Evenness and Abundance
Models
James A. Danoff-Burg
Dept. Ecol., Evol., & Envir. Biol.
Columbia University
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
All series / models will have some associated
worked examples and computer time
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Diversity of Diversities
Difference between the diversities is usually one of
relative emphasis of two main envir. aspects
Two key features


Richness
Abundance – our emphasis today
Each index differs in the mathematical method of
relating these features


One is often given greater prominence than the other
Formulae significantly differ between indices
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Evenness
Definition of Evenness


How equally abundant are each of the species?
A simple way to combine abundance and richness
Rarely are all species equally abundant

Some are better competitors, more fecund, more
abundant in general than others
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Evenness increases diversity
Increasing evenness greater diversity

True for all indices
S=4
N=8
S=4
N=8
Higher
Evenness,
Diversity
Site 1
Lecture 3 – Evenness & Species Abundance Models
Site 2
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Evenness as an Indicator
For many ecosystems, high evenness is a sign of
ecosystem health



Don’t have a single species dominating the ecosystem
Often invasives dominate
Paradox of enrichment
• E.g., polluted / enriched Lake Okeechobee, Florida

Disturbed areas are mostly edge species
• Simple biodiversity
• Dominance of a few species ecologically, numerically
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Evenness Across Locations
 Between ecosystem comparability is usually not possible

Some areas have lower biodiversity naturally than others
• Tiaga is naturally much less even than the deciduous forest
• Tiaga is often dominated by a single species (e.g., Blue Spruce)

Seasonality may confound the comparison as well
• Earlier in temperate growing season, less even than later

This is a general principle for most all indices this term
 When would you want to compare across locations?


Trying to prioritize areas for conservation
Based largely on biodiversity (not ecol. uniqueness)
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
New Ideas on Evenness
Types of evenness
Consequences of those types of evenness (a.k.a.
species abundance models)
Methods of testing and evaluation
Introduction to each series
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Types of Evenness
 Types of evenness patterns are called Species
abundance models
 Have four main types of abundance models
1.
2.
3.
4.
Geometric series
Log series
Log-normal series
Broken stick
 Decreasing dominance of a single species from
#1 to #4

Possibly both numerical and ecological dominance
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
How would this
appear if:
Maximal
evenness?
Minimal
evenness?
100
10
Minimal Evenness (one species)
Maximal Evenness
1
Per
Species
Abundance 0.1
0.01
Next to Minimal Evenness (two species)
0.001
10
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
Log Series
0.001
Geometric Series
10
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Abundance Models
Use all the available data

Most thorough representation of the data
Observations



Evenness increases from Geometric  Log  Lognormal  Broken Stick models
Dominance of any one species decreases from
Geometric  Log  Log-normal  Broken Stick
models
Broken stick is the closest nature gets to maximal
evenness
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Other Methods of Abundance
Curves
How else could these graphs be constructed?

How would the data thereby be interpreted?
Possible ideas:


Biomass
Number of species per trophic level
• Trophic level  Species
• Number of species  Abundance

Number of species per feeding guild
• Feeding Guild  Species
• Number of species  Abundance

Others?
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Testability
Simple visual inspection of a single curve is
insufficient
How to test each abundance model differs
between each model




Geometric – simple rank / abundance plots with
abundance on log scale on Y axis (as already seen)
Log – frequency distribution of # of spp vs. abundance
Log-normal – similar to Log, but use a log scale on X
axis
Broken stick – rank / abundance plot, using ranks,
rather than abundance (more in a moment)
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
Log Series
0.001
Geometric Series
10
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Geometric Series
Niche pre-emption is structuring the ecosystem

Species 1 takes a certain percentage of the resources
and prevents others from using them
• Assumes competitive exclusion and resource exhaustion


Species 2 takes a bit more
Continues with other species until all resources are
used and all species are included
Minimal cooperation in ecosystem
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Geometric Series
Assumes that the species abundance is roughly
proportional to total resource use


Linear increase in abundance  linear increase in
resource use
Interspecific per-individual resource use is comparable
Mostly commonly found in species poor
communities



Early succession
Degraded ecosystems (Enriched, Invaded)
Harsh Ecosystems
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Testing the Geometric Series
Model – An Exercise
Should have: straight line species plot & statistical
test
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
Log Series
0.001
Geometric Series
10
20
30
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
40
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Worked Example 2
Magurran p. 130-131
Use the plant feeder data from one of the gardens
Work individually
Create a Rank / Abundance graph as in Fig. 2.4 of
Magurran

Only a gross approximation of whether it actually fits
Estimate k

Use Excel and do so iteratively
Conduct Chi-square goodness of fit test (GOF)

Use SPSS to do this
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
0.001
Geometric Series
10
Log Series
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log Series Model
Closely related to geometric series


Some studies have found both fitting the same data
Similar to geometric in hypotheses about origin of
community
• Arrival of species to a novel environment
• Both say that a few factors predominantly structure the
community
• Both say that one (geometric) or a few (log) species dominate a
species

Log differs from Geometric in assumptions about arrival
• Arrivals are randomly arranged
– Can get some clumped, some long intervals between arrivals
• In geometric series, the arrivals are regular and continual
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log Series Model Mathematics
Base equations




Best fit to a GOF test between expected & obs data
Base form of log series: , (x2 / 2), (x3 / 3), … (xn / n)
Observed S = [-ln(1-x)]
x is calculated iteratively
•
•
•
•

Using the following equation S / N =(1-x) / x[-ln(1-x)]
Solve S / N, then plug numbers in for x to determine its value
0.9 > x > 1.0
If N / S > 20, then x > 0.99
 (a diversity index) = N(1-x) / x
• Plug in x once obtained and get 
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Testing the Log Series Model –
Worked Example 3
Procedure:

Construct a rank abundance plot as in the geometric
• Estimate rough estimate of how well it fits to theoretical







Determine log2 based classes (octaves = doubling abd)
Determine numbers of each class in observed data
Estimate x
Solve for 
Calculate expected abundance for each abd level
Group into classes
Conduct GOF test
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
Log Series
0.001
Geometric Series
10
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log Normal Series
Most communities fit the Log Normal Series


Usually large, mature communities
E.g., temperate forest trees
Ubiquity may be b/c of simple mathematics


Normal distribution is often a consequence of large
numbers
Central Limit Theorem
• Large # factors  random variation will result in normal
distribution
• Central assumption behind parametric statisticss
•  probability with  # of factors
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log Normal Series
Species are grouped into classes


Octaves – most common (Log2)
Any log base can be used
16
2
4
8
16
32
64
128
256
512
14
12
Number
of
Species
10
8
6
4
2
0
Number of Individuals
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log Normal Community
Assembly
Assumption about community formation

Sequential breaking of empty niche space (Sugihara
1980)
• Each species that arrives splits the niche space
• Occupies a niche space proportional to its relative abundance
• Probability of niche space being subdivided is independent of its
sizes
• Breakages occur successively

Mechanism can be through an ecological or
evolutionary process
Fit to model: not necessarily supports assumption
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Other Explanations
Central Limit Theorem

Not necessarily a biological explanation (May 1981)
Ugland & Gray 1982

Species can be divided into three abundance classes
• Rare (65%), Intermediate (25%), Common (10%)



Communities are composed of patches
Abundance of species = sum of abd in all patches
 enough to result in Log Normal distribution
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Log-Normal Miscellany
Missed species




Very rare species will not be sampled
Those less abundant than the critical number are beind
the veil line
Need to estimate how many there should be there
Smaller the sample  increased number behind veil
line
• Because have a higher veil line, relative to larger samples
Simplicity of calculations


Would be there, but for the veil line
Pielou (1975) created a fit to truncated log normal
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Testing the Log-Normal Series
Model – Worked Example 4
 Use Pielou’s Truncated Log Normal process

Estimating # of spp missed to the left of veil line
 Process (14 Steps!)










Sort species, from most to least abundant, ln transform
Calculate mean and variance of community
Determine observed class abundance
Calculate Gamma & Sobs
Estimate Theta
Calculate Mu, Vx, zo
Lookup po of zo
Estimate total S (including those behind veil line)
Lookup po of zo of each class abundance
Conduct GOF test
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Today: Evenness and
Abundance Models
Evenness – a review
Evenness – new issues
Introduction to the Models
Geometric Series
Log Series
Log-Normal Series
Broken-Stick Model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Hypothetical Model Curves
100
10
Broken Stick Model
1
Per
Species
Abundance 0.1
Log-Normal Series
0.01
Log Series
0.001
Geometric Series
10
20
30
40
Species Addition Sequence
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Broken Stick Model
Sometimes called random niche boundary
hypothesis

“broken stick” – MacArthur (1957)
A stick randomly and simultaneously broken into S
pieces
No real relationship between earlier species
presence and niche size of subsequent arrivals

Unlike all earlier models
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Examples of fits to Broken
Stick
Successfully fit in past


Passerine birds (MacArthur 1960)
Minnows and Gastropods (King 1964)
In General:

Best fit in narrowly defined communities of
taxonomically related organisms
No adequate diversity index needed if data fit
Broken Stick

S is adequate measure of diversity
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Broken Stick Model
Most equitable species abundance as ever
happens naturally

Most biologically realistic “uniform” distribution
Theoretically, only looking at one resource

E.g., space
Strongly subject to sample size

Don’t have crowding limitations between species
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Limitations of Broken Stick
Model
Not really applicable to a single sample


Usually conceived of as the average spp abd.
Distribution
Can be misleading to test fit of a single sample to theory
of equal resource partitioning
Fine to use as we will use it

Adherence to a species abundance model
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Testing the Broken Stick Model
– An Exercise
Procedure (5 steps)





Calculate N and S
Determine Observed species in each Log2 abundance
classes
Calculate Expected species for each abundance level
(1-2000 or so)
Determine Expected species in each Log2 abundance
class
Conduct GOF test
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Our Data This Term I
Relationship between plant biodiversity, pest
insect biodiversity, and beneficial insect
biodiversity





Read website at
http://www.columbia.edu/itc/cerc/danoff-burg/webpages/gardens_main.htm
Has a pretty good amount of background on the topic
Field sites were in Manhattan and Brooklyn community
gardens
Data collected during summer 2001
I will also email you the data matrix
• Please begin looking it over so that you are comfortable with it
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Our Data This Term II
Influence of Hemlock Woolly Adelgid on carrion
beetle biodiversity






Separated by many (at least 3) trophic levels
Adelgid is a phloem-feeding insect
Carrion beetles are detritivores or predators on fly
larvae on carrion
Field sites at Black Rock Brook, Black Rock Forest
Data collected during summer 2001
I will also email you these data
• Please begin looking over the data set
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu
Next week:
Abundance, An Introduction
Read



Magurran Ch 2
Magurran Worked Examples 1-6
Southwood & Henderson 2.1, 2.2, 13.1
We will conduct a few evenness and species
abundance models next week
Decide which of the two projects on which you are
interested in working collaboratively

3 people per group
Lecture 3 – Evenness & Species Abundance Models
© 2003 Dr. James A. Danoff-Burg, jd363@columbia.edu