Patterns in diversity and abundance

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• Patterns in diversity and abundance
• A. Species number (richness)
• 1. Methodological concerns with counting
• a. what is a species?
• b. rarefaction and maximum species number
• c. species density
• 2. Application to biogeographic patterns
• a. latitudinal gradients
• b. topographic relief
• c. Peninsulas
• d. Islands

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• Common explanations for higher diversity in the
  tropics
• -- time for evolutionary diversification
• -- time for migration to appropriate habitats
• -- spatial heterogeneity (topographic or plant
  hosts)
• -- competition more intense in the tropics,
  leading to more niche evolution
• -- predation more intense in the tropics,
  allowing more coexistence
• -- environmental stability allows more niche
  evolution
• -- greater environmental instability in the
  tropics, reducing competition and creating more
  niche space
• -- greater productivity in tropics, Eltonian
  pyramid

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Exceptions abound, including for ants, bats, and
pitcher plant communities.
Latitude
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• Patterns in diversity and abundance
• A. Species number (richness)
• 1. Methodological concerns with counting
• a. what is a species?
• b. rarefaction and maximum species number
• c. species density
• 2. Application to biogeographic patterns
• a. latitudinal gradients
• b. topographic relief
• c. Peninsulas
• d. Islands

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Diversity in Peninsulas
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Island Biogeography Theory (MacArthur and
Wilson)
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Simberloffs Experiments on Mangrove Islands.
Simberloff covered the islands with tarps and
fumigated them. After 250 days, the number of
species had returned to its original number, with
smaller, isolated islands having fewer species.
Abundances of each species, however, did not
recover as rapidly. Flying species and those
more adapted to variable environments returned
first, followed by more competitive species
(e.g., ants) later.
Simberloff and Wilson 1969. Ecology 50278-296
and 1970. Ecology 51934-937.
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Mountain tops as Islands
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This, in turn, led to the SLOSS debate. -- If
only a limited amount of habitat can be protected
in parks and reserves, which strategy is better
a Single Large reserve Or Several Small
reserves -- This remains a somewhat bitter and
certainly oversimplified debate in conservation
biology.
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Ant data from Plots in Costa Rica
Can species number adequately describe community
patterns?
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Does species number adequately describe
communities? -- No, and it should probably never
be used alone. -- But, in the context of a known
community, it may have some value. -- There is no
denying that broad patterns in species number
exist and may have meaning, but the true
mechanisms may be obscured using species
richness. Avoid. Use only in conjunction with
other measures. Such as . . . .
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• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• 1. Indices of diversity
• 2. Relative abundance figures
• 3. Rank-abundance figures

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• Diversity indices
• H' Shannon Wiener diversity (sometimes
  mislabeled as Shannon-Weaver).
• This value goes up as either the number of
  species increases or the relative evenness of
  abundances increases.

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• Other Diversity indices
• Simpsons Reciprocal Index (sometimes D is used
  other ways)
• The lowest possible value would be 1 and the
  maximum value would be s.
• Alpha (?), from Fishers original work
• Where SN is the number of species represented by
  N individual, X is a fit number, generally just
  less than one, and a is an index of diversity.
• The Brillouin index is given by
• where ni is the individual in species i and N is
  total number of individuals.
• Others include Margalef index, McIntoshs measure
  of diversity, the Berger-Parker index, Evar
  Index, and Hurlbert's PIE (probability of an
  interspecific encounter). Perhaps Simpsons is
  safest, Shannon-Weiner is most used.

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Examples of Diversity Measures Bats of the Black
Volta
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• Evenness

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Clark and Warwicks taxonomic distinctness
index -- describes the average taxonomic
distance between two randomly chosen organisms
through the phylogeny of all species in the
assemblage. Where s the number of species
in the study and ?ij is the taxonomic path length
between species i and j.
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Now, we are left with 3 different simple measures
of communities -- species richness -- species
diversity -- species evenness
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Ant data from Plots in Costa Rica
Can species number adequately describe community
patterns? Can a diversity index adequately
describe community patterns?
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• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• 1. Indices of diversity
• 2. Abundance category figures
• 3. Rank-abundance figures

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Corbets original data on butterflies from
Malaysia
number of species
number of individuals/species
Corbet, A. S. Proc. Roy. Entom. Soc., ser. A.
16101-16.
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Another well-known relative abundance graph
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• Mathematical explanation for Prestons canonical
  distribution
• Integrating over time gives
• Generally, future populations sizes are expected
  to be relatively independent of their first
  population size, so we can ignore the first term.
  In a non-equilibrium regime, r will vary
  randomly, so the sum of r will be normally
  distributed (Central Limit Theorem says that
  additive distributions are normal). This means
  that ln Nt will be normally distributed,
  resulting in a log normal distribution for Nt.
  In brief, populations are expected to increase
  exponentially, so a bunch of different
  populations will present a log-normal
  distribution of abundances.

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• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• 1. Indices of diversity
• 2. Abundance category figures
• - inductive log series and log normal
  distributions
• 3. Rank-abundance figures
• - deductive approaches

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Motomuras geometric series (1932. Jap. J. Zool.
44379-383)
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Whittaker popularized the dominance-diversity
curve or what we now usually call the
rank-abundance curve
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MacArthur looked for a niche-based theory and
came up with the idea of randomly sharing some
resource arrayed along a continuum. This is the
broken-stick hypothesis. Where Nr is the
abundance of the r-th rarest species in a
community of Ss species and Ns individuals.
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• Tokeshi tested 7 different models, using data on
  stream chironomids. These included
• Dominance decay (large-stick)
• MacArthur fraction (broken-stick)
• Random fraction (odd b-s)
• Random assortment (odder b-s)
• Composite model
• Dominance re-emption (geometric)
• Tokeshi, M. 1993. Species abundance patterns and
  community structure. Adv. Ecol. Res. 24112-186.

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• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• 1. Indices of diversity
• 2. Abundance category figures
• - inductive log series and log normal
  distributions
• 3. Rank-abundance figures
• - deductive approaches
• - geometric, broken stick, many others
• - Hubbells neutral theory

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probability of having S number of species in the
community
Expected abundance ri of the ith ranked species
Hubbell, S. J. 1997. Coral Reefs 16 S9-S21.
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Hubbell, S. J. 1997. Coral Reefs 16 S9-S21.
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Hubbell, S. J. 1997. Coral Reefs 16 S9-S21.
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Species sampling from 131 plots on dunes in
England, with 100 sample plots, each 2 x 2 m.
There were 101 species across these plots. We
could plot this in n-dimensional space, with 101
axes and 100 points. But, if there are strong
species associations, fewer axes can provide most
of the relevant information.
Orloci, L. 1966. J. Ecology 54193-215.
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Introduction
A Short Primer on Ordination -- mostly from M.
Palmer (http//www.ikstate.edu/artsci/botany/ordin
ate)

• Quantitative community ecology is one of the most
  challenging branches of ecology. Community
  ecologists typically need to analyze the effects
  of multiple environmental factors on dozens (if
  not hundreds) of species simultaneously, and
  statistical errors (both measurement and
  structural) tend to be huge and ill behaved. It
  is not surprising, therefore, that ecologists
  have employed a variety of multivariate
  approaches for community data to try to simplify
  their view of community data. These approaches
  have been both endogenous and borrowed from other
  disciplines. The majority of techniques fall into
  two main groups classification and ordination.
• Classification is the placement of species
  and/or sample units into groups
• ordination is the arrangement or ordering of
  species and/or sample units along gradients.
• I will be providing an introduction to this
  second approach, ordination, today.

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Properties of Community data
A Short Primer on Ordination -- mostly from M.
Palmer
A community data matrix has taxa (usually
species) as rows and samples as columns plot
1 plot 2 plot 3 plot 4 . . . . Species
A 34 5 0 0 Species B 12 5 23 1 Species
C 0 4 3 0 . . . . . .. .. .. .. (Note that rows
and columns could be reversed). In most studies
of vegetation, the sample is a quadrat or
transect though it can consist of a number of
subsamples. Samples in animal ecology may
consist of traps, seine sweeps, or survey routes.
Biogeographic studies may rely on the cells of
large grids or political units as samples. The
elements in community data matrices are
abundances of the species. Abundance is a
general term that can refer to density, biomass,
cover, or even incidence (presence/ absence) of
species. The choice of an abundance measure will
depend on the taxa and the questions under
consideration. Species composition is frequently
expressed in terms of relative abundance i.e.
constrained to a constant total such as 1 or
100.
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General Properties of Community Data
A Short Primer on Ordination -- mostly from M.
Palmer

• They tend to be sparse a large portion of
  entries can be zeros.
• Most species are typically present in a minority
  of locations, and contribute little to the
  overall abundance.
• The number of factors influencing species
  composition is potentially very large. For
  example, forest tree density can be influenced by
  time since fire, elevation, nutrients, soil
  depth, soil texture, water availability and many
  other factors.
• The number of important factors is typically few.
  That is, a few factors can explain the majority
  of the explainable variation. Another way of
  saying this is that the intrinsic dimensionality
  is low.
• There is much noise. Even under ideal
  circumstances, replicate samples will vary
  substantially from each other. This is largely
  due to stochastic events, though observer error
  may also be appreciable.
• There is much redundant information species
  often share similar distributions. It is this
  property of redundancy that allows us to make
  sense of compositional data by looking for
  general patterns of co-occurrence.

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Properties of Community data
A Short Primer on Ordination -- mostly from M.
Palmer
A simple first way of thinking about community
data is to consider each plot to be represented
in multispecies space, such that the data
matrix plot 1 plot 2 plot 3 plot 4 . . .
. Species A 34 40 0 0 Species B 12 10 23 1 Speci
es C 0 2 3 0 . . . . . .. .. .. .. can be
represented as Euclidean distances Between
points plot 1 plot 2 plot 3 plot 4 plot
1 0 12 56 32 plot 2 12 0 29 17 plot
3 56 29 0 4 plot 4 32 17 4 0
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Principal Components Analysis (PCA)
A Short Primer on Ordination -- mostly from M.
Palmer
Consider now the abundances of 3 species in plots
a through z, plotted in our Euclidean distance
plot. What PCA does is that it takes your cloud
of data points, and rotates it such that the
maximum variability is visible. Another way of
saying this is that it identifies your most
important gradients. In this example, you might
bea able to tell that X1 and X2 are related to
each other, but it is less clear whether X3 is
related to X1 or X2.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
The first stage in rotating the data cloud is to
standardize the data by subtracting the mean and
dividing by the standard deviation. Thus, the
centroid of the whole data set is zero. We label
these standardized axes S1, S2, and S3. The
relative location of points remains the same
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
From looking at the last two figures, one can
already identify a gradient from the lower left
front to the upper right back. In other words,
there appears to be an underlying gradient along
which species 1 and species 2 both increase. Let
us now draw a line along this gradient
Principal Components Analysis chooses the first
PCA axis as that line that goes through the
centroid, but also minimizes the square of the
distance of each point to that line. Thus, in
some sense, the line is as close to all of the
data as possible. Equivalently, the line goes
through the maximum variation in the data.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
The second PCA axis also must go through the
centroid, and also goes through the maximum
variation in the data, but with a certain
constraint It must be completely uncorrelated
(i.e. at right angles, or "orthogonal") to PCA
axis 1.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
If we rotate the coordinate frame of PCA Axis 1
to be on the X-axis, and PCA Axis 2 to be on the
Y-axis, then we get the diagram at right. We can
see that samples a, b, c, and d are at one
extreme of species composition, and samples t, w,
x, y, and z are at the other extreme. But there
is a secondary gradient of species composition,
from samples b, m, n, u, r and t up to samples l,
q, w, and y.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
What is the underlying biology behind such a
gradient? PCA, and any other indirect gradient
analysis, is silent with respect to this
question. This is where the biological
interpretation comes in. The scientist needs to
ask, what is special about the samples on the
right which make them fundamentally different
from those samples on the left? What is it about
the biology of species 1 that makes it occur in
the same locations as species 2?
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
We have only plotted two PCA Axes. Why did we
not plot the third? Well, we could, but in this
example, it wouldnt really be worthwhile If we
were going to plot three axes, then why even
bother to perform PCA in the first place?
Remember we only started with 3 species and 27
plots. Of course, this is just an example. We
would normally use PCA for much more complex data
sets where ordination is used to simplify the
data from many-species axes to a small number of
PCA axes.
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PCA is an Eigenvalue-based Ordination Method
A Short Primer on Ordination -- mostly from M.
Palmer
Eigenanalysis is central to the mathematical
discipline of linear (matrix) algebra, and a
thorough understanding of ordination methods
requires a training in linear algebra. However,
for our present purposes, it suffices to know
that Eigenanalysis is a mathematical
operation on a square, symmetric matrix.
Distance and similarity matrices should be square
and symmetric. It is possible to perform a
eigenanalyses analytically (that is, get exact
results) only for very small matrices (e.g. three
rows and columns). For large matrices,
eigenanalysis requires an iterative approach
which eventually "closes in" on the answer (in
most cases). The result of an eigenanalysis
consists of a series of eigenvalues and
eigenvectors. Each eigenvalue has an eigenvector,
and there are as many eigenvectors and
eigenvalues as there are rows in the initial
matrix. Eigenvalues are usually ranked from the
greatest to the least. The first eigenvalue is
often called the "dominant" or "leading"
eigenvalue. Eigenvalues are also called "latent
values". The eigenvalue is a measure of the
strength of an axis, the amount of variation
along an axis, and ideally the importance of an
ecological gradient. The precise meaning depends
on the ordination method used.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
Remember that the distance matrix is square and
so we can apply eigenanalysis. Every PCA axis
has an eigenvalue associated with it, and they
are ranked from the highest to the lowest. The
largest three eigenvalues correspond to the first
three axes in the above example and are 1.8907,
0.9951, and 0.1142 respectively. These are
related to the amount of variation explained by
the axis. Note that the sum of the eigenvalues is
3, which is also the number of variables. It is
usually typical to express the eigenvalues as a
percentage of the total PCA Axis 1 63 PCA
Axis 2 33 PCA Axis 3 4
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
How do we know which species contribute to which
axes? We look at the component loadings Species
PCA 1  PCA 2 PCA 3  S1 0.9688  0.0664 -0.2387  S2
0.9701  0.0408 0.2391  S3 -0.1045  0.9945 0.0061 
This means that the value of a sample along the
first axis of PCA is 0.9688 times the
standardized abundance of species 1 PLUS 0.9701
times the standardized abundance of species 2
PLUS -0.1045 times the standardized abundance of
species 3. We can interpret Axis 1 as being
highly positively related to the abundances of
species 1 and 2, and weekly negatively related to
the abundance of species 3. Axis 2, on the other
hand, is positively related to the abundance of
all species, but mostly species 3.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
Finally, we can begin to determine what forces
are structuring the community by correlating
different environmental variables with the PCA
axes. For example, if data were available for
soil N, moisture, or temperature, we could
observe the correlations between these factors
and each of the PCA for each plot. This is
pure hypothesis creating --gt we cant prove that
any of these environmental factors create the
observed community patterns, especially since
many environmental factors co-vary. But, it may
be a first step for creating experimentally
testable hypotheses.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
Bluntly, these all seems incredibly useful and
too easy so far. What is wrong with PCA? Well,
PCA often produces an artifact known as the
Horseshoe Effect, in which the second axis is
curved and twisted relative to the first, and
does not represent a true secondary gradient.
This misleading effect is due to an assumption of
PCA that there is a linear, rather than unimodal,
relationship between dominant environmental
variables and species abundance. Do note,
however, that if we only sampled a small enough
section of the gradient the data might be linear
enough to allow the use of PCA. So, PCA may be
most appropriate for short environmental
gradients, where there are fewer 0s in the data.
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Principal Components Analysis (cont.)
A Short Primer on Ordination -- mostly from M.
Palmer
Another method is Correspondence Analysis (CA).
CA assumes that species have unimodal response
curves to underlying environmental gradients.
And, in this case, both species and samples are
placed in some ordination space in what is called
a biplot. Instead of maximizing the variance
explained as with PCA, CA maximizes the
correspondence between species scores and sample
scores. CA is one of a variety of methods that
can be used, each of which has certain advantages
and disadvantages. If you end up needing these
methods, I recommend Bill Parkers course in
Geology and/or Mike Palmers web site
(http//www.ikstate.edu/artsci/botany/ordinate).
See, also, Legendre and Gallagher, 2001.