Scaling Relationships in Biology
(including Community Ecology)
Big fleas have little fleas
on their back to bite them,
and little fleas have lesser fleas
and so ad infinitum.
Swift 1733 (?)
Photo of fiddler crabs from Gilbert (2000) Developmental Biology, 6th ed.; other photos from Wikipedia
Scaling Relationships
Organisms range over 21 orders of magnitude in body size!
Statistic from West et al. (1997); Fig. from Bonner (1988)
Scaling Relationships
Biologically relevant processes operate over an enormous
range of spatial & temporal scales
Figure from Levin (1992)
Scaling Relationships
For example… gas exchange through individual stomata & global warming
represent phenomena that occur at vastly different scales of space & time
Processes are naturally linked across scales, so how can we extrapolate
from one scale to another (e.g., leaf  forest  globe)?
What are the mechanistic links among patterns and processes across scales?
Photos from Wikipedia
Scaling Relationships
A good starting point is to identify scaling relationships
Scaling often assesses how attributes change
with changes in a fundamental dimension
(e.g., length, mass, time)
The attributes of the organism, community or ecosystem are generally the
dependent variables (Y), whereas the fundamental dimension
is the independent variable (X)
Scaling Relationships
Many scaling relationships can be expressed as power laws:
Y = c Xs
X is the independent variable – measured in units of a
fundamental dimension; c is a constant of proportionality;
and s is the exponent (or “power” of the function)
The relationship is a straight line on a log-log plot:
Log10(Y) = Log10(c) + s  Log10(X)
…and by rearranging, this is the form of the familiar equation for a straight line:
y = mx + b
Scaling Relationships
Consider the scaling of squares & cubes as functions of the
length of a side (the fundamental dimension)
Area = Length2
Area  Length2
Surface area = 6 * Length2
Surface area  Length2
Volume = Length3
Volume  Length3
Scaling Relationships
120
2
Y = X2 y = x
100
(accelerating
function)
Area
80
60
40
20
0
0
2
4
6
Length
8
10
12
Scaling Relationships
120
2
Y = X2 y = x
100
(accelerating
function)
Area
80
60
40
20
0
0
2
4
6
Length
8
10
12
Scaling Relationships
120
2
Y = X2 y = x
100
(accelerating
function)
Area
80
60
40
20
0
0
2
4
6
Length
8
10
12
Scaling Relationships
3.5
120
2
Y = X2 y = x
(accelerating
function)
Area
80
Y = 2X
3
Log10(Area)
100
60
40
20
2.5
y = 2x
2
1.5
1
0.5
0
0
0
2
4
6
8
10
0
12
Length
0.2
0.4
0.6
0.8
Log10(Length)
Etc…
1
1.2
Scaling Relationships
700
Log10(Surface Area)
3
2
Y = 6 * yX=26x
Surface area
600
500
(accelerating
function)
400
300
200
100
= 2x + 0.7782
Y = y2X
+ 0.778
2.5
2
1.5
1
0.5
0
0
0
2
4
6
8
10
0
12
Length
0.2
0.4
0.6
0.8
Log10(Length)
Etc…
1
1.2
Scaling Relationships
3.5
1200
3
Y = X3 y = x
(accelerating
function)
800
600
400
2.5
2
1.5
1
200
0.5
0
0
0
2
4
Y = 3Xy = 3x
3
Log10(Volume)
Volume
1000
6
8
10
0
12
Length
0.2
0.4
0.6
0.8
Log10(Length)
Etc…
1
1.2
Scaling Relationships
Consider the ways in which surface area & volume
of a sphere scale with its radius
Surface area = 4  r2
Surface area  r2
Volume = 4/3  r3
Volume  r3
Scaling Relationships
Surface-to-volume ratio:
Surface area  r2
Volume  r3


Surface area1/2  r
Volume1/3  r
Surface area1/2  Volume1/3

Surface area  Volume2/3
Scaling Relationships
Slope = 1
1400
Log10(Surface area)
Y=4.83 * X
1200
Surface area
3.5
y = 4.8352x0.6667
0.667
1000
800
600
(decelerating
function)
400
200
y = 0.6667x* +X0.6844
Y=0.667
+ 0.68
3
2.5
2
1.5
1
0.5
0
0
0
1000
2000
3000
4000
0
5000
Volume
1
2
3
Log10(Volume)
Etc…
Volume increases
proportionately faster
than surface area
4
Scaling Relationships
Slope = 1
1400
Log10(Surface area)
Y=4.83 * X
1200
Surface area
3.5
y = 4.8352x0.6667
0.667
1000
800
600
(decelerating
function)
400
200
y = 0.6667x* +X0.6844
Y=0.667
+ 0.68
3
2.5
2
1.5
1
0.5
0
0
0
1000
2000
3000
4000
0
5000
Volume
1
2
3
Log10(Volume)
Etc…
This simple fact has
myriad important
implications for biology
(e.g., Bergmann’s “rule”?)
4
Scaling Relationships
Slope = 1
1400
Log10(Surface area)
Y=4.83 * X
1200
Surface area
3.5
y = 4.8352x0.6667
0.667
1000
800
600
(decelerating
function)
400
200
y = 0.6667x* +X0.6844
Y=0.667
+ 0.68
3
2.5
2
1.5
1
0.5
0
0
0
1000
2000
3000
4000
0
5000
Volume
1
2
3
Log10(Volume)
Etc…
For example, endoparasite S
should increase more rapidly
than ectoparasite S as host
body size increases
4
Surface area / Volume
Scaling Relationships
3.5
-1 -1
Y =y =3 3x
*X
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
Radius
Etc…
As you could infer from the
earlier figures, the surface
area to volume ratio changes
with the radius of the sphere
3.5
Log10(Surface area / Volume)
Surface area / Volume
Scaling Relationships
-1 -1
Y =y =3 3x
*X
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
0.6
Y= -1
0.48
y =*-xX++0.4771
0.4
0.2
0
-0.2
-0.4
-0.6
0
12
Radius
0.2
0.4
0.6
0.8
1
1.2
Log10(Radius)
Etc…
…and the rate of change of
the ratio is constant in log-log
plotting space
Scaling Relationships
Allometry – Coined by Julian Huxley (1932) for the study of size
& its relationship to characteristics within individuals
(due to ontogenetic changes) & among organisms
(due to size-related differences in shape, metabolism, etc.)
For example, size is related
allometrically to basal
metabolic rate in birds
& mammals:
B  M3/4
The red line’s slope = 1
Scaling Relationships in Community Ecology
Metabolic Ecology Theory
Geoff West, James Brown & Brian
Enquist proposed that many
allometric relationships in biology
are governed by the physical
properties of branching distribution
networks (e.g., blood vessels,
xylem & phloem)
Figure from West et al. (1997)
Scaling Relationships
Allometric relationship: Height vs. diameter in trees
The critical buckling height
for cylinders is:
Hcritical = k * (E/)1/3 * D2/3
Therefore, if trees maintain
“elastic similarity”:
H  D2/3
Giant
sequoia
See Greenhill (1881); Figure from McMahon (1975)
Douglas Ponderosa
fir
pine
Scaling Relationships
Allometric relationship: Height vs. diameter in trees
If trees maintain
“elastic similarity”:
H  D2/3
Both lines have slopes = 2/3;
the broken line is 1/4 the
magnitude of the complete line
Trees avoid buckling under their
own weight, with a 4x
safety factor
See Greenhill (1881); Figure from McMahon (1975)
Dataset for U.S. record trees.
Scaling Relationships in Community Ecology
Species-area relationships
The Arrhenius equation describes a
power-law scaling relationship:
S = cAz
log (S) = log (c) + z * log (A)
Figure from Rosenzweig (1995)
Scaling Relationships in Community Ecology
Mainland vs. island
size relationships (Foster’s
“rule”)
a. Insular races of mammals
compared to their nearest mainland
relatives; the scaling relationship
suggests an “optimum size of
100 g”
Figure from Brown’s Macroecology (1995)
Scaling Relationships in Community Ecology
Mainland vs. island
size relationships (Foster’s
“rule”)
a. Insular races of mammals
compared to their nearest mainland
relatives; the scaling relationship
suggests an “optimum size of
100 g”
b. The largest (solid circles) &
smallest (open circles) mammals of a
landmass as a function of area; as
area – and thus the number of
species – decreases, the sizes of the
mammals converge on 100 g
Figure from Brown’s Macroecology (1995)
Scaling Relationships in Community Ecology
Size vs. density in plant communities
The relationship between size
& number for plants grown in
monoculture gave rise to the
empirical “self-thinning rule”,
i.e., the mean size of
individuals in the stand is
proportional to their density
raised to the -3/2 power
Self-thinning rule:
m = k N -3/2
Plantago asiatica
Figure from Yoda et al. (1963)
Scaling Relationships in Community Ecology
Size vs. density in plant communities
The similarity to geometric
constraints suggested this
possibility:
Self-thinning rule:
m = k N -3/2
Area  Volume2/3
Area  Density-1
Volume  Mass
Density-1  Mass2/3
Mass  Density-3/2
Plantago asiatica
Figure from Yoda et al. (1963)
Scaling Relationships in Community Ecology
Size vs. density in plant communities
Enquist & colleagues have
challenged the traditional -3/2
thinning rule by re-examining sizedensity relationships & by providing
a new, mechanistic way to
approach the problem
Enquist et al.’s prediction:
m = k N -4/3
Figure from Enquist et al. (1998)
Scaling Relationships in Community Ecology
Size vs. density in plant communities
Enquist & Niklas (2001)
suggested Metabolic Ecology
Theory as the explanation for
the apparent consistency of
size-density relationships
across forests
Their prediction
translates into:
N = k DBH -2
Figure from Enquist & Niklas (2001)
Scaling Relationships in Community Ecology
Empirical curves do not exactly
match predictions (slope =
-2; shaded region) from
Metabolic Ecology Theory
(Enquist & Niklas
2001 )
6.00
1000000
100000
5.00
10000
4.00
BCI - Panama
EDO - Dem. Rep. Congo
3.00
1000
N / km2
HKK - Thailand
KRP - Cameroon
2.00
100
LAM - Malaysia
LAP - Colombia
10
1.00
LEN - Dem. Rep. Congo
MUD - India
0.00
1
PAL - Philippines
PAS - Malaysia
SIN - Sri Lanka
0
-1.00
0.00
1
YAS - Ecuador
0.50
1.00
10
1.50
2.00
100
2.50
Diameter (mm)
Figure redrawn from Muller-Landau, Condit, Harms et al. (2006) Ecology Letters
3.00
1000
3.50
Scaling Relationships in Community Ecology
Species-genus & species-family ratios
Enquist et al. (2001) suggested
three hypotheses for the relationship
between species richness and
number of higher taxa within a local
community
a. A positive relationship with a shallow
slope; as species are added they come
from an increasingly limited subset of
higher taxa
b. A slope of unity represents the upper
constraint boundary; addition of new
species occurs only upon addition of
higher taxa
Figure from Enquist et al. (2002)
c. Communities could be scattered
within the shaded region below the
constraint line, such that the variance in
abundance of higher taxa increases with
S; higher taxa abundance would be
effectively unpredictable from S
Scaling Relationships in Community Ecology
Species-genus & species-family ratios
Enquist et al. (2001) found
surprising similarity among
tropical forests worldwide
Figure from Enquist et al. (2002)
Scaling Relationships – Fractals
Fractal models describe the geometry of a wide variety of natural objects
E.g., the branching distribution networks of organisms…
Within an object, as a fundamental dimension changes, fractal
properties of the object obey scaling (power function) relationships
Figure from West et al. (1997)
Scaling Relationships – Fractals
Fractal objects may also exhibit the property of self similarity
(self-similar objects maintain characteristic properties over all scales)
The Sierpinski Triangle
If you are curious, visit: http://www.arcytech.org/java/fractals/sierpinski.shtml
Scaling Relationships – Fractals
“In the natural world, there is no guarantee that… elegant selfsimilar properties will apply” Sugihara & May (1990)
Even so, fractal properties (and self-similarity over finite
scales) appear throughout the natural world:
Barnsley’s fractal fern
Scaling Relationships – Fractals
“In the natural world, there is no guarantee that… elegant selfsimilar properties will apply” Sugihara & May (1990)
Even so, fractal properties (and self-similarity over finite
scales) appear throughout the natural world:
Clematis fremontii –
Fremont’s leather flower;
endemic to KS, NE, MO
(Original from Erickson 1945)
Figure from Brown’s Macroecology (1995)
Scaling Relationships – Fractals
It has become customary to introduce fractals with reference to measuring
the coast of Britain (e.g., Mandelbrot 1983) , a project that first suggested
the intriguing fact that as the scale of the ruler decreases, the length of the
coast increases:
L = K δ1-D
L = Total length
δ = Length of the ruler
D = Fractal dimension
Figure from Sugihara & May (1990)
Self-similarity characterizes the
object of interest if D is constant
over all scales (δ), i.e., if the power
term of the function is constant
Scaling Relationships – Fractals
The fractal dimension (D) can be thought of as the
“crookedness,” “tortuosity” or “complexity” of the object:
D=1
D = 1.26
D = 1.5
(d)
D=2
Figure from Sugihara & May (1990)
Scaling Relationships – Fractals
In practice there are many ways to estimate D, and to use D in
community ecology (see Sugihara & May 1990).
Morse et al. (1985) used the boundary-grid method to show that the areas of
leaf surfaces in nature display fractal properties, and that D changes with δ
D ≈ 1.5 for the boundaries of
vegetation surfaces
Morse et al. (1985) show that
this means that for an order
of magnitude decrease in
body length, there is 3.16
more area to occupy!
Figure from Morse et al. (1985, Nature)
Scaling Relationships – Fractals
Morse et al. (1985), used
their analysis to suggest an
explanation for the
observation that as the body
sizes of arthropods increase,
their numbers (densities)
decrease more rapidly than
expected if available area
remained constant
Figure from Morse et al. (1985, Nature)