Principal Components Analysis cont'

1

2
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.
3
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.
4
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.
5
St. George Island, DCA (spp data only), 1999,
only species in greater than 2 of sites (N
21), 267 sites were not empty (DCA ignores empty
sites). Axis 1 eigenvalue 0.639, Axis 2
eigenvalue 0.454. From H. Buckley, 9/06
6
Some Common Ordination Methods
A Short Primer on Ordination -- mostly from M.
Palmer
Indirect gradient analysis Distance-based
approaches Polar ordination, PO (Bray-Curtis
ordination) Principal Coordinates Analysis,
PCoA (Metric multidimensional scaling) Nonmetric
Multidimensional Scaling, NMDS Eigenanalysis-bas
ed approaches Linear model Principal
Components Analysis, PCA Unimodal model
Correspondence Analysis, CA (Reciprocal
Averaging) Detrended Correspondence Analysis,
DCA Direct gradient analysis Linear
model Redundancy Analysis, RDA Unimodal
model Canonical Correspondence Analysis,
CCA Detrended Canonical Correspondence
Analysis, DCCA
7

• 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
• 4. Ordination approaches such as PCA
• C. Problems with scale
• 1. alpha, beta, and gamma diversity
• 2. measures of gamma diversity

8
Why Use Ordination?
A Short Primer on Ordination -- mostly from M.
Palmer
According to Gauch (1982) "Ordination primarily
endeavors to represent sample and species
relationships as faithfully as possible in a
low-dimensional space". But why? 1) It is
impossible to visualize multiple dimensions
simultaneously. While physicists grumble if space
exceeds four dimensions, ecologists typically
grapple with horrible complexity with dozens of
dimensions (species and/or samples). 2) Ideally
and typically, dimensions of this low
dimensional space will represent important and
interpretable environmental gradients. 3) A
single multivariate analysis saves time, in
contrast to a separate univariate analysis for
each species. 4) If statistical tests are
desired, problems of multiple comparisons are
diminished when species composition is studied in
its entirety 5) Community patterns may differ
from population patterns. 6) The graphical
results from most techniques often lead to ready
and intuitive interpretations of
species-environment relationships. In other
words, it seems to work!
9

• 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
• 4. Ordination approaches such as PCA
• C. Problems with scale
• 1. alpha, beta, and gamma diversity
• 2. measures of gamma diversity

10
A B C
A B C
D E F
A B C
G H I
A B C
Imagine different local communities within some
larger landscape. These could be fish species on
coral heads, or insects on trees, or plants on
nearby islands. In the above case, the local
richness (alpha diversity) is 3 for both
landscapes. But, clearly the regional richness
is very different. Whittaker proposed gamma
diversity to describe this landscape-level
diversity.
11
Whittaker, R. 1960. Vegetation of the Siskiyou
Mountains, Oregon and California. Ecological
Monographs 30279-338.
12
Whittaker distinguished quantifying diversity at
different spatial scales, such that alpha
diversity was the property of defined spatial
unit, such as a discrete forest or pond, while
beta diversity was the extent to which diversity
of two or more spatial units changed. In
particular, he viewed beta diversity as a measure
of how diversity changed over an environmental
gradient, and referred to beta diversity as a
measure of species turnover.
Scale measure of diversity Within sample
point diversity Within habitat alpha
diversity Among habitats beta
diversity Within landscape gamma diversity To
make this all a bit more confusing, Whittaker
recognized that there must be a relationship
between alpha and gamma. If alpha is high and/or
beta is high, then gamma must be high. So, he
proposed that alpha x beta gamma.
13
This has caused a lot of confusion, because many
people mistakenly assume that alpha, beta, and
gamma diversity are all measures of the same
thing and have the same units, but at different
spatial scales. This is not true. Alpha and
Gamma diversity can be measured in all the ways
we have been discussing for the last week, such
as species richness, Simpsons index, or
rank-abundance curves. Beta diversity is a
different beast, as it quantifies variation among
sites, rather than characterizing a give site.
But, useful as this concept is, how can we
measure it? Several people suggested that
instead of it being multiplicative, it is just
additive, such that alpha beta gamma
14
Cornell and Lawton proposed a simple graphical
approach for looking at diversity as a function
of scale. By plotting regional richness (gamma
diversity) against local richness (alpha
diversity), one can clearly see how the residual
diversity changes with scale. If local
processes dominate, then an asymptote will occur.
However, if regional (e.g., migration)
dominates, then the local community will be just
some constant proportion of the regional
community. But, is the left-over bit beta?
gamma
alpha
15
Russ Lande noted that if these measures were just
additive, then a very simple statistical method,
ANOVA, could quantify the variance in diversity
at each level. That is, we can consider gamma as
a measure of the total variation in diversity.
Then, we can simply partition that into a
contribution from within local communities
(alpha) and among local communities (beta).
Lande, R. 1996. Statistics and partitioning
of species diversity, and similarity among
multiple communities. Oikos 765-13.
16
This has been used and extended in several
studies. Gering et al., for example, note that
there can be many more spatial scales than just
the local/regional perspective. They looked at
herbivorous insects found on individual trees,
trees within stands, stands within sites, sites
within ecoregions, and then ecoregions across a
region. The definitions of all these scales are
arbitrary. They use alpha and beta then in a
relative sense. Landes partitioning allows them
to assume that each level is additive so that
total diversity alpha beta1 beta2 beta3
beta4. Gering, J. C., T. O. Crist, and J. A.
Veech. 2003. Additive partitioning of species
diversity across multiple spatial scales
implications for regional conservation of
biodiversity. ConservationBiology 17488-499.
17
Consider the following two regions. Here the
number of species per local community is always
the same (alpha 3). The regional diversity is
also the same (gamma 4 total species).
Landes method would say that the beta diversity
is equal in these communities. That seems crazy.
But,the additive method ignores species identity
and only quantifies variation in the numbers.
18

• 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
• 4. Ordination approaches such as PCA
• C. Problems with scale
• 1. alpha, beta, and gamma diversity
• 2. measures of gamma diversity

19

• 2. measures of gamma diversity
• -- beta gamma/alpha (Whittaker 1960)
• -- beta (S/a)-1/(N-1) 100 (n2, Harrison
  et al. 1992)
• -- beta g(H) l(H)/2, determined by the
  number of species gained (g(H)) or lost (l(H))
  along a gradient (Cody 1975)
• -- beta S2/(2r S) -1 (Routlenge 1977)
• -- beta g(H) l(H)/2S (Wilson and Shmida
  1984)

20

• 2. more measures of gamma diversity, but as
  complementarity or similarity
• -- C 1-a/(abc), where a is the number of
  species in both sites, b is the number only in
  site 1, and c is only the number in site 2.
  (Colwell and Coddington, 1994).
• -- C 2a/(2abc) (Sorensen, 1948)
• -- beta 1 - a/(a min(b,c)) (Lennon et al.
  2001)
• -- Landes additive method.

21
Pablo Munguia used this approach for studying pen
shells communities. He generally found a
positive slope, suggesting that migration was
important for determining species richness. He
also noted that there is also a successional
effect, making the standard local-regional curve
difficult to interpret. Munguia, P. 2004.
Successional patterns on pen shell communities at
local and regional scales. J. Animal Ecology
7364-74.
22

• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• C. Problems with scale
• D. Describing trophic webs
• 1. Types of webs
• 2. Describing webs
• 3. Laws for food webs
• 4. Can trophic webs really be described?

23

• Top predators -- species that get eaten by
  nothing else in the food web
• Basal species -- species that feed on nothing
  within the web (usually plants)
• Omnivores -- species that feed at more than one
  trophic level
• Trophic species -- groups of species that have
  the same predator and prey
• Cannibalism -- a cycle in which a species feeds
  upon itself
• Connectance -- number of actual interactions
  divided by the number of possible interactions
• Compartments -- suites of species with strong
  linkages among group members but weak linkages to
  other species

24
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25
connectedness webs emphasize the feeding
relationships among organisms
energy flow webs show energy flow between
resource and consumers
functional webs links that maintain the integrity
of the community
26

• D. Describing trophic webs
• 1. Types of webs
• 2. Describing webs
• -- length of food chains (usually maximum)
• -- connectance ( actual links / possible
  links)
• -- link density ( links / species)
• -- top (no predators)
• -- intermediate (w/ predators and prey)
• -- basal (autotroph and detritus)
• -- top/intermediate
• -- intermediate/basal
• -- top/basal
• -- omnivores

27

• D. Describing trophic webs
• 1. Types of webs
• 2. Describing webs
• 3. Laws for food webs
• In a series of papers in the 1980s, Briand and
  Cohen suggested that some of these descriptors
  showed constant patterns across communities.
  They suggested, for example, that food chain
  length was independent of primary productivity
  and is unaffected by environmental variability.
  In particular, they proposed that the proportions
  of species in each trophic level were constants.
  
• This has been a dynamic and contentious area,
  with really only a few scientists involved.
  Fredrick Briand and Joel Cohen collected data on
  a large number of food webs (gt 113 the
  community club) to test many of their ideas.
  Others who promote these laws include George
  Sugihara and Neo Martinez.
• Briand, F., and J. E. Cohen. Science
  238956-960

28
Cohen, J., and F. Briand. 1984. Trophic links
of community food webs. PNAS 814105-4109.
29

• D. Describing trophic webs
• 1. Types of webs
• 2. Describing webs
• 3. Laws for food webs
• Is this invariance or simply uncorrelated? The
  invariance of the food web ratios remains
  controversial. Others have found there are
  relationships between these ratios and species
  number, if the type of community is taken into
  account.
• Murtaugh and Kollath (1997, Ecology
  781382-1387) analyzed another large pool of
  community data collected by Schoenly. They also
  generally found that there was no relationship
  between trophic fractions and connectance and
  species number when full data set is analyzed.
  However, they frequently found correlations when
  the data were divided into different types of
  communities.

30
Murtaugh and Kollath. 1997. Ecology 781382-1387
31
Murtaugh and Kollath. 1997. Ecology 781382-1387
32

• Patterns in diversity and abundance
• A. Species number (richness)
• B. Relative and absolute abundance patterns
• C. Problems with scale
• D. Describing trophic webs
• 1. Types of webs
• 2. Describing webs
• 3. Laws for food webs
• 4. Can trophic webs really be described?

33
A number of people had concerns about the
food-web approach. In particular, there was
concern about the quality of the food-web data
it seemed oversimplified and incomplete. Gary
Polis illustrated these problems by trying to
quantify the food web in what would seem a
simple, depauperate community in the Coachella
Valley desert.
He found that this simple community had 174
species of vascular plants, 138 vertebrates, 55
arachnids, and 2,000 to 3,000 insect species, not
to mention the microorganisms, mites, and
nematodes. More importantly, the food web
structure was incredibly tangled. Besides the
obvious large number of interactions, he also
found significant incidences of omnivory,
ontogenetic shifts in diet, looping in the food
chains, and a severe lack of compartmentalization.

34
Gary Polis research where he attempted to
construct the full food web for a simple real
community in a California desert. This is only
part of the food web.
35
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37
Much of what Polis found contradicts food-web
theory. He found much longer chains, greater
connectivity, top predators being rare or
nonexistent, and prey to predator ratios that are
much greater than 1.0 From Polis (1991) I
argue that most cataloged webs are oversimplified
caricatures of actual communities. That
cataloged webs depict so few species, absurdly
low ratios of predators on prey and prey eaten by
predators, so few links, so little omnivory, a
veritable absence of looping, and such a high
proportion of top predators, argues strongly that
they poorly represent real biological
communities. Consequently, the practice of
abstracting empirical regularities from such
catalogs yields an inaccurate and artifactual
view of trophic interactions within communities.
Contrary to strong assertions by many theorists,
patterns form food webs of real communities
generally do not support predictions arising from
dynamic and graphic models of food-web structure.
38
In several later papers, Polis continued to pound
the food-web theory folks. His work in
Coachella, as well as on the islands in the Gulf
of California, found much more complex scenarios
than those in the simple theories. He noted that
his real-world view of communities also caused
problems for other current concepts about
communities. Polis and Strong suggested that
the multiple, reticulate connections between
consumers and resources are contrary to the
normal practice of simplifying webs into linear
trophic levels, as required by HSS. Further,
they challenged the oversimplification of trophic
cascades, bottom-up regulation, and top-down
regulation that are inherent in many current
views of communities. They feel that such
patterns are relatively uncommon in
nature. Polis, G. A., and D. R. Strong. 1996.
Food web complexity and community dynamics.
American Naturalist 147813-846.
39
So, the food-web theory may not be especially
practical. Like much theory, perhaps there are
still general principles to be learned from this
approach. However, there direct application to
real communities seems questionable. Finally,
we havent even mentioned how communities may
change through time. This lead to another whole
discussion about quantifying communities.
Proposed measures of stability include Stability
the ability to return to an initial point
following a perturbation. Resilience the time
require for a system to return to an initial
point following a perturbation. Resistance the
degree to which a variable (e.g. species
abundance or diversity) is changed following a
perturbation. Variability the variability
(e.g., sd, variance, or CV) exhibited in some
community measure over time.