FORUM is intended for new ideas or new ways of interpreting existing information. Itprovides a chance for suggesting hypotheses and for challenging current thinking onecological issues. A lighter prose, designed to attract readers, will be permitted. Formalresearch reports, albeit short, will not be accepted, and all contributions should be concisewith a relatively short list of references. A summary is not required.
FORUMFORUMFORUM
On the conceptual basis of the self

thinning rule
Jose´

Leonel Torres
,
Instituto de Fı´sica y Matema´ticas
,
Uni
ersidad Michoacana
,
Apartado Postal
2

82
,
58040
Morelia
,
Michoaca´n
,
Mexico
(
jltorres@ifm
1
.
ifm
.
umich
.
mx
)
. –
Vinicio J
.
Sosa and Miguel Equihua
,
Instituto deEcologı´a
,
A
.
C
.,
km
2
.
5
ant
.
carretera Coatepec
,
91000
Xalapa
,
Veracruz
,
Mexico
. –
Leonel Torres
,
Instituto deIn
estigaciones Biolo´gicas
,
Uni
ersidad Veracruzana
,
Apartado Postal
294
,
91000
Xalapa
,
Veracruz
,
Mexico
.
We show a widely accepted proof of the selfthinning ruleoffered by Enquist et al. to be mathematically incomplete, as itdoes not identify the plant mass distributions that satisfy acondition implicitly used in the proof. We propose a method toguide the search for such mass distributions, based on a requirement of maximum mass diversity under the appropriate constraints. This generic method allows construction of a probabilitydensity that incorporates the available information on a givenstochastic variable, and we illustrate its use through the calculation of a continuous mass distribution for the selfthinning rulethat satisﬁes the implicit condition mentioned above. We suggesta biological justiﬁcation of maximum mass diversity, as a corollary to the random and unbiased nature of the source of diversityin Darwin’s principle.
The selfthinning rule is one of the few longlivedquantitative propositions in ecology, despite recurrentepisodes of criticism of its empirical and conceptualfoundations (Weller 1989, Lonsdale 1990). It is anallometric relation that connects average mass
M
withplant density
N
max
under conditions of maximum productivity (or maximum total biomass):
N
max
(
M
)
−
,where ‘
’ denotes proportionality. There has beensome controversy about the precise value of the exponent
, traditionally taken as 2
/
3, but closer to 3
/
4according to more recent evidence (Enquist et al. 1998).In a loglog plot this produces straight lines that ﬁt thedata from mature single and mixedspecies stands reasonably well, for average mass values spanning severalorders of magnitude (Gorham 1979, White 1981).Plants build and maintain themselves from the energy and materials available in their environment, andso must share a limited ﬂux of resources. The maximumvalue of this ﬂux is determined by the fraction of solarfree energy that can be ﬁxed through photosynthesis,and by the maximal possible ﬂow of CO
2
from theatmosphere and of water with dissolved minerals fromthe soil. An argument based on crowding as a limit togrowth was srcinally proposed to justify the value of the exponent in the selfthinning rule, with severalsubsequent reﬁnements offered in the same vein (White1981, Norberg 1988, Franco and Kelly 1998). A morefundamental approach starts with the allometry
Q
M
3
/
4
, valid for individual plants and animals, where
Q
is the ﬂux of sap or blood and
M
the mass of anorganism. This rule has a ﬁrm empirical basis and wasderived by West et al. (1997), from a requirement of biological optimality in the process of carrying nutrients to all parts of the body (see, however, Dodds et al.2001). They assumed transport of the appropriate ﬂuid(sap or blood) in a network of pipes that ramiﬁes infractal fashion and allows minimal power loss againstfriction. Applying the relationship
Q
M
3
/
4
to eachplant in a given plot, Enquist et al. (1998) offered asimple counting argument to show that it leads to theselfthinning rule, if one assumes that plants grow andproliferate until limited by the availability of energeticand material resources. There exists for mammals aformal analog to the selfthinning rule (Damuth 1981),so the argument by Enquist et al. seemed to suggest acommon srcin for the massdensity relationship observed in plants and animals (Damuth 1998).However, from a strictly mathematical perspectivethe proof of the selfthinning rule offered by Enquist etal. cannot be considered complete. Calling
Q
tot
the totalavailable ﬂux of energy and materials in a plot (takenfor simplicity with unit area) and
N
max
the number of plants in the regime where they use the whole of
Q
tot
,we have
Q
tot
=
j
n
j
Q
j
N
max
j
(
n
j
/
N
max
)
M
j
3
/
4
, where
n
j
is the number of plants with ﬂux usage
Q
j
, and theallometry
Q
j
M
j
3
/
4
was employed in the last expression. As
n
j
/
N
max
is the fraction of plants that consumea ﬂux
Q
j
(and have mass
M
j
), we thus have
Q
tot
544
OIKOS 95:3 (2001)
N
max
M
3
/
4
, where the bar indicates average value. Enquist et al. (1998) wrote instead
Q
tot
N
max
(
M
)
3
/
4
, andusing the constancy of
Q
tot
they got the selfthinningrule,
N
max
(
M
)
−
3
/
4
. The quantities
M
and
M
canbe very different for an arbitrary mass probabilitydistribution
P
(
M
); for example, (
M
2
−
M
2
) is the variance of
M
. Hence in order to complete this proof onemust identify those plant mass distributions that implythe condition
M
3
/
4
(
M
)
3
/
4
.The mathematical expression of the selfthinning ruledescribes two different types of observations (Harper1977), that we label
‘
context A
’
and
‘
context B
’
:
Context A
. Seedlings from the same species areplanted at high density and removed by thinning as thestand matures (intraspeci
ﬁ
c case). The densitymassrelationship obtained from successive snapshots of thestand
ﬁ
rst approaches and then traces out a trajectorywell
ﬁ
t by a power law. In the interspeci
ﬁ
c version, theobserved combination of plant density and averagemass that maximizes their product in a monospeci
ﬁ
cstand de
ﬁ
nes a point for each species, and a collectionof such points satis
ﬁ
es a power law with the sameexponent as in the intraspeci
ﬁ
c case.
Context B
. The massdensity relationship is measuredin mature stands in a steadystate regarding age andsize structure, where births balance deaths and plantgrowth into a new size class is offset by mortality. Datafrom a set of stands of this kind ful
ﬁ
ll a power law withthe same exponent as in context A. This rule has alsobeen con
ﬁ
rmed for mixed stands of forest trees, considering all species collectively (White 1981).For the selfthinning rule referred to context A, thereis (approximately) only one individual mass value ateach stage, so
P
(
M
) is a constant and
M
3
/
4
(
M
)
3
/
4
istrivially satis
ﬁ
ed; hence the proof by Enquist et al.turns out to be correct. But in context B one has arange of possible plant mass values, and the question of which distributions
P
(
M
) lead to
M
3
/
4
(
M
)
3
/
4
mustbe considered. Posed in this way the problem is perhapsimpossible to solve in full mathematical generality. Oneway to look for suitable mass distributions is by trialand error, but a search of this type would be botharbitrary and inef
ﬁ
cient. What is needed is a biologically justi
ﬁ
ed argument that will guide our quest. Wepropose an argument of this kind, based on the (constrained) maximization of an entropylike function thatgives a measure of mass diversity. It allows incorporation of empirical evidence like the known moments of the desired mass distribution in the process of constructing it. As an illustrative example, using ourpresent information about the selfthinning rule in context B, it leads to
P
(
M
)
M
−
1
/
4
as a suitable massdistribution. Numerical exploration then shows that
P
(
M
)
M
−
, with
centered about 1
/
4 but not tooclose to 0 or 1, is also compatible with the selfthinningrule.Below we present our method of searching for candidate mass distributions, describe its biological basis andillustrate its use with the example just mentioned.Mathematical details are included in the Appendix.
Mass distribution of plants
A general mathematical recipe to calculate probabilitydistributions that incorporate constraints is the maximum entropy method (Montroll and Schlesinger 1983,Buck 1991). Suppose we want a probability density
p
(
x
) that satis
ﬁ
es a number of constraints determinedby functions {
F
i
(
x
);
i
=
1,
…
,
k
} through the conditions(assuming a continuous variable
x
to simplify thenotation),
F
i
(
x
)
p
(
x
)d
x
=
i
(1)with {
i
} some constants and
F
1
(
x
)
=
1
=
1
, so the
ﬁ
rst condition simply describes the normalization requirement
p
(
x
)d
x
=
1. If we assume in addition that
p
(
x
) maximizes the entropylike function
S
=−
p
(
x
) ln[
p
(
x
)]d
x
, the solution to our problem is
p
(
x
)
=
C
exp
−
i
i
F
i
(
x
)
n
(2)where the constant
C
=
e
−
1
and the parameters
i
aredetermined using the
k
constraints (1).Our biological adaptation of this method centers onthe above constraints and the extremal condition on theentropylike function
S
, which has been used as ageneric measure of diversity in various contexts, including population analysis (Fleming 1973, Buzas andHayek 1996). The conditions (1) become more familiarin the special case where
F
i
(
x
)
=
x
r
i
, with
r
i
an integer,as
F
i
(
x
)
p
(
x
)d
x
=
i
then becomes the
r
i
th moment of
x
. Hence if one knows one or more moments of thebiological variable under consideration, using thismethod one can construct a function that incorporatessuch information. Without the maximum entropy requisite, determination of
p
(
x
) would demand knowledgeof all the moments of
x
. For example, if nothing about
p
(
x
) is known beyond the fact that it must be normalized, maximum entropy implies
p
(
x
)
=
constant. If onlythe
ﬁ
rst moment
x
¯
is available, the resulting distribution becomes
p
(
x
)
e
−
x
. Similarly, if one only knowsthe second moment of
x
, its distribution
p
(
x
) turns outto be a Gaussian, and if there is information on severalmoments,
p
(
x
) will be the exponential of some polynomial function of
x
. Depending on the available empirical and theoretical information we thus obtain speci
ﬁ
cdistributions that are symmetrical or skewed, unimodal,multimodal, etc. The range of possibilities increases if 545
OIKOS 95:3 (2001)
more general forms than simple powers are consideredfor the functions
F
i
(
x
) in eq. (1).To discuss the biological content of the maximumentropy requirement it is convenient to revert to adiscrete variable, concretely one we are using for theselfthinning rule, namely
Q
j
, the cost of a plant withmass
M
j
in terms of resources, where
Q
j
M
j
3
/
4
. It isshown in the Appendix that maximizing
S
=−
j
f
j
ln
f
j
, with
f
j
=
n
j
/
N
, is equivalent to maximizing themultinomial function
F
({
n
j
})
=
N
!
/
n
1
!
n
2
!
n
3
!
…
, where
n
!
=
1.2
…
(
n
−
1) .
n
is the factorial of
n
. The quantity
F
({
n
j
}) numbers the different ways in which we canpartition
N
, i.e., choose groups of size
n
j
among
N
equivalent objects, with no bias or restriction in theprocess besides
j
n
j
=
N
. Imagine now that we partition
N
a huge number of times
N
in this fashion, andcount how many occasions each particular set {
n
j
}turns up in the resulting ensemble of partitions; callingthis quantity
R
({
n
j
}), it will be proportional to
F
({
n
j
}).To add a pictorial effect, let us imagine
N
identicalislands scattered at random, one for each partition inthe ensemble, and paint the
R
({
n
j
}) of them associatedwith a given set {
n
j
} the same color. For large values of
n
1
,
n
2
,
n
3
, etc.,
F
({
n
j
})
=
N
!
/
n
1
!
n
2
!
n
3
!
…
is very narrowlypeaked at a distinct partition {
n
j
*}. Thus if we paint the
R
({
n
j
*}) islands red the whole archipelago will look red,despite a fractionally insigni
ﬁ
cant admixture of othercolors.From our mathematical discussion above, the partition that maximizes the multinomial function (hencethe entropy
S
=−
t j
=
1
f
j
ln
f
j
) with no additional constraint besides
t j
=
1
f
j
=
1, is one where all portions {
n
j
}are identical, and this implies
f
1
=
f
2
=
…
=
f
t
. Identifying
t
with the number of plant types (i.e., possible massvalues), this result reduces to the constant mass distribution in context A for the selfthinning rule, whenthere is only one type of plant (
t
=
1), i.e., a single valueof individual mass at each stage in the life of a stand.One shows in the same way that, if we add the constraint of limited resources (or, equivalently, full use of
Q
tot
), but keep intact our potential to make any kind of plant among the
t
allowed types, the quantities
f
j
e
−
Q
j
, with
a constant, will maximize the entropy
S
=−
t j
=
1
f
j
ln
f
j
, subject to the constraints
t j
=
1
n
j
=
N
max
and
tj
=
1
n
j
Q
j
=
Q
tot
,What can be the srcin of the assumed unlimitedpotential to make any kind of plant among the possibletypes (given availability of the necessary resources), asimplied by the unconstrained maximization of themultinomial function or, equivalently, of the entropy
S
? Our suggestion is that it can be justi
ﬁ
ed as acorollary to the random and unbiased nature of thesource of biological diversity according to Darwin
’
sprinciple, and we offer this as the biological foundationof our mathematical procedure.We now consider in more detail the selfthinning rulein context B, assuming a continuous mass distributionfrom seedlings to trees. In the discrete case, maximummass (or cost) diversity plus full use of available resources imply
f
j
e
−
Q
j
, where
can in principle becalculated from
j
n
j
Q
j
=
Q
tot
. As the solution to thisequation is both nonunique and impossible to obtainin practice, we regard
as a parameter to be
ﬁ
xed usingempirical information. We now convert our discreteprobabilities {
f
j
} into a continuous mass probabilitydensity
P
(
M
) (cf. Appendix),
P
(
M
)
M
−
1
/
4
e
−
M
3
/
4
(3)where
is a constant proportional to
and
M
−
1
/
4
comes from d
Q
/
d
M
, using the allometry
Q
M
3
/
4
.In the special case when the mass range is largeenough (
M
max
M
min
) our mass distribution can besimpli
ﬁ
ed as follows. Without loss of generality wemeasure all masses in units of
M
min
, the minimumpossible mass, supposed common to all seedlings. Fromeq. (3) the ratio of the number of largest plants to thatof smallest ones is of order e
−
M
max3
/
4
, or essentially zeroexcept for very small values of
, as typically
M
max
10
6
for trees, in units of
M
min
(we do not include algaland similar systems (Agusti et al. 1987), where thiscondition is not ful
ﬁ
lled). In the stationary regime largetrees are indeed observed, and we use the freedom westill have on the value of
to choose
so small that theexponential factor in eq. (3) is approximately constantin the interval 1
M
M
max
, and the mass probabilitydistribution reduces to
P
(
M
)
M
−
1
/
4
(4)Using this expression for
P
(
M
) it is shown in theAppendix that
M
3
/
4
M
3
/
4
constitutes a good approximation, thus providing a continuous mass distributionthat will satisfy the selfthinning rule in context B.Numerical exploration of
P
(
M
)
M
−
shows that theapproximation
M
3
/
4
M
3
/
4
holds for a range of valuesaround
=
1
/
4, if this parameter does not come tooclose to 0 or 1.
Discussion
We obtained our continuous plant mass distribution forthe selfthinning rule from the following assumptions:1. Mass and nutrient
ﬂ
ux are the only relevant attributes for plant density.2. The allometry
Q
M
3
/
4
, valid for individuals.3. Plant growth and proliferation are limited by theavailability of energetic and material resources.4. The desired distribution satis
ﬁ
es a criterion of maximum mass diversity, under stationary conditions of greatest productivity in a plot.
OIKOS 95:3 (2001)
546
5. The plant distribution implies
M
3
/
4
(
M
)
3
/
4
, givena large range of possible masses (from seedlings totrees).Discussion of our problem just in terms of a massdistribution, disregarding species abundance and diversity, constitutes a drastic simpli
ﬁ
cation in the naturallyarising stationary regime considered here. Body size,however, is correlated with many important features of organisms. In the case of animals, for example, it isassociated with physiological attributes like metabolic,birth, growth and mortality rates, and life history features like longevity and age at
ﬁ
rst reproduction(Calder 1984, Blueweiss et al. 1987, LaBarbera 1989,West et al. 1997). Although less information of thiskind is available for plants, we work on the assumptionthat body size occupies a comparable central positionin their context.Apart from the proposed connection between (unconstrained) maximum entropy and the random andunbiased nature of the source of diversity, the biological input in our approach stems from the constraints.Changing the functions
F
i
(
x
) in eq. (1) above leads tovery different probability distributions. Besides reducing arbitrariness in our search, this procedure allowsincorporation of the available empirical and theoreticalinformation into the resulting distributions. However,the method has signi
ﬁ
cant limitations. It does not guarantee, for example, that a candidate mass distributionfor the selfthinning rule will automatically satisfy
M
3
/
4
(
M
)
3
/
4
, and to achieve this generally requiressome tuning of parameters. In our particular case weconsidered a range of sizes from seedlings to trees, and
M
3
/
4
(
M
)
3
/
4
was satis
ﬁ
ed for very small values of theparameter
in
P
(
M
)
M
−
1
/
4
e
−
M
3
/
4
For plant communities where the mass range is small (an algal system,for example), the tuning of
in order to ful
ﬁ
ll
M
3
/
4
(
M
)
3
/
4
would likely require a larger effort of numericalexperimentation. Moreover, the method does not determine the range of allowed individual masses, the latterbeing
ﬁ
xed by the experimentalist in context A, and bygenetic factors in context B (cf. Introduction). Nevertheless, as the constraints are so transparently codedinto the resulting probability distribution (cf. eqs (1)and (2)), it is conceivable that through the latter ourprocedure will help identify relevant life history anddemographic factors at work in certain situations.Recently Enquist and Niklas (2001) carried out a vaststatistical analysis of tree abundance as a function of diameter at breast height in various environments, andof abundance and total biomass as functions of speciesdiversity, latitude and elevation in treedominated communities. Although their data are widely spread aboutthe mean, they were able to
ﬁ
t the latter rather wellwith simple power laws. The authors present theirstatistical
ﬁ
ndings as successful tests of theoretical predictions, ultimately based on the same extrapolationfrom the individual relation
Q
M
3
/
4
to the collectivelevel that produced the selfthinning rule (Enquist et al.1998). Our results in this paper do not support such aprocedure, at least in the case of mixedmass communities considered in their analysis. The theoretical understanding of their plots must then be considered an openproblem, regarding a size
–
frequency distribution thatvaries like
D
−
2
with tree diameter
D
, and the relativeindependence of total biomass and tree density withrespect to elevation, latitude and species diversity.Our proposed mass distribution,
P
(
M
)
M
−
1
/
4
e
−
M
3
/
4
incorporates the requirement of maximum productivity (full use of available resources
Q
tot
) This maybe contrasted with the mass distribution in context A,which turns out to be constant and related to a singlevalue of the individual mass (at each stage in thehistory of a stand). From the perspective of the maximum entropy method, which generated the latter distribution without using the maximum productivitycondition, it embodies less information than the continuous one for context B.A certain parallelism suggests itself between ourmathematical scheme and Darwin
’
s principle, withmaximum entropy appearing as a manifestation of theunbiased and random character of the source of biological diversity, and the constraints expressing the effectof the environment. The method may thus be usefulbeyond its original context of the selfthinning rule.From this wider perspective it is the extremal propertiesof the resulting probability distributions that mightfurnish clues about good biological design in particularsituations (Thompson 1942, Rosen 1967, Torres 1991,Enquist et al. 1998, Torres 2001). Although never popular with biologists wary of
‘
just so
’
explanations, whenapplied to processes that involve allometries, the searchfor testable criteria of good design constitutes a routetowards the ideal of macroecology (Brown 1995, Gaston and Blackburn 1999, Lawton 1999), which seeks toidentify a scale where description becomes feasible interms of a small set of global variables that satisfygeneral laws impervious to noise from the myriad contingent factors at work in detailed community studies.
Acknowledgements
–
J.L. Torres gratefully acknowledges support from Universidad Michoacana and CONACYT, Mexico,that made possible a sabbatical leave at Instituto de Ecolog
ı´
aA.C. in Xalapa, Mexico, where most of this work was carriedout. He also thanks J. Brown for suggestions that de
ﬁ
nitelyimproved this paper.
References
Agusti, S., Duarte, C. M. and Kalf, J. 1987. Algal cell size andthe maximum density and biomass of phytoplankton.
–
Limnol. Oceanogr. 32: 983
–
986.
547
OIKOS 95:3 (2001)
Blueweiss, L. H., Kudzma, V., Nakashima, D. et al. 1987.Relationships between body size and some life historyparameters.
–
Oecologia 37: 257
–
272.Brown, J. H. 1995. Macroecology.
–
Univ. of Chicago Press.Buck, B. (ed.) 1991. Maximum entropy in action: a collection of expository essays.
–
Oxford Univ. Press.Buzas, M. A. and Hayek, L. C. 1996. Biodiversity resolution: an integrated approach.
–
Biodiv. Letters 3: 40
–
43.Calder, W. A. 1984. Size, function and life history.
–
Harvard Univ. Press.Damuth, J. 1981. Population density and body size in animals.
–
Nature 290: 699
–
700.Damuth, J. 1998. Common rules for animals and plants.
–
Nature 395: 115
–
116.Dodds, P. S., Rothman, D. H. and Weitz, J. S. 2001. Reexamination of the
‘
3
/
4law
’
of metabolism.
–
J. Theor.Biol. 209: 9
–
27.Enquist, B. J. and Niklas, K. J. 2001. Invariant scaling relations across treedominated communities.
–
Nature 410:655
–
660.Enquist, B. J., Brown, J. H. and West, G. B. 1998. Allometric scaling of plant energetics and population density.
–
Nature 395: 163
–
165.Fleming, T. H. 1973. Number of mammal species in Northand Central American forest communities.
–
Ecology 54:555
–
563.Franco, M. and Kelly, C. K. 1998. The interspeci
ﬁ
c mass
–
density relationship and plant geometry.
–
Proc. Natl.Acad. Sci. USA 95: 7830
–
7835.Gaston, K. J. and Blackburn, T. M. 1999. A critique formacroecology.
–
Oikos 84: 353
–
368.Gorham, E. 1979. Shoot height, weight and standing crop inrelation to density of monospeci
ﬁ
c stands.
–
Nature 279:148
–
150.Harper, J. L. 1977. Population biology of plants.
–
Academic Press (pp. 171
–
194).LaBarbera, M. 1989. Analyzing body size as a factor inecology and evolution.
–
Annu. Rev. Ecol. Syst. 20: 97
–
117.Lawton, J. H. 1999. Are there general laws in ecology?
–
Oikos 84: 177
–
192.Lonsdale, W. M. 1990. The selfselfthinning rule: dead oralive?
–
Ecology 71: 1373
–
1385.Montroll, E. W. and Schlesinger, M. F. 1983. Maximumentropy formalism, fractals, scaling and 1
/
f
noise: a taleof tails.
–
J. Stat. Physics 32: 209
–
230.Norberg, R. A. 1988. Theory of growth geometry of plantsand selfthinning of plant populations: geometric similarity, elastic similarity, and different growth modes of plantparts.
–
Am. Nat. 131: 220
–
256.Rosen, R. 1967. Optimality principles in biology.
–
Butterworths.Thompson, D
’
A. W. 1942. On growth and form.
–
Cambridge Univ. Press.Torres, J.L. 1991. Natural selection and thermodynamic optimality.
–
Il Nuovo Cimento D 13: 177
–
185.Torres, J.L. 2001. Biological power laws and Darwin
’
s principle.
–
J. Theor. Biol. 209: 223
–
232.Weller, D. E. 1989. The interspeci
ﬁ
c size
–
density relationship among stands and its implications for the
−
3
/
2power rule of selfthinning.
–
Am. Nat. 133: 20
–
41.West, G. B., Brown, J. H. and Enquist, B. J. 1997. A general model for the origin of allometric scaling laws inbiology.
–
Science 276: 122
–
126.White, J. 1981. The allometric interpretation of the selfthinning rule.
–
J. Theor. Biol. 89: 475
–
500.
Appendix
We include in the Appendix our main approximationsand other mathematical details.(1) On the relationship between entropy and themultinomial function. Maximizing a positive function isthe same as maximizing its logarithm. Consider
G
({
n
j
})
=
ln[
N
!
/
n
1
!
n
2
!
n
3
!
…
], and use the standard approximation ln[
n
!]
n
ln[
n
]
−
n
(Stirling
’
s approximation, valid for large
n
); this immediately leads to
G
({
n
j
})
−
j
f
j
ln
f
j
, with
f
j
=
n
j
/
N
.(2) From a discrete to a continuous mass distribution(cf. eq. (3)). Consider the normalization condition1
j
e
−
Q
j
=
j
e
−
Q
j
j
j
max
l
e
−
Q
(
j
)
d
j
=
Q
max
Q
min
e
−
Q
d
j
d
Q
d
Q
,where
j
=
1 is the increment in the index
j
and we usedthe RiemannStieltjes de
ﬁ
nition for an integral. Measuring the
ﬂ
ux
Q
in a linear scale d
j
/
d
Q
becomes a constantand we get from the expressions at both ends of this chain1
Q
max
Q
min
e
−
Q
d
Q
=
M
max
M
min
e
−
M
3
/
4
d
Q
d
M
d
M
M
max
M
min
e
−
M
3
/
4
M
−
1
/
4
d
M
,with
a constant, using the allometry
Q
M
3
/
4
.(3) On the approximation
M
3
/
4
M
3
/
4
. Assuming
P
(
M
)
M
−
in the interval 1
M
M
max
and zerooutside it, with 0
1 and
M
max
1, we obtain
M
1
−
2
−
M
max
,and
M
1
−
1
−
+
M
max
.Writing
M
=
M
(
+
)
this implies
=
/
ln
M
, with
=
ln
1
−
1
−
+
2
−
1
−
n
.For
=
3
/
4 and
1
/
4,
=−
0.06 and ln
M
13 for
M
max
10
6
; hence
−
0.005 and the approximation
M
3
/
4
M
3
/
4
is justi
ﬁ
ed.
OIKOS 95:3 (2001)
548