Environmental Engineering Reference
In-Depth Information
Box 2.16
Defining growth: the von Bertalanffy growth function.
The von Bertalanffy growth function (von Bertalanffy 1957) is the most widely
used model of physical growth. This relates linear measures of size (such as length
or height),
s
, to age,
a
, by:
e
g
(
a
0
-
a
)
s
s
max
1
where
s
max
is the asymptotic maximum size,
g
is the rate of growth towards maxi-
mum, and
a
0
shifts the curve along the age axis (so allowing size at age 0 to be
greater than zero). If growth is measured by weight,
w
, a modification is required
to take account of the scaling of weight with size (usually a cubic power function).
Allowing the weight-size scaling exponent,
b
, to vary, a weight-based growth
curve is given by:
e
g
(
a
0
-
a
)
b
w
w
max
1
A non-linear procedure, such as the
nls
(non-linear least squares) function in R soft-
ware, can be used to fit this model to age and size or weight data for a cross section
of individuals. However, it is important to have some observations near the largest
and smallest possible sizes if these model parameters are to be estimated precisely.
The addax
Addax nasomaculatus
is a Saharan antelope critically endangered by
hunting, with probably fewer than 200 now remaining in the wild (Wacher,
et al
.
2005; Figure 2.15). However, since the mid-1980s, a herd of 50 addax has been
held in a 10 km
2
enclosure in Bou-Hedma National Park, Tunisia, with a view to
eventually re-establishing a wild population in the northern part of the historic
range. As part of an assessment of the demographic performance of this herd, its
growth curve was compared with that of a fully captive herd in St Louis Zoo,
USA. From the Tunisian herd, 19 addax were captured, aged from tooth eruption
and wear patterns, and weighed, while ages and weights were available for 11 ani-
mals in the zoo population. Using a fixed weight exponent (
b
) of 2.6 (estimated
from weight and chest girth data), models were fitted to the data for each popula-
tion separately, and for all animals combined, yielding the following estimates
(standard errors in brackets):
w
max
g
a
0
Tunisia
116.9 (6.7)
0.75 (0.17)
0.82 (0.43)
St Louis
108.3 (4.5)
2.22 (3.58)
0.19 (1.34)
Combined
111.2 (3.9)
0.85 (0.18)
0.8 (0.41)
Fitting separate curves for each population did not provide a significantly better
fit (
F
3, 24
0.19), suggesting identical growth curves. However, lack
data for very young animals from the zoo population meant that the parameters
governing initial growth (
g
and
a
0
) were very poorly estimated for this popula-
tion, giving little power to detect differences.
1.74,
p
