“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)

Relative-Abundance Patterns

Empirical distributions of relative abundance (two graphical representations)

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

1. Frequency histogram(Preston plot)

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Log2 abundance

Fre

quen

cy (

no.

of s

pp.)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 50 100 150 200 250 300 350

2. Dominance-diversity (or rank-abundance)

diagram, after Whittaker (1975)

Rank

Log 1

0 a

bund

ance

Relative-Abundance Patterns

1. Descriptive (inductive):

Fitting curves to empirical distributions

2. Mechanistic (deductive):

Creating mechanisticmodels that predict theshapes of distributions

Approaches used to better understand relative abundance distributions

A combined approach: create simple, mechanistic models that predictrelative abundance distributions; adjust the model to maximize

goodness-of-fit of predicted distributions to empirical distributions

Relative-Abundance Patterns

Preston (1948) advocated log-normal distributions

Num

ber

of s

peci

es

1 16 256 4096

60

40

20

Log normal fit to the data

Figure from Magurran (1988)

Log2 moth abundance (doubling classes or “octaves”)

The left-hand portion of the curve may be missing beyond the “veil line” of a small sample

1 16

20

Preston (1948) advocated log-normal distributions

Figure from Magurran (1988)

Num

ber

of s

peci

es

Log2 moth abundance (doubling classes or “octaves”)

1 16 256

40

20

The left-hand portion of the curve may be missing beyond the “veil line” of a small sample

Preston (1948) advocated log-normal distributions

Figure from Magurran (1988)

Num

ber

of s

peci

es

Log2 moth abundance (doubling classes or “octaves”)

1 16 256 4096

60

40

20

Log-normal fit to the data

The left-hand portion of the curve may be missing beyond the “veil line” of a small sample

Preston (1948) advocated log-normal distributions

Figure from Magurran (1988)

Num

ber

of s

peci

es

Log2 moth abundance (doubling classes or “octaves”)

Figure from Magurran (1988)

Rank

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)Variation among hypothetical rank-abundance curves

Figure from Magurran (1988)

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Rank

Examples of relative-abundance data; each data set is best fit by one of the theoretical distributions

geometric series

broken-stick

log-normal

Log-normal distribution:

Log of species abundances are normally distributed

Why might this be so?

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

May (1975) suggested that it arises from the statistical properties of large numbers and the Central Limit Theorem

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)”

Figure from Magurran (1988)

Rank

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Figure from Magurran (1988)

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Rank

geometric series

broken-stick

log-normal

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

…

Sp. 1

Sp. 2

Sp. 3

Sp. 4

Total Available Resources

Does this seem like a reasonable mechanism, or just a metaphor with uncertain relevance to the real world?

Figure from Magurran (1988)

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Rank

geometric series

broken-stick

log-normal

Geometric series:

This pattern of species abundance is found primarily in species-poor (harsh) environments or in early stages of succession

Remember, however, that pattern-fitting alone does not necessarily mean that the mechanistic metaphor is correct!

Figure from Magurran (1988)

Rank

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Figure from Magurran (1988)

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Rank

broken-stick

log-normal

geometric series

Broken Stick:

The sub-division of niche space among species may be analogous to randomly breaking a stick into S pieces (MacArthur 1957)

Sp. 1

Sp. 2

Sp. 3

Sp. 4

Total Available Resources

Sp. 5

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

Figure from Magurran (1988)

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Rank

broken-stick

log-normal

geometric series

Broken stick:

Uncommon pattern

In any case, remember that pattern-fitting alone does not necessarily mean that the mechanistic metaphor is correct!

Figure from Magurran (1988)

Rank

Abu

ndan

ce (

% o

f tot

al in

divi

dual

s)

Num

ber

of s

peci

es

1 16 256 4096

60

40

20

Figure from Magurran (1988)

Log2 moth abundance (doubling classes or “octaves”)

Log series fit to the data

Note the emphasis on rare species!

Log normal fit to the data

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 communityand 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

Neutral model of localcommunity dynamics

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

Neutral model of localcommunity dynamics

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

Neutral model of localcommunity dynamics

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

The source pool (metacommunity) relative abundance isgoverned by its size, speciation and extinction rates

Local community relative abundance is additionallygoverned 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

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 communityand 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

Theta () = 2JM

JM = metacommunity size

= speciation rate

The model provides predictions that match

nearly all observed distributions of relative

abundance

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

Theta () = 2JM

JM = metacommunity size

= speciation rate

m = immigration rate

Model predictions can be fit to the observed relative-abundance

distribution of trees on BCI

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

Theta () = 2JM

JM = metacommunity size

= speciation rate

m = immigration rate

Does the good fit of the model to real data mean that we now understand

relative abundance distributions

mechanistically?

Figure from Hubbell (2001)

Hubbell’s Neutral Theory

Low Plateau 24.80 ha High Plateau 6.80 ha

Swamp 1.20 haStream 1.28 ha

Slope 11.36 ha

Mixed 2.64 ha Young 1.92 ha

7 topographically / hydrologically / successionallydefined habitats

Barro Colorado Island 50-ha plot, Panama

0

1

2

3

4

5

0 50 100 150 200

Relative abundance rank

Lo

g10

(N)

Positiveassociationwith swamp

Neutralassociationwith swamp

Negativeassociationwith swamp

171 species,each with> 60 stems

3355 total stems in the 1.2-ha swamp

Relative abundance is often a function of “Biology”

0

1

2

3

4

5

0 50 100 150 200 250 300 350

Relative abundance rank

Lo

g10

(N) Ficus &

Cecropia

Non-Moraceae

OtherMoraceae

303 species,each with atleast 1 stemIn 1990

Relative abundance is often a function of “Biology”