1. 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 S. P. Hubbell from UCLA; quotes from Hubbell (2001, pg. 30)

2. Relative-Abundance Patterns
Empirical distributions of relative abundance (two graphical representations)
1. Frequency histogram (Preston plot)
Log2 abundance
Frequency (no. of spp.)
Dominance-diversity
(or rank-abundance)
diagram, after
Whittaker (1975)
Rank
Log10 abundance
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

3. 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
the shapes of relative abundance distributions; then examine the
goodness-of-fit of the predicted distributions to empirical distributions

4. Preston (1948) advocated log-normal distributions
Log normal fit to the data 60
Number of species 40
Dashed line = log-normal
Solid line = log series 20
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)

5. Preston (1948) advocated log-normal distributions
The left-hand portion of the curve may be missing beyond the
“veil line” of a small sample
Number of species
Dashed line = log-normal
Solid line = log series 20
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)

6. Preston (1948) advocated log-normal distributions
The left-hand portion of the curve may be missing beyond the
“veil line” of a small sample
Number of species 40
Dashed line = log-normal
Solid line = log series 20
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)

7. 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 60
Number of species 40
Dashed line = log-normal
Solid line = log series 20
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)

8. Variation among hypothetical rank-abundance curves
Abundance (% of total individuals)
Rank
Figure from Magurran (1988)

9. Examples of relative-abundance data; each data set is best fit by
one of the theoretical distributions
broken-stick
Abundance (% of total individuals)
log-normal
geometric series
Rank
Figure from Magurran (1988)

10. 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
See: “Central Limit Theorem Applet” by R. Todd Ogden, Dept. of Statistics, Univ. of South Carolina at:
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)”

11. Abundance (% of total individuals)
Geometric series – straight line on rank-abundance plot.
Rank
Figure from Magurran (1988)

12. Abundance (% of total individuals)
broken-stick
Abundance (% of total individuals)
log-normal
geometric series
Rank
Figure from Magurran (1988)

13. 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

14. Geometric series:
This pattern of species abundance is found primarily in species-poor (harsh) environments or in early stages of succession
broken-stick
Remember, however, that pattern-fitting alone does not necessarily mean that the mechanistic metaphor is correct!
Abundance (% of total individuals)
log-normal
geometric series
Rank
Figure from Magurran (1988)

15. Abundance (% of total individuals)
Rank
Figure from Magurran (1988)

16. Abundance (% of total individuals)
broken-stick
Abundance (% of total individuals)
log-normal
geometric series
Rank
Figure from Magurran (1988)

17. Even so, there are not many examples of communities
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. 5
Sp. 4
Sp. 3
Sp. 2
Sp. 1
Total Available Resources

18. Broken stick: Uncommon pattern
In any case, remember that pattern-fitting alone does not necessarily mean that the mechanistic metaphor is correct!
broken-stick
Abundance (% of total individuals)
log-normal
geometric series
Rank
Figure from Magurran (1988)

19. Abundance (% of total individuals)
Geometric series – straight line on rank-abundance plot.
Rank
Figure from Magurran (1988)

20. Note the emphasis on rare species!
Log series fit to the data
Log normal fit to the data 60
Number of species 40
Dashed line = log-normal
Solid line = log series 20
Log2 moth abundance (doubling classes or “octaves”)
Figure from Magurran (1988)

21. First described mathematically by Fisher et al. (1943)
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 ( ) from Wikimedia Commons

22. 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
Highlight importance of Motoo Kimura’s neutral theory in population genetics and Island Biogeography for the development of Hubbell’s ideas!

23. Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)

24. Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)

25. Hubbell’s Neutral Theory
Neutral model of local
community dynamics
Figure from Hubbell (2001)

26. 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

27. 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)

28. 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)

29. 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)

30. Barro Colorado Island 50-ha plot, Panama
7 topographically / hydrologically / successionally
defined habitats
Stream 1.28 ha
Swamp 1.20 ha
Mixed 2.64 ha
Young 1.92 ha
Low Plateau ha
High Plateau 6.80 ha
Slope ha

31. Relative abundance is often a function of “Biology”
3355 total stems in the 1.2-ha swamp
171 species, each with
> 60 stems

32. Relative abundance is often a function of “Biology”
303 species, each with at
least 1 stem
In 1990