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dW
b
W
b
=
dW
a
W
a
,
β
(1.45)
indicating that the percentage change in the body weight, its growth, is directly
proportional to the percentage change in the weight of the organ. The constant of propor-
tionality is dimensionless and indicates how the relative growth of the entire organism
is related to the relative growth of the part. This is the scaling index sought in a wide
variety of empirical studies.
Perhaps the most famous allometric relation does not concern the relative growth
within a given animal, but concerns a property interrelating a variety of species. This
is the allometric relation between the metabolic rate and the body mass of multiple
mammalian species. The metabolic rate
R
refers to the total utilization of chemical
energy for the generation of heat by the body of an animal. In Figure
1.10
the “mouse-
to-elephant” curve depicts the metabolic rate for mammals and birds plotted versus body
weight on log-log graph paper. The straight-line segment on this graph indicates that
the metabolic rate is of the power-law form
W
β
,
R
=
α
(1.46)
where again
are empirical constants. The value of the dimensionless power-law
index in (
1.46
) is given by the slope of the curve in Figure
1.10
to be
α
and
β
75; however,
the exact value of the power-law index is still the subject of some controversy.
β
≈
0
.
Elephant
1000
Bull
Horse
Boar
Cow and steer
Man
Sow
100
Chimpanzee
Dog
Goose
Woman
Sheep
Goat
Cassowary
Wild birds
Condor
10
Cock
Macaque
Cat
Hen
Rabbit
Guinea pig
Marmot
Giant rats
Small birds
Rat
1
Pigeon and dove
Mouse
0.1
0.01
0.1
1
10
Body mass (kg)
100
1000
10,000
The mouse-to-elephant curve. Metabolic rates of mammals and birds are plotted versus the body
weight (mass) on log-log graph paper. The solid-line segment is the best linear regression to the
data from Schmidt-Neilson [
27
]. Reproduced with permission.
Figure 1.10.
