The
Canonical Distribution of Commonness and Rarity: Part I
Preston explains how populations
containing many individuals of numerous species are distributed under lognormal
law. For the abscissa or X axis number of individual representing a species are
used. For the ordinate or Y axis the numbers of unique species are plotted. Log
base 2 or “octaves” are used in the X axis, this setup results in a “species
curve” that take the shape of a bell curve where the rarest species are found
to the far left of the curve and most common found the right. When a second
graph is plotted similar to the first replacing number of unique species in the
Y axis with number of individuals those species comprise an “Individual Curve”
is formed. It is explained that the descending limb of the “Individual Curve”
“…terminates over the last observed point of the Species Curve…” which
corresponds to or near the crest of the Individuals Curve. The shape of these
curves are referred to as canonical.
The paper explains that although the
graph of the Species Curve extends infinitely left and right the areas under
the graph more than 9 octaves from the mode is hardly enough to represent a
single species, when using a 178 species as an example, and in such a situation
+/- 9 octaves can be used as the finite distribution. It states it is best to
make this position at “that point where the remaining area under the tail of
the curve corresponds to half a species.” Preston states that the canonical
equation implies a relationship between the total individuals and total
species.
Preston determined a relationship
between species and area could be made if the distributions are canonical. The
number of species would be equal to the area raised to the constant z multiplied by a second constant. It was
determined that the value of z is
near 0.262, far less when dealing with samples within a larger area.
He tests this relationship with a
number of data sets for various “isolates”. An isolate being an area of land
where flora and fauna are in equilibrium but minimally affected by other
populations, Islands where used in many examples. These test included breeding
populations of birds in England, Wales, and Finland, populations of mammals,
birds, reptiles, amphibians, “land vertebrates” and plants on various oceanic
and lake islands. In all cases the z value
remained close to the theoretic value of 0.262, ranging from 0.220 to 0.325.
It is explained how the use of
“samples”, or a small random sampling of a larger presumably canonical
“universe”, differ from “isolates”. A truncated lognormal distribution is
created, with the same modal height and standard deviation as the “universe”.
From this the total number of species can be calculated from a smaller subset
of observed/collected ones. As an example he uses moths being collected by
means of a light trap. Not every moth is captured, only a smaller sample from a
large geographic range based on power of flight. Differences between the
universe and the sample are noted. “The ratio of species to individuals is
vastly higher in the sample the universe and second, there are vastly more
species represented by a single specimen…” At the beginning species are
collected at the same rate individuals are captured this slowly tapers off
until new individuals are captured but new species are now rarely collected. It
is not that all unique species are collected but that the chances of
encountering the rarest species from random sampling are quite low. The part of
the distribution on the left side of the lognormal distribution graph that
cannot be seen in observed data due to the rarity of the species found there is
known as the “veil line”. It is shown that by doubling the observed or captured
individuals an additional octave can be uncovered which not only doubles the count
of all previously visible octaves but adds an individual for each species in
the new octave “unveiled”.
When using small samples, under 100
species, individuals may be diverse, however if species aggregate the sample
numbers will be distorted compared to the universe, skewing the canonical
distribution graph. Examples of how these non-random distribution patterns
affect these canonical graphs are shown with various data sets of plants and
animals.
This paper applies statistics to ecological data in an attempt to explain and expand on the basic principles of species abundance and distribution that are still followed and studied today. The species-area curve, species vs population abundance, and even rarefaction curves are explored in this paper concluding the following:
ReplyDelete1. Species-Area: Positive relationship between area and number of different species; number of species increases with increasing area.
2. Species numbers vs number of individuals: There are more species with smaller relatively abundances and few species with large abundances.
3. Rarefaction Curve: The more you sample the less number of new species you will find until you’ve eventually found all species and this curve plateaus.
So not only did this paper explore and solidify fundamental ecological concepts but the fact that they did so mathematically also helped to establish and push forward the practice of mathematical ecology which is the norm today.
This paper does an exceptional job of establishing some accepted and very fundamental mathematical analyses still used today. The ideas presented, especially the Rarefaction Curve, form the basis of many experiments, and are almost always taken into account when designing different methods for population studies.
ReplyDeletePreston makes a point multiple times to distinguish between animal populations and plants populations. This emphasizes the differences that need to be taken into account when surveying populations of plants, and when predicting a Species-Area curve. An area can be very large, i.e. all of the Rocky Mountains of Colorado, but overall, the species count will increase slowly over an increasing area and eventually come to a plateau. There are clearly areas of higher plant species diversity than others, but distinguishing between the differences of plant vs. animal populations is so essential in being able to predict populations over space and time.
I agree that Preston does an excellent job of incorporating mathematical concepts to explore the ecological concepts concerning population. However, the explanation as to why the observational evidence appears to fit with the theoretical calculations somehow gets lost in the multitude of graphs and mathematical equations.
ReplyDeleteI understand that he was attempting to show the complex relationships between samples and "isolates" and observed population sizes can be reflected in mathematical theory, but why does it do this? In essence, the observations made by Real and Levan in the "Introduction to Part II" for Preston's (1948) earlier paper as descriptive rather than explicative still seems to hold true. (Although, I could have just gotten lost in the math.)
I'm here as well, although I'm more than reasonably certain that I got lost in the math... I got the distinct feeling that a goodly portion of the paper was about how Preston does his math differently from Fisher and Merikallio. I did, however, find the mathematical explanation of the species curve mildly interesting.
DeleteI think that Preston examines the data on species richness from all the island "isolates" to show that there is empirical support in the real world for his theoretical species-area relationship (i.e. that empirical zs are close to 0.27). It's interesting that while he does calculate regressions lines to fit the data, he doesn't use any statistics to show how close the fit actually is to theory, or whether there's a significant departure. I nonetheless thought the data he showed was convincing in providing support for the theory. He also doesn't get into empirical values for the constant in the species-area relationship (theoretically 2.07(rho/m)^0.0262)
DeleteThis is why I'm so glad that computers can give us regression lines instead of us having to calculate and accurately draw them out.
DeleteBecause, frankly, I distinctly remember just "eyeballing" them in arithmetic classes as a "young'un".. Haha.
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DeleteI had a very difficult time understanding what this paper was getting at. I did see some basic population/sampling information, like the idea that initially each individual sampled is likely to be a new species where later on after many individuals are sampled you are unlikely to encounter novel species very often. However it seems that this statement is contradicted later in the paper. He also makes a keen distinction between the population and sample, which I think is always good to keep in mind when interpreting findings based on a sample. I like the examples used but I felt Preston did a lousy job of tying his examples to his thesis/hypothesis. There were also so many examples that it made keeping up with them a challenge. Perhaps one or two per section (with more interpretation) would have been more useful. I had a hard time understanding what an octave was perhaps this is something the group can enlighten me on?
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DeleteFrom what I got out of it, "octave" is an arbitrary term for a unit that is used to represent a scale. The scale of the octave is double that of the representative number. Octave is represented by a logarithmic unit “R” that tends to form a “lognormal” distribution or Gaussian curve, which is the “Species Curve”.
DeleteOh, man. I had to remind myself of some math terms, like abscissa and ordinate. I think we just tend to use more layman terms such as x- and y-axis when describing graphs. I don't think I totally got it until I looked at the legends to the graphs, which I very much appreciate. I probably got a bit fuddled with all of the math, but I like seeing concepts displayed with a visual representation.
ReplyDeleteI also appreciated seeing actual islands used to demonstrate isolates.
I was a bit confused with the Species and Individuals curves. I did not understand why the Individuals curve was placed next to the Species curve. If the Species curve is comprised of species that are comprised of individuals then wouldn't the Individuals curve be nested inside of the Species curve? Or is his way of drawing the curves just for the sake of demonstrating the imaginary part of the Individuals curve?
"...math terms, like abscissa and ordinate. I think we just tend to use more layman terms such as x- and y-axis when describing graphs"
DeleteI totally agree Kristen! I'm not sure I ever learned the terms "abscissa" and "ordinate"
I'm not sure that I did either! But it is completely reasonable that I also forgot. Haha.
DeleteI, like most of the others, felt a little lost in some of the mathematical concepts. However, I feel that after going over the models several times, I have a better grasp on these concepts. I really enjoy how these papers are progressing into more modernistic papers, using concepts that we see everyday in ecology papers. One question I feel that Preston really gets at is: how do we know that the patterns we see in collecting are not caused by random sampling? I am always curious as to how the size of sampling areas are determined, and how sampling can be made less random.
ReplyDeleteI found this to be a pretty interesting piece, though it was a bit of a slog. It seems to be written assuming the reader is familiar with Preston 1948, so we get thrown right into things from the beginning without as much context as would be helpful. Some pretty cool insights nonetheless. Here are some thoughts:
ReplyDelete- With all the examples given of species richness on islands and some of the discussion of isolates v. islands that are "samples" of mainland diversity, there seems to be a lot in here that is a precursor to island biogeography theory (which I think we get to later). One thing missing, though, is consideration of extinctions and speciation.
- The species-area relationship must not only have implications for sampling biodiversity, but also for conservation. The shape of the 4th root law curve would seems to imply that we could conserve the vast majority of species with reserves of limited area.
- I was interested in the parameter m - the minimum number of individuals necessary for a species to be viable. It appears to be important in the canonical distribution of species/individuals, but might be difficult to estimate empirically, and Preston didn't seem to give much theoretical insight into possible values (thought I might've missed this). m would also seem to have conservation implications, and I know there's been some work on 'population viability analysis' in the conservation world, but I'm not very familiar with this or if it has connections to Preston. Anyone else know something about this?
- I like how Preston brought in the concept of contagion - though it was a bit hard to figure out at first. This, along with a few other discussion points seem to foreshadow the development of landscape ecology, where spatial questions are central.
I really liked this paper (surprise) and I thought it had a lot of really good concepts that are more mathematical-biology than pure math. I really liked his usage of equations to describe these systems but I also really liked how he was very keen to allow for lots of leeway in his numbers, especially when comparing his equations to real examples. He also demonstrated a really good understanding that large sample size will eventually lean towards the mean rather than trying to cram the sample into a little box described by his mathematical equations.
ReplyDeleteSome of the things I thought were hilarious is his arbitrary drawing of lines to some data and how he uses his personal interpretation a lot. Those things wouldn't really fly in today's research journals but I feel like it gives a certain feel to his paper that isn't really there anymore. He uses the word "I" a lot which is also kind of cool to see in a paper. Overall, I really liked it.
Preston’s methods interest me. He exercises systematic caution in plotting data points on the same graph with his theoretical line to see how well they fit, rather than starting out with calculating a regression line for the points. He expresses skepticism when the data fits the line almost too well. He also exercises care in his exploration of the relationship of sample to population. I am especially interested in his comparison of log-log (Gleasonian) to semi-log (Arrhenian) plots of number of species vs. years of observation (or area). If the range of the x-value (years or area) is small enough (but not too small), it doesn’t matter whether the number of species is plotted on a log scale or not—the relationships are both linear. Why?
ReplyDeleteThe Canonical Distribution of Commonness and Rarity: Part 1.
ReplyDeletePreston (1962) is providing a good insight about the theory of species-area relationship. The species area curve is following a log normal distribution. This theory is divided by the author in isolates ( islands that represent discontiguous habitats) and samples ( mainland contigous habitats). The specie-area relationship is better understood if we use the power function that was defined by the Arrhenius Equation, where N=KA^z.
N= number of species
A= Area
Z= slope defined in the log space
K= constant which values depende on the unit of measurement for the area