated log-normal distributions
The left-hand portion of the curve may be missing beyond the
“veil line” of a small sample
Log-normal fit to the data
Number of species
60
40
20
1
16
256
4096
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)
Abundance (% of total individuals)
Variation among
hypothetical rankabundance curves
Rank
Figure from Magurran (1988)
Abundance (% of total individuals)
Examples of relativeabundance data; each
data set is best fit by
one of the theoretical
distributions
broken-stick
log-normal
geometric series
Rank
Figure from Magurran (1988)
Log-normal distribution:
Log of species abundances are normally distributed
Why might this be so?
May (1975) suggested that it arises from the statistical properties of
large numbers and the Central Limit Theorem
Central Limit Theorem: When a large number of factors combine to
determine the value of a variable (number of individuals per species),
random variation in each of those factors (e.g., competition, predation,
etc.) will result in the variable being normally distributed
J. H. Brown (1995, Macroecology, pg. 79):
“…just as normal distributions are produced by additive combinations of
random variables, lognormal distributions are produced by multiplicative
combinations of random variables (May 1975)”
Rank
Figure from Magurran (1988)
Abundance (% of total individuals)
Abundance (% of total individuals)
broken-stick
log-normal
geometric series
Rank
Figure from Magurran (1988)
Geometric series:
The most abundant species usurps proportion k of all available
resources, the second most abundant species usurps proportion k
of the left-overs, and so on down the ranking to species S
Does this seem like a reasonable mechanism, or just a
metaphor with uncertain relevance to the real world?
Sp. 4
Sp. 3
Sp. 2
Sp. 1
Total Available Resources
…
Abundance (% of total individuals)
Geometric series:
This pattern of species
abundance is found
primarily in species-poor
(harsh) environments or
in early stages of
succession
broken-stick
log-normal
geometric series
Rank
Figure from Magurran (1988)
Remember, however,
that pattern-fitting
alone does not
necessarily mean that
the mechanistic
metaphor is correct!
Rank
Figure from Magurran (1988)
Abundance (% of total individuals)
Abundance (% of total individuals)
broken-stick
log-normal
geometric series
Rank
Figure from Magurran (1988)
Broken Stick:
The sub-division of niche space among species may be analogous to
randomly breaking a stick into S pieces (MacArthur 1957)
This results in a somewhat more even distribution of abundances among
species than the other models, which suggests that it should occur when
an important resource is shared more or less equitably among species
Even so, there are not many examples of communities
with species abundance fitting this model
Sp. 4
Sp. 3
Sp. 2
Sp. 1
Total Available Resources
Sp. 5
Broken stick:
Abundance (% of total individuals)
Uncommon pattern
In any case, remember
that pattern-fitting
alone does not
necessarily mean that
the mechanistic
metaphor is correct!
broken-stick
log-normal
geometric series
Rank
Figure from Magurran (1988)
Rank
Figure from Magurran (1988)
Abundance (% of total individuals)
Note the emphasis on rare species!
Log series fit to the data
Log normal fit to the data
Number of species
60
40
20
1
16
256
4096
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)
Log series:
First described mathematically by Fisher et al. (1943)
Log series takes the form: x, x2/2, x3/3,... xn/n
where x is the number of species predicted to
have 1 individual, x2 to have 2 individuals, etc...
Fisher’s alpha diversity index () is usually not biased by sample size
and often adequately discriminates differences in diversity among
communities even when underlying species abundances do not exactly
follow a log series
You only need S and N to calculate it… S =  * ln(1 + N/)
But do we understand why the distribution of relative abundances
often takes on this form?
Photo of R. A. Fisher (1890-1962) from Wikimedia Commons
Hubbell’s Neutral Theory
An attempt to predict relative-abundance distributions from neutral models of
birth, death, immigration, extinction, and speciation
Assumptions: individuals play a zero-sum game within a community
and have equivalent per capita demographic rates
Immigration from a source pool occurs at random; otherwise species
composition in the community is governed by community drift
Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)
Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)
Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)
Hubbell’s Neutral Theory
An attempt to predict relative-abundance distributions from neutral models of
birth, death, immigration, extinction, and speciation
Assumptions: individuals play a zero-sum game within a community
and have equivalent per capita demographic rates
Immigration from a source pool occurs at random; otherwise species
composition in the community is governed by community drift
The source pool (metacommunity) relative abundance is
governed by its size, speciation and extinction rates
Local community relative abundance is additionally
governed by the immigration rate
By adjusting these (often unmeasurable) parameters, one is able to fit predicted
relative abundance distributions to those observed in empirical datasets
Hubbell’s Neutral Theory
The model provides
predictions that match
nearly all observed
distributions of relative
abundance
Theta () = 2JM
JM = metacommunity
size
 = speciation rate
Figure from Hubbell (2001)
Hubbell’s Neutral Theory
Model predictions can be
fit to the observed
relative-abundance
distribution of trees on
BCI
Theta () = 2JM
JM = metacommunity
size
 = speciation rate
m = immigration rate
Figure from Hubbell (2001)
Hubbell’s Neutral Theory
Does the good fit of the
model to real data mean
that we now understand
relative abundance
distributions
mechanistically?
Theta () = 2JM
JM = metacommunity
size
 = speciation rate
m = immigration rate
Figure from Hubbell (2001)
Barro Colorado Island 50-ha plot, Panama
7 topographically / hydrologically / successionally
defined habitats
Stream 1.28 ha
Swamp 1.20 ha
Low Plateau 24.80 ha
Mixed 2.64 ha
High Plateau 6.80 ha
Young 1.92 ha
Slope 11.36 ha
Relative abundance is often a function of “Biology”
5
4
Log10(N)
3355 total stems in the 1.2-ha swamp
3
Positive
association
with swamp
2
Neutral
association
with swamp
171 species,
each with
> 60 stems
1
Negative
association
with swamp
0
0
50
100
150
Relative abundance rank
200
Relative abundance is often a function of “Biology”
5
Log10(N)
4
3
Ficus &
Cecropia
2
NonMoraceae
303 species,
each with at
least 1 stem
In 1990
1
Other
Moraceae
0
0
50
100
150
200
250
Relative abundance rank
300
350