3. ENERGY REQUIREMENTS OF CAPTIVE WILD ANIMALS S31
deposited, and these vary with stage of growth and
between species. To assist in understanding the way
in which these variables influence energy requirements
during growth, and thus in predicting energy require
ments, it is helpful to develop a model. The model
described here is similar to that developed previously
(23)but the impact of different scaling procedures and
of variation in the energy density of tissue deposited
are also examined in this case.
The basic assumptions and development of the
model are as follows:
Assumption 1. That the pattern of growth in body
mass follows a Gompertz curve
M = Mae-e""'"'' (1)
where M = mass (kg),Ma = adult mass (kg),B - growth
rate constant (1/d), t = age (d) and t' = age at the in
flexion point of the sigino id curve (d).
Various mathematical functions approximate the
typically sigmoid pattern of weight gain of vertebrates.
In addition to the Gompertz equation shown above,
the logistic and Von Bertalanffy equations have often
been used to summarize data and as models (24-26).
These curves differ in the position of their inflexion
points. The inflexion of the logistic curve occurs at
0.5 of the asymptote, that of the Gompertz at 0.37
and that of the Von Bertalanffy at 0.3 of the asymptote.
Assumption 2. That between species the growth
rate constant (B)is inversely related to adult body mass
raised to an exponent of ~0.25:
= xM -0.25
(2)
Comparing time taken to grow among domesticated
mammals, using data from Brody (27), Taylor (28)
found that it increased with adult body mass raised to
the 0.27 power between species. Ricklefs (24) found
the same exponent when investigating the relationship
between time taken to grow and adult mass in birds.
Since then others, for example, Case (29) and Calder
(30)have presented further evidence that time to grow
tends to increase with about the quarter power of adult
mass (and thus that the growth rate constant B is in
versely related to M0025as shown in Eq. 2). However,
the value found for the exponent in allometric rela
tionships is dependent on the species included within
the sample and on the type of regression analysis used
(31),and other values have been reported. For example,
Zullinger et al. (26) derived an exponent of 0.30 in
their analysis of growth rates among mammals and
Ricklefs (32) derived an exponent of 0.34 in his anal
ysis of growth rates in birds.
Kirkwood (23) proposed using the time taken to
grow from 25 to 75% of adult weight (125-75d) as an
index of time taken to grow when undertaking com
parisons between species. The times at which these
proportions of adult size are reached can be quite ac
curately pinpointed because growth rate is relatively
high at these stages of growth, so tis_75can be easily
measured from growth curves.
If Õ25-75is dependent on Ma°25between species, then
comparisons of time taken to grow can be made be
tween species of different adult masses by comparing
values of tïs_75/Ma°-2S.Kirkwood and Webster (33) la
beled this quantity 0. It is a body-mass-independent
index of time taken to mature. Values of 6 are shown
for a variety of species in Table 3. These show that
considerable variation in time taken to grow remains
after taking body mass into account. Birds generally
take a shorter time to grow than mammals, and mam
mals generally take a shorter time to grow than rep
tiles. Marsupials tend to grow more slowly than av
erage for eutherian mammals, but there is wide vari
ation among eutherians. The Old World monkeys and
apes (especially humans) have very prolonged growth
periods.
Assumption 3. That the energy cost of each unit
weight gain is 8400 kj/kg throughout growth. In prac
tice, the energy cost of weight gain cannot be much
less than this but is often considerably more.
Taking Assumption 1 it follows that growth rate
(GR, kg/d) is described by the differential of the Gom
pertz equation:
TABLE 3
Values for 6(abodymass-independent index of time taken to
grow) for a variety of species of vertebrates
Species B Source
ReptilesMediterranean
tortoiseRussell's
viperBirdsFairy
penguinSalvin's
prionDouble-crested
cormorantAndean
condorKestrelDomestic
broilerfowlJapanese
quailNicobar
pigeonScarlet
macawAfrican
greyparrotScarlet
cock-of-thè-rockStarlingMammalsQueensland
koalaMasked
shrewRufous
elephantshrewLesser
mouselemurCommon
marmosetRhesus
macaqueHumanGuinea
pigMongolian
gerbilCatDog
(Labradorbitch)Giant
pandaFriesian
cow211064018261029113525211831211118528674823461912609395105408
34Ref.
35Ref.
36Ref.
37Ref.
38Ref.
39Ref.
40Ref.
41Ref.
42Ref.
43Ref.
44Ref.
44Ref.
45Ref.
46Ref.
47Ref.
48Ref.
49Ref.
50Ref.
51Ref.
52Ref.
53Ref.
54Ref.
55Ref.
56Ref.
57Ref.
58Ref.
59
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4. S32 KIRKWOOD
GR = -BM In (u) (3)
where stage of maturity, u = M/Ma.
It is a property of the Gompertz curve that
B = 1.57/125-75 (4)
and therefore:
GR = (-1.57/t25.75)Mln(u) (5)
We have defined 0 as t2S>,75/Maoli,and it follows that:
GR = (-1.57/0M025u°25)MIn (u) (6)
this simplifies to:
GR = (-1.57/0)M°75u025In (u) (7)
and therefore the growth rate per metabolic mass (GR/
M075) is related to time taken to grow and stage of
maturity as follows:
GR/M°7S= (-1.57/0)u°2SIn (u) (8)
The form of this expression, relating growth rate per
metabolic mass to stage of growth as 0varies, is shown
in Figure 1A. It is dependent on the value of the ex
ponent, and if Õ25-75is scaled by dividing by M°33(in
stead of M°25)the equation becomes:
GR/M0«7= (-1.57/0)u033 In (u) (9)
This equation is illustrated in Figure IB. If the model
is based on the logistic curve instead of the Gompertz
curve, the equations corresponding to equations 8 and
9 are:
GR/M075 = (2.2/0)u°25(l -u) (10)
and
GR/M067 = (2.2/0)u033 (1 - u) (11)
These equations are illustrated, solved for 0 values of
20 and 100, in Figures 1C, D.
The energy required to sustain the rate of tissue
deposition (Pg, kj/d per metabolic mass) can be esti
mated as:
Pg/M075 = -1.57c/0)u°25In (u) (12)
where c is the energy cost per unit mass gain.
The total energy requirement can then be estimated
by adding an approximation of the maintenance com
ponent of the energy budget. This can be based on the
maintenance requirements of adults or, if no specific
information is available, on allometric equations such
as those in Table 1.
The pattern of energy requirement for growth that
this model predicts in relation to stage of growth and
time taken to grow is shown in Figure IE. In Figure
IE, equation 12 is solved for when 0 = 20 and c = 8400
kj/kg. This model and the variants shown in Figure 1
indicate that growth rate and energy requirements rel
ative to metabolic mass are likely to be highest in the
early stages of growth and to decline as adult mass is
10O
40
0-2 0-4 0-6 0-8 1-0
80
60
40
30
0-2 0-4 0-6 0-8 1-0
d
0-2 0-4 0-6 0-8 1-0 0-2 0-4 0-6 0-8 1-0
f
= 0-2 0«0-6 0-8 1-0 O-2 0-4 0-6 0-8 1-0
Stage of maturity
FIGURE 1 Predicted growth rates and energy require
ments of homeothermic animals in relation to stage of ma
turity (u, proportion of adult mass attained) and a body-
mass-independent index of time taken to grow (6, see text).
The upper and lower lines are solutions to the model (Eq. 8)
when 8 values are 20 and 100, respectively. (A) Predicted
growth rate per metabolic mass when growth follows a
Gompertz curve and when 8 is 20 (upper line) or 100 (lower
line). (B) As for 1A but assuming growth follows a logistic
curve. (C) As for 1A but assuming growth rate scales with
M°67(rather than M°75| and that time taken to grow scales
with M033 (rather than M°"). (D) As for 1C but assuming
growth follows a logistic curve. (£) Predicted minimum en
ergy requirement of homeothermic animals in relation to
stage of maturity and time taken to grow. In this case 6 is
20. The solid line is derived assuming the energy cost of
tissue deposition is constant throughout growth at 8400 kj/
kg, the broken line is derived assuming that the energy cost
of tissue deposition increases with stage of maturity at (8400
+ 8400 u) kj/kg. The maintenance requirement is assumed
to be 500 kj/d per kg075 throughout growth. (F) As for IE
but assuming growth follows a logistic curve.
approached. They also illustrate the impact of varia
tion in 0 on growth rate and energy requirements in
relation to metabolic mass during the growth period.
The energy requirements that the models predict can
be considered as minimum estimates.
In reality the situation is complicated by several
factors and I will mention some here. First, growth
does not exactly follow Gompertz or logistic curves.
However, these often provide remarkably good fits and
differences are unlikely to change the shape of the
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