What are the future works in "How many species have mass m?" ?

The similarity of these distributions to those of other taxonomic groups suggests that this explanation may be universal, although further empirical work is necessary to substantiate this hypothesis. The authors conclude by noting that the model ’ s good agreement with data suggests that it may be a useful way to establish null expectations in the study of general trends in body mass evolution ( much like diffusion models in population genetics ; Hartl and Clark 2007 ) in the absence of factors such as interspecific competition, population dynamics, geography, predation, and so on.

(Open Access) How Many Species Have Mass M (2009) | Aaron Clauset

Q: What are the contributions mentioned in the paper "How many species have mass m?" ?

Here the authors introduce and analytically solve a simplified cladogenetic diffusion model. Under this model, the authors show that a specific trade-off between the strength of within-lineage drift toward larger masses ( Cope ’ s rule ) and the increased risk of extinction from increased mass is necessary to produce realistic mass distributions for both taxa.

vol. 173, no. 2 the american naturalist february 2009 

How Many Species Have Mass M?

Aaron Clauset,

David J. Schwab,

and Sidney Redner

1,3

1. Santa Fe Institute, Santa Fe, New Mexico 87501; 2. Department of Physics and Astronomy, University of California, Los Angeles,

California 90024; 3. Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215

Submitted June 16, 2008; Accepted August 25, 2008; Electronically published December 17, 2008

Online enhancements: color versions of ﬁgures.

abstract: Within large taxonomic assemblages, the number of spe-

cies with adult body mass M is characterized by a broad but asym-

metric distribution, with the largest mass being orders of magnitude

larger than the typical mass. This canonical shape can be explained

by cladogenetic diffusion that is bounded below by a hard limit on

viable species mass and above by extinction risks that increase weakly

with mass. Here we introduce and analytically solve a simpliﬁed

cladogenetic diffusion model. When appropriately parameterized, the

diffusion-reaction equation predicts mass distributions that are in

good agreement with data on 4,002 terrestrial mammals from the

late Quaternary and 8,617 extant bird species. Under this model, we

show that a speciﬁc trade-off between the strength of within-lineage

drift toward larger masses (Cope’s rule) and the increased risk of

extinction from increased mass is necessary to produce realistic mass

distributions for both taxa. We then make several predictions about

the evolution of avian species masses.

Keywords: macroevolution, species body mass distribution, diffusion,

Cope’s rule, mammals, birds.

Introduction

For a large taxonomic group under stable macroevolu-

tionary conditions, how many species have a mass M? This

question has wide implications for the evolution and dis-

tribution of many other species characteristics that cor-

relate strongly with body mass, including life span, life

history, habitat, metabolism, and extinction risk (Calder

1984; Brown 1995; Bennett and Owens 1997; Cardillo et

al. 2005).

Extant species, including mammals, birds, ﬁsh, and in-

sects, exhibit a canonical form of the species mass distri-

bution (Stanley 1973; Kozłowski and Gawelczyk 2002; Al-

len et al. 2006), in which the typical mass is an intermediate

value; for example, in mammals, the typical mass is that

of the common Paciﬁc rat (Rattus exulans; 40 g). Larger

or smaller species, in turn, are signiﬁcantly less common

* Corresponding author; e-mail: aaronc@santafe.edu.

Am. Nat. 2009. Vol. 173, pp. 256–263. 䉷 2009 by The University of Chicago.

DOI: 10.1086/595760

but asymmetrically so; the largest species, such as the ex-

tinct imperial mammoth (Mammuthus imperator;10

for terrestrial mammals, are many orders of magnitude

larger, while the smallest species are only a little smaller,

for example, the Remy’s pygmy shrew (Suncus remyi; 1.8

g).

The ubiquity of this distribution of species masses (ﬁg.

1A) suggests the existence of a universal evolutionary

mechanism. A theoretical explanation of this distribution

may shed light both on the interaction between ecological

and macroevolutionary processes (Stanley 1975) and on

long-term trends in species mass (Alroy 2000a, 2000b),

including Cope’s rule, the oft-studied notion that species

mass tends to increase within a lineage over evolutionary

time (Stanley 1973; Alroy 1998). Clauset and Erwin (2008)

recently showed, by comparing extensive computer sim-

ulations with empirical data, that cladogenetic diffusion

in the presence of a taxon-speciﬁc lower limit on mass

min

and extinction risks that grow weakly with mass can

explain both the canonical form described above and the

precise form of the distribution of terrestrial mammal

masses.

Here, we present a simpliﬁed three-parameter version

of the Clauset and Erwin (CE) model for the species mass

distribution and solve this model in the steady state. Com-

paring the predictions of this simpliﬁed model with species

mass data for the same 4,002 terrestrial mammals from

the late Quaternary (Smith et al. 2003, 2007; henceforth

denoted “Recent” mammals, which includes species that

have become extinct during the Holocene), we reproduce

Clauset and Erwin’s results. We then show that the model’s

predictions, when appropriately parameterized, are also in

good agreement with data for 8,617 extant avian species

(Dunning 2007).

The Clauset-Erwin Model

Many models of the variation of species body mass over

evolutionary time assume a cladogenetic diffusion process

(Stanley 1973; McKinney 1990; McShea 1994; Kozłowski

and Gawelczyk 2002), where each descendant species’ mass

How Many Species Have Mass M? 257

Figure 1: A, Illustration of the similarity of the body mass distributions

for different taxonomic groups, using data for 4,002 Recent terrestrial

mammals (Smith et al. 2003, 2007) and 8,617 extant birds (Dunning

2007). For visual clarity, the empirical distributions have been smoothed

using a Gaussian kernel (Wasserman 2006) and rescaled to lay on the

same abscissa interval. B, Schematic showing the basic cladogenetic dif-

fusion process by which species mass varies over evolutionary time. The

mass of a descendant M

is related to the mass of its ancestor M

by a

random multiplicative factor l, which represents the total selective in-

ﬂuence on species mass from all sources; that is, . A colorM p lM

version of this ﬁgure is available in the online edition of the American

Naturalist.

is related to its ancestor’s mass M

by a random mul-

tiplicative factor l; that is, (ﬁg. 1B), where lM p lM

represents the total selective inﬂuence on the descendant

species’ mass from all sources. Clauset and Erwin (2008)

studied a family of such diffusion models, whose form and

parameters could be estimated empirically. Under their

model, the evolution of species mass is bounded on the

lower end by hard physiological limits, perhaps from met-

abolic (Pearson 1948, 1950; West et al. 2002) or morpho-

logical (Stanley 1973) constraints, and on the upper end

by an extinction risk that increases weakly with mass (Liow

et al. 2008). Near the lower limit, however, within-lineage

changes to body mass become increasingly biased toward

larger masses; that is, Aln lS increases as .M r M

min

Using model parameters estimated from ancestor-

descendant data on extinct North American terrestrial

mammals since the Cretaceous-Paleogene boundary (Al-

roy 2008), Clauset and Erwin simulated 60 million years

of mammalian body mass evolution and found that the

predicted distribution closely matches that of 4,002 Recent

terrestrial mammal species. They also found that simpler

diffusion models—for example, those that omitted the

lower limit, an increased extinction risk from increased

mass, or the increased bias toward larger masses as M r

—predicted signiﬁcantly different distributions.M

min

Thus, these three mechanisms can explain not only the

canonical form of the species mass distribution but also

its taxon-speciﬁc shape.

A Mathematically Solvable Version

The CE model, however, remains too complex for math-

ematical analysis, even though it omits many ecological

and microevolutionary processes such as interspeciﬁc

competition, predation, and population dynamics. By sim-

plifying its assumptions slightly, we can formulate a dif-

fusion-reaction model of species body mass evolution,

which retains most of the key features of the CE model

and which can be mathematically analyzed. This model

can then be used to make inferences about body mass

evolution without resorting to laborious simulations.

Let denote the number of species with logarithmicf(x, t)

mass at time t; that is, we transform the CEx p lnM

model’s multiplicative diffusion process into an additive

diffusion process on a logarithmic scale. The cladogenesis

mechanism leads to two offspring species in which x r

. In the continuum limit, a value (Cope’sx ⫹ lnl Aln lS

1 0

rule) corresponds to positive drift velocity in the equation

of motion for . In the CE model for terrestrial mam-f(x, t)

mals, the drift velocity was estimated from fossil data to

increase like M

1/4

near the lower limit on species mass, a

feature that Clauset and Erwin (2008) found was necessary

to accurately predict the number of small-bodied mam-

mals. (Whether this quarter-power form is related to the

quarter-power scaling commonly found elsewhere in the

body size literature [Savage et al. 2004] remains to be seen.)

For mathematical simplicity, however, we will ignore this

dependence at the expense of possibly misestimating the

number of small-bodied species. We also assume that se-

lection pressures on body mass are roughly independent,

implying that the distribution of changes l is approxi-

mately lognormal (but see Clauset and Erwin 2008).

258 The American Naturalist

The feature that two offspring are produced at each

update step corresponds to a population growth term in

the equation of motion that is proportional to f itself. The

extinction probability may also be represented by a loss

term that is proportional to f. For terrestrial mammals,

recent empirical studies (Liow et al. 2008) support the

assumption that the probability per unit time of a species

becoming extinct p

(x) grows weakly with its mass. The

CE model uses a simple parameterization of this behavior:

, where , which corresponds to anp (x) p A ⫹ Bx B ≥ 0

extinction probability that grows logarithmically with spe-

cies mass.

Combining these three elements—diffusion, cladogen-

esis, and extinction—we may write the continuum equa-

tion of motion for the number of species with massf(x, t)

at time t asx p ln M

⭸f ⭸f ⭸ f

⫹ v p D ⫹ (1 ⫺ A ⫺ Bx)f,(1)

⭸t ⭸x ⭸x

where is the drift velocity (strength of Cope’s

v p Aln lS

rule) and the variance is the diffusion coef-

D p Aln lS

ﬁcient. (In physics, eq. [1] is called the convection-

diffusion equation or the Fokker-Planck equation.) This

equation, however, omits the lower limit on species body

mass, which we incorporate momentarily.

The time-dependent equation of motion itself may be

useful for studying evolutionary trends in species body

mass or for making inferences about correlated extinction

or speciation events. For our purposes, however, we are

mainly interested in its stationary solution. Such a steady

state should exist whenever all species within the taxon

experience roughly the same set of macroevolutionary se-

lective pressures, that is, under stable macroevolutionary

conditions. To derive this solution, we set the time deriv-

ative in equation (1) to zero to obtain

 

f ⫺ mf ⫹ (a ⫺

, where , , , andbx)f p 0 m p

v/D a p (1 ⫺ A)/D b p B/D

the prime denotes differentiation with respect to x.We

now eliminate the ﬁrst derivative term by introducing

to transform the steady state equation to

mx/2

f p e w



w ⫹ a ⫺⫺bx w p 0.

[( ) ]

This equation can be brought into the form of the standard

Airy’s differential equation (Abramowitz and Stegun

1972):



w ⫺ zw p 0, (2)

where we introduce the new variable

1/3

z p b x ⫺

and the prime now denotes differentiation

⫺2/3 2

b (a ⫺ m /4)

with respect to z. The general solution to equation (2) is

, where Ai(z) and Bi(z) are thew(z) p c Ai(z) ⫹ c Bi(z)

Airy functions. Since there can be no species with inﬁnite

mass, we may set . The parameter c

is then deter-c p 0

mined by the normalization of f(x).

Thus, we can now write the species mass distribution

mx/2 1/3 ⫺2/3

f(x) ∝ eAib x ⫺ ba⫺ .(3)

[()]

Including now the taxon-speciﬁc lower limit x

min

on spe-

cies mass implies the constraint for the steadyf(x ) p 0

min

state solution and allows us to eliminate one parameter

from equation (3). Using the fact that the ﬁrst zero of the

Airy function is located at , which we nowz p ⫺2.3381 …

require to coincide with , gives the constraintx p x

min

1/3 ⫺2/3

z p b x ⫺ ba⫺ ,

0min

()

which we may solve for a. Inserting this result into equa-

tion (3) yields

mx/2 1/3

f(x) ∝ eAi[b (x ⫺ x ) ⫹ z ](4)

min 0

as the steady state solution for the species mass distribution.

If the lower limit x

min

is known, this simple-minded

model has only two parameters: m, associated with the

biased diffusion process, and b, associated with the ex-

tinction process. A positive bias in the diffusion m

1 0

(Cope’s rule) has several systematic effects on the distri-

bution of species masses: it (1) pushes the left tail of the

distribution away from the lower limit at x

min

, (2) shifts

the modal mass toward slightly larger values, and (3) ex-

tends the right tail of the distribution. In contrast, in-

creasing b implies that species become extinct with greater

probability for a given mass M, which contracts the right

tail of the distribution. For a given bias m and number of

species n, the parameter b also sets an effective upper limit

on the expected maximum observed mass within the taxon

without invoking a hard boundary, for example, from bio-

mechanical constraints (McMahon 1973).

Mammalian Body Mass Evolution

We now test the predictions of this simple mathematical

model using empirical data for 4,002 Recent terrestrial

mammals (Smith et al. 2003, 2007) and 8,617 birds (Dun-

ning 2007). In the former case, we take , theM p 1.8 g

min

size of the smallest known mammal, and we estimate

and (SE) from Al-

v p 0.109 Ⳳ 0.021 D p 0.508 Ⳳ 0.027

How Many Species Have Mass M? 259

Figure 2: A, Solution of the diffusion-reaction model of species body

mass evolution (eq. [4]) with (smoothed) empirical data for 4,002 ter-

restrial mammals from the late Quaternary. Parameters m and x

min

were

estimated from empirical data on extinct terrestrial mammals since the

Cretaceous-Paleogene boundary, while b was chosen to minimize the

tail-weighted Kolmogorov-Smirnov (wKS) distance between the model

and the empirical distribution. B, The plane of wKS distance of(m, b)

the speciﬁed model and the empirical data for terrestrial mammals

(lighter values correspond to smaller wKS). The distinctive groove (line)

demonstrates that a systematic relationship between the bias parameter

m and the extinction parameter b is necessary to produce realistic mass

distributions (small wKS). The parameter pair used in A is marked.

Dashes demarcate the line of no within-lineage bias ( ). A colorm p 0

version of this ﬁgure is available in the online edition of the American

Naturalist.

roy’s (2008) ancestor-descendant data for North American

terrestrial mammals. Incorporating these values into equa-

tion (4) leaves only b unspeciﬁed. A strong test of this

model would estimate b from fossil data; however, while

studies of extinction among mammals suggest that b

1 0

(Liow et al. 2008), current data do not appear to be suf-

ﬁciently detailed to give a precise estimate for mammals.

Instead, following Clauset and Erwin (2008), we choose

b by minimizing the tail-weighted Kolmogorov-Smirnov

(wKS) goodness-of-ﬁt statistic (Press et al. 1992) for the

predicted and empirical distributions:

FS(x) ⫺ P(x)F

wKS p max ,

冑

P(x)(1 ⫺ P(x))

where S(x) is the empirical distribution function and P(x)

is the predicted cumulative distribution function. Thus,

small values of wKS correspond to a model that is statis-

tically close to the empirical data everywhere. We ﬁnd that

two alternative methods of choosing b, by numerically

matching the modal masses of the model and the empirical

data or by matching the expected maximum mass of the

model with the observed maximum in the empirical data,

produce similar results.

For the interested reader, details on the preparation of

the empirical data are discussed at length by Alroy (1998)

and Smith et al. (2003, 2007) for mammalian fauna and

by Dunning (2007) for avian fauna. In general, body mass

estimates were derived using conventional techniques (e.g.,

dentition techniques for mammals; Damuth 1990). For

simplicity, differences due to sexual dimorphism, geo-

graphic variation, and so on were ignored or averaged out.

Although such differences can be critical for smaller stud-

ies, given the scale of our data in terms of the number of

species studied and the wide range of body masses, mild

misestimates of body masses are unlikely to change our

conclusions unless they are widespread and systematic.

The resulting ﬁt (ﬁg. 2A) is in good agreement with the

empirical data, except for a slight overestimate of the num-

ber of species with mass near 1 kg, an underestimate of

the number near 300 kg, and a slight misestimate of the

number of very small-bodied species. The deviations in

the right tail are also seen in the CE model and may be

due to, for example, phylogenetically correlated speciation

or extinction events in the recent past. The deviations in

the left tail may be due to our omission of the mass de-

pendence in the drift term m identiﬁed by Clauset and

Erwin (2008); however, incorporating this behavior into

our diffusion-reaction model is technically nontrivial.

Thus, the evolution of mammalian species body masses

can largely be viewed as a simple diffusion process char-

acterized by (1) a slight within-lineage drift toward larger

masses over evolutionary time (Cope’s rule), (2) a hard

lower boundary on how small body masses can become,

and (3) a very soft constraint on large body masses in the

form of increased extinction risk. Phrased more concep-

tually, the left tail of the mammalian body mass distri-

260 The American Naturalist

Figure 3: A, Steady state solution of the diffusion-reaction model of

species body mass evolution (eq. [4]), with , , and bm p 0 M p 2g

min

chosen to minimize the tail-weighted Kolmogorov-Smirnov (wKS) dis-

tance between the model and the empirical distribution, with (smoothed)

empirical data for 8,617 avian species. B, The plane of wKS dis-(m, b)

tances (lighter values correspond to smaller wKS), showing a systematic

relationship between m and b for producing realistic (small wKS) avian

mass distributions. The form of this relationship is similar to that of

mammals (ﬁg. 2B), suggesting a similar overall macroevolutionary mech-

anism. A color version of this ﬁgure is available in the online edition of

the American Naturalist.

bution is mainly controlled by the lower limit on mass,

while the right tail is the result of an evolutionary trade-

off at different timescales: over the short-term, within-

lineage increases in body mass offer selective advantages—

such as better tolerance of resource ﬂuctuations, better

thermoregulation, better predator avoidance, and so on

(Calder 1984; Brown 1995)—while they also increase the

long-term risk of extinction, a trade-off previously iden-

tiﬁed in the more speciﬁc case of carnivorous mammals

(Van Valkenburgh 1999; Van Valkenburgh et al. 2004).

Avian Body Mass Evolution

Unlike mammals, data on most other taxonomic groups

are generally not sufﬁcient to yield accurate estimates of

the parameters m and b (but see Novack-Gottshall and

Lanier 2008). The evolutionary history of mammalian

body masses is relatively clear, in part because mammalian

fossils are relatively plentiful and are often sufﬁciently well

preserved that body mass estimates can be made (Damuth

1990; Van Valkenburgh 1990) and because the distribution

of species body masses during an apparently stable evo-

lutionary period is known. Avian species, however, present

an interesting case for study using our model; the distri-

bution of extant avian body masses (ﬁg. 3A) is relatively

well characterized (Dunning 2007), and evidence of a min-

imum species body mass M

min

is reasonable (Pearson

1950). However, the avian fossil record may be too sparse

to yield accurate estimates of m and b (Fountaine et al.

2005; Hone et al. 2008).

Even without estimates of m and b, however, the dif-

fusion model can be used to make quantitative statements

about the general character of avian body mass evolution.

To demonstrate this, we consider which combinations of

the parameters m, b produce “realistic” mass distributions,

that is, those with a small distributional distance to the

empirical distribution. In particular, we compute the wKS

distance between the model and the empirical data over

the plane and determine the regions that yield the(m, b)

best ﬁts. To illustrate this technique in a better-understood

context, we ﬁrst apply it to the data on Recent terrestrial

mammals, disregarding for the moment that we have an

estimate of m from fossil data.

The result of this exercise (ﬁg. 2B) shows that realistic

mammalian mass distributions can be produced by a wide,

but not arbitrary, variety of biases m and extinction risks

b, including no bias at all; that is, . This degeneracy,m p 0

which forms a groove in the plane following roughly(m, b)

with , suggests that part of the difﬁculty in

m ∝ bv≈ 0.5

determining for a particular taxon whether mass evolution

is biased toward larger sizes (see, e.g., Maurer et al. 1992;

Maurer 1998; Bokma 2002) is that a positive bias is not

a necessary condition for the evolution of realistic mass

distributions. Indirect tests of the sign of m, based either

on the mass distribution within subclades (McShea 1994;

Wang 2001) or on changes to minimum and maximum

masses within a subclade over geologic time (Jablonski

1997), may be adequate if confounding hypotheses can be

eliminated or if an appropriate null model is available. On

How Many Species Have Mass M

Figures

Citations

宏生态学（Macroecology）及其研究

Geographic and temporal correlations of mammalian size reconsidered: a resource rule

Determinants of inter-specific variation in basal metabolic rate

Allometric estimation of metabolic rates in animals.

Occupancy, range size, and phylogeny in Eurasian Pliocene to Recent large mammals

References

CRC handbook of avian body masses

Size and shape in biology.

Multiple Causes of High Extinction Risk in Large Mammal Species

All of Nonparametric Statistics

The predominance of quarter-power scaling in biology

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