For taxonomic levels higher than species, the abundance distributions of
number of subtaxa per taxon tend to approximate power laws, but often show
strong deviationns from such a law. Previously, these deviations were
attributed to finite-time effects in a continuous time branching process at the
generic level. Instead, we describe here a simple discrete branching process
which generates the observed distributions and find that the distribution's
deviation from power-law form is not caused by disequilibration, but rather
that it is time-independent and determined by the evolutionary properties of
the taxa of interest. Our model predicts-with no free parameters-the
rank-frequency distribution of number of families in fossil marine animal
orders obtained from the fossil record. We find that near power-law
distributions are statistically almost inevitable for taxa higher than species.
The branching model also sheds light on species abundance patterns, as well as
on links between evolutionary processes, self-organized criticality and
fractals.Comment: 10 pages, 4 Fig

Burlando

C. Adami

J. Chu

Raup

Sneppen

English

arXiv

For taxonomic levels higher than species, the abundance distributions of the number of subtaxa per taxon tend to approximate power laws but often show strong deviations from such laws. Previously, these deviations were attributed to finite-time effects in a continuous-time branching process at the generic level. Instead, we describe herein a simple discrete branching process that generates the observed distributions and find that the distribution's deviation from power law form is not caused by disequilibration, but rather that it is time independent and determined by the evolutionary properties of the taxa of interest. Our model predicts—with no free parameters—the rank-frequency distribution of the number of families in fossil marine animal orders obtained from the fossil record. We find that near power law distributions are statistically almost inevitable for taxa higher than species. The branching model also sheds light on species-abundance patterns, as well as on links between evolutionary processes, self-organized criticality, and fractals

Chu, Johan

Adami, Christoph

Caltech Authors - Main

A simple explanation for taxon abundance patterns
Johan Chu and Christoph Adami*
W. K. Kellogg Radiation Laboratory 106-38, California Institute of Technology, Pasadena, CA 91125
Edited by Burton H. Singer, Princeton University, Princeton, NJ, and approved November 1, 1999 (received for review July 13, 1999)
For taxonomic levels higher than species, the abundance distribu-
tions of the number of subtaxa per taxon tend to approximate
power laws but often show strong deviations from such laws.
Previously, these deviations were attributed to finite-time effects
in a continuous-time branching process at the generic level. In-
stead, we describe herein a simple discrete branching process that
generates the observed distributions and find that the distribu-
tion’s deviation from power law form is not caused by disequili-
bration, but rather that it is time independent and determined by
the evolutionary properties of the taxa of interest. Our model
predicts—with no free parameters—the rank-frequency distribu-
tion of the number of families in fossil marine animal orders
obtained from the fossil record. We find that near power law
distributions are statistically almost inevitable for taxa higher than
species. The branching model also sheds light on species-abun-
dance patterns, as well as on links between evolutionary pro-
cesses, self-organized criticality, and fractals.
Taxonomic abundance distributions have been studied sincethe pioneering work of Yule (1), who proposed a continuous-
time branching process model to explain the distributions at the
generic level and found that they were power laws in the limit of
equilibrated populations. Deviations from the geometric law
were attributed to a finite-time effect, namely, to the fact that the
populations had not reached equilibrium. Much later, Burlando
(2, 3) compiled data that seemed to corroborate the geometric
nature of the distributions, even though clear violations of the
law are visible in his data also. In this article, we present a model
that is based on a discrete branching process whose distributions
are time independent and in which violations of the geometric
form reflect specific environmental conditions and pressures to
which the assemblage under consideration was subjected during
evolution. As such, the model holds the promise that an analysis
of taxonomic abundance distributions may reveal certain char-
acteristics of ecological niches long after their inhabitants have
disappeared.
The model described herein is based on the simplest of
branching processes, known in the mathematical literature as the
Galton–Watson process. Consider an assemblage of taxa at one
taxonomic level. This assemblage can be all the families under a
particular order, all the subspecies of a particular species, or any
other group of taxa at the same taxonomic level that can be
assumed to have suffered the same evolutionary pressures. We
are interested in the shape of the rank-frequency distribution of
this assemblage and the factors that influence it.
We describe the model by explaining a specific example: the
distribution of the number of families within orders for a
particular phylum. The adaptation of this model to different
levels in the taxonomic hierarchy is obvious. We can assume that
the assemblage was founded by one order in the phylum and that
this order consisted of one family that had one genus with one
species. We further assume that new families in this order are
created by means of mutation in individuals of extant families.
This creation of new families can be viewed as a process by which
existing families can ‘‘replicate’’ and create new families of the
same order, which we term daughters of the initial family. Of
course, relatively rarely, mutations may lead to the creation of a
new order, a new class, etc. We define a probability Pi for a family
to have i daughter families of the same order (true daughters).
Thus, a family will have no true daughters with probability P0,
one true daughter with probability P1, and so on. For the sake
of simplicity, we initially assume that all families of this phylum
share the same Pi. We show later that variance in Pi among
different families does not significantly affect the results, in
particular the shape of the distribution. The branching process
described above gives rise to an abundance distribution of
families within orders, and the probability distribution of the
branching process can be obtained from the Lagrange expansion
of a nonlinear differential equation (4). Using a simple iterative
algorithm (http:yyxxx.lanl.govyabsycond-maty9903085) in place
of this Lagrange expansion procedure, we can calculate rank-
frequency curves for many different sets of Pi. It should be
emphasized here that we are mostly concerned with the shape of
this curve for n # 104 and not the asymptotic shape as n 3 ‘,
a limit that is not reached in nature.
For different sets of Pi, the theoretical curve can be close to
a power law, a power law with an exponential tail, or a purely
exponential distribution (Fig. 1). We show herein that there is a
global parameter that distinguishes among these cases. Indeed,
the mean number of true daughters (i.e., the mean number of
different families of the same order to which each family gives
rise in the example above),
m 5 O
i50
‘
i z Pi [1]
is a good indicator of the overall shape of the curve. Universally,
m 5 1 leads to a power law for the abundance distribution. The
further m is away from 1, the further the curve diverges from a
power law and toward an exponential curve. The value of m for
a particular assemblage can be estimated from the fossil record
allowing for a characterization of the evolutionary process with
no free parameters. Indeed, if we assume that the number of
families in this phylum existing at one time is roughly constant
or that this number varies slowly compared with the average rate
of family creation (an assumption the fossil record seems to
vindicate; ref. 5), we find that m can be related to the ratio RoyRf
of the rates of creation of orders and families by
m 5 S1 1 RoRfD
21
[2]
to leading order (http:yyxxx.lanl.govyabsycond-maty9903085).
In general, we cannot expect all the families within an order
to share the same m. Interestingly, it turns out that even if the
Pi and m differ widely between different families, the rank-
frequency curve is identical to that obtained by assuming a fixed
m equal to the average of m across the families (Fig. 2), i.e., the
variance of the Pi across families seems to be completely
immaterial to the shape of the distribution—only the average
m [ ^m& counts.
In Fig. 3, we show the abundance distribution of families
within orders for fossil marine animals (6), together with the
prediction of our branching model. The theoretical curve was
This paper was submitted directly (Track II) to the PNAS office.
*To whom reprint requests should be addressed. E-mail: adami@krl.caltech.edu.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.
PNAS u December 21, 1999 u vol. 96 u no. 26 u 15017–15019
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obtained by assuming that the ratio RoyRf is approximated by the
ratio of the total number of orders to the total number of
families,
Ro
Rf
>
No
Nf
, [3]
and that both are very small compared with the rate of muta-
tions. The prediction m 5 0.9(16) obtained from the branching
process model by using Eq. 3 as the sole parameter fits the
observed data remarkably well (P 5 0.12; Kolmogorov–Smirnov
test; see Fig. 3 Inset). Alternatively, we can use a best fit to
determine the ratio RoyRf without resorting to Eq. 3, yielding
RoyRf 5 0.115(20) (P 5 0.44). Fitting abundance distributions to
the branching model thus allows us to determine a ratio of
parameters that reflect dynamics intrinsic to the taxon under
consideration and the niche(s) it inhabits. Indeed, some taxa
analyzed in refs. 2 and 3 are better fit with 0.5 , m , 0.75,
pointing to conditions in which the rate of taxon formation was
much closer to the rate of subtaxon formation, indicating either
a more ‘‘robust’’ genome or richer and more diverse niches.
In general, however, Burlando’s data (2, 3) suggest that a wide
variety of taxonomic distributions are fit quite well by power laws
(m 5 1), implying that actual taxonomic abundance patterns
from the fossil record are characterized by a relatively narrow
range of m near 1. Such distributions are indeed likely within the
model description advanced here. It is obvious that m cannot
remain above 1 for significant time scales, because it would lead
to an infinite number of subtaxa for each taxon. What about low
m? We propose that low values of m are not observed for large
(and therefore statistically important) taxon assemblages for the
following reasons. A small value of m implies either a small
number of total individuals for this assemblage or a very low rate
of beneficial taxon-forming (or niche-filling) mutations. The
former might lead to this assemblage not being recognized at all
in field observations. Either case will lead to an assemblage with
too few taxa to be statistically tractable. Also, because such an
assemblage contains a small number of individuals or is less
suited for further adaptation or both, it would seem to be
susceptible to early extinction.
The branching model can—with appropriate care—also be
applied to species-abundance distributions, even though these
are more complicated than those for higher taxonomic orders for
several reasons. Among these are the effects of sexual repro-
duction and the localized and variable effects of the environment
and other species on specific populations. Still, because the
arguments for using a branching process model essentially rely
on mutations that may produce lines of individuals that displace
others, species-abundance distributions may turn out not to be
qualitatively as different from taxonomically higher-level rank-
frequency distributions as is usually expected.
Historically, species abundance distributions have been char-
acterized by using frequency histograms of the number of species
in logarithmic abundance classes. For many taxonomic assem-
blages, this procedure was found to produce a humped distri-
bution truncated on the left—a shape usually dubbed lognormal
Fig. 1. Predicted abundance pattern P(n) (probability for a taxon to have n
subtaxa) of the branching model with different values of m. The curves have
been individually rescaled.
Fig. 2. Abundance patterns obtained from two sets of numerical simulations
of the branching model, each with m 5 ^m& 5 0.5. Different values for m were
chosen from a uniform probability distribution of width 1 for the runs repre-
sented by crosses and from a distribution of width 0.01 for those represented
by circles. Simulations in which m and Pi are allowed to vary significantly and
those where they are severely constricted are impossible to distinguish if they
share the same ^m&.
Fig. 3. The abundance distribution of fossil marine animal orders (6)
(squares) and the predicted curve from the branching model (solid line). The
fossil data have been binned above n 5 37 with a variable bin size (http:yy
xxx.lanl.govyabsycond-maty9903085). The predicted curve was generated by
using RoyRf 5 NoyNf 5 0.115, where No and Nf were obtained directly from the
fossil data. (Inset) Kolmogorov–Smirnov significance levels P obtained from
comparison of the fossil data to several predicted distributions with different
values of RoyRf, showing that the data are best fit by RoyRf 5 0.135. The arrow
points to our prediction RoyRf 5 0.115 where P 5 0.12. A Monte Carlo analysis
shows that for a sample size of 626 (as we have here), the predicted RoyRf 5
0.115 is within the 66% confidence interval of the best fit RoyRf 5 0.135 (P 5
0.44). The Kolmogorov–Smirnov tests were done after removal of the first
point, which suffers from sampling uncertainties.
15018 u www.pnas.org Chu and Adami
(7–10). In fact, this distribution is not incompatible with the
power law type distributions described above. Indeed, plotting
the fossil data of Fig. 3 in logarithmic abundance classes pro-
duces a lognormal shape (Fig. 4). For species, m is the mean
number of children each individual of the species has. (Of
course, for sexual species, m would be half the mean number of
children per individual.) In the present case, m , 1 implies that
extant species’ populations decrease on average, whereas m 5 1
implies that average populations do not change. An extant
species’ population can decline because of the introduction of
competitors andyor the decrease of the size of the species’
ecological niche. Let us examine the former more closely. If a
competitor is introduced into a saturated niche, all species
currently occupying that niche would temporarily see a decrease
in their m until a new equilibrium was obtained. If the new
species is significantly fitter than the previously existing species,
it may eliminate the others. If the new species is significantly less
fit, then it may be the one eliminated. If the competitors are
about as efficient as the species already present, then the
outcome is less certain. Indeed, it is analogous to a nonbiased
random walk with a possibility of ruin. The effects of introducing
a single competitor are transient. However, if new competitors
are introduced more or less periodically, then these added
competitors would act to push m lower for all species in this
niche, and we would expect an abundance pattern closer to the
exponential curve as opposed to the power law than otherwise
expected. We have examined this question in simulations of
populations in which new competitors were introduced into the
population by means of neutral mutations—mutations leading to
new species of the same fitness as extant species—and found that
these distributions are fit very well by the branching model. A
higher rate of neutral mutations and thus of new competitors
leads to distributions closer to exponential. We have performed
the same experiment in more sophisticated systems of digital
organisms (artificial life; refs. 11 and 12) and found the same
result (http:yyxxx.lanl.govyabsycond-maty9903085).
If no new competitors are introduced but the size of the niche
is gradually reduced, we expect the same effect on m and on the
abundance distributions. Whether it is possible to separate the
effects of these two mechanisms in ecological abundance pat-
terns obtained from field data is an open question. An analysis
of such data to examine these trends would certainly be very
interesting.
Thus far, we have sidestepped the difference between histor-
ical and ecological distributions. For the fossil record, the
historical distribution we have modeled here should work well.
For field observations where only currently living groups are
considered, the nature of the death and extinction processes for
each group will affect the abundance pattern. In our simulations
and artificial-life experiments, we have universally observed a
strong correlation between the shapes of historical and ecolog-
ical distributions. We believe this correspondence will hold in
natural distributions as well when death rates are affected mainly
by competition for resources. The model’s validity for different
scenarios is an interesting question, which could be answered by
comparison with more taxonomic data.
Our branching process model allows us to reexamine the
question of whether any type of special dynamics—such as
self-organized criticality (13, 14)—is at work in evolution (15,
16). Although showing that the statistics of taxon rank-frequency
patterns in evolution are closely related to the avalanche sizes in
self-organized criticality sandpile models, the present model
clearly shows that instead of a subsidiary relationship in which
evolutionary processes may be self-organized critical, the power
law behavior of both evolutionary and sandpile distributions can
be understood in terms of the mechanics of a Galton–Watson
branching process (http:yyxxx.lanl.govyabsycond-maty9903085
and ref. 17). The mechanics of this branching process are such
that the branching trees are probabilistic fractal constructs.
However, the underlying stochastic process responsible for the
observed behavior can be explained simply in terms of a random
walk (18). For evolution, the propensity for near power law
behavior is found to stem from a dynamical process in which m ’
1 is selected for and highly more likely to be observed than other
values, whereas the ‘‘self-tuning’’ of the self-organized criticality
models is seen to result from arbitrarily enforcing conditions that
would correspond to the limit RoyRf 3 0 and therefore m 3 1
(http:yyxxx.lanl.govyabsycond-maty9903085).
We thank J. J. Sepkoski for kindly sending us his amended data set of
fossil marine animal families. Access to the INTEL PARAGON XPyS was
provided by the Center of Advanced Computing Research at the
California Institute of Technology. This work was supported by a grant
from the National Science Foundation.
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2. Burlando, B. (1990) J. Theor. Biol. 146, 99–114.
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4. Harris, T. E. (1963) The Theory of Branching Processes (Springer, Berlin).
5. Raup, D. M. (1985) Paleobiology 11, 42–52.
6. Sepkoski, J. J. (1992) A Compendium of Fossil Marine Animal Families
(Milwaukee Public Museum; Milwaukee, WI), 2nd Ed.
7. Preston, F. W. (1948) Ecology 29, 255–283.
8. Preston, F. W. (1962) Ecology 43, 410–432.
9. Preston, F. W. (1962) Ecology 43, 185–215.
10. Sugihara, G. (1980) Am. Nat. 116, 770–787.
11. Adami, C. (1998) Introduction to Artificial Life (Springer, New York).
12. Chu, J. & Adami, C. (1997) in Artificial Life V: Proceedings of the Fifth
International Workshop on the Synthesis and Simulation of Living Systems, eds.
Langton, C. G. & Shimohara, K. (MIT Press, Cambridge, MA), pp. 462–469.
13. Bak, P., Tang, C. & Wiesenfeld, K. (1987) Phys. Rev. Lett. 59, 381–384.
14. Bak, P., Tang, C. & Wiesenfeld, K. (1988) Phys. Rev. A 38, 364–374.
15. Sneppen, K., Bak, P., Flyvbjerg, H. & Jensen, M. H. (1995) Proc. Natl. Acad.
Sci. USA 92, 5209–5213.
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18. Spitzer, F. (1964) Principles of Random Walk (Springer, New York).
Fig. 4. The abundance distribution of fossil marine animal orders in loga-
rithmic abundance classes (the same data shown in Fig. 3). The histogram
shows the number of orders in each abundance class (left scale), whereas the
solid line depicts the number of families in each abundance class (right scale).
Species rank-abundance distributions where the highest abundance class
present also has the highest number of individuals (as in these data) are
termed canonical lognormal (8, 9).
Chu and Adami PNAS u December 21, 1999 u vol. 96 u no. 26 u 15019
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A simple explanation for taxon abundance patterns

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A Simple Explanation for Taxon Abundance Patterns

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