The best
allometric model
, where the exponent is assumed to be
a
= 3, is
W
= 0
.
0079791
L
3
,
with
J
3
= 3
.
9113
×
10
6
.
Clearly, the second model is the best fitting model from a unbiased perspective having the least
square errors. However, the last model is not that far away and makes more sense from a physical
argument.
The script also produced the graph below, which shows that all of the three models are fairly
similar. This gives visual evidence that using the third model in the next section is reasonable.
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0
10
20
30
40
50
60
70
80
90
100
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Length (cm)
Weight (g)
Allometric Models for Lake Trout
Allometric
Nonlinear Fit
Cubic Fit
Data
Accumulation of Mercury with Age
The model for accumulation of Mercury as the fish ages is discussed in the lecture notes:
Linear
Differential Equations
. It begins with a best fitting model for the weight vs age data. Below are
two ways to create the best fitting cubic model with the weight vs age data. The first technique uses
the cubic model described above with the von Bertalanffy model. A one parameter search is applied
to the cube of the von Bertalanffy model, so find the best
α
1
for
W
(
t
) =
α
1
(
92
.
401
(
1

e

0
.
14553
t
))
3
.
When this approach is taken with
fminsearch
, the result is
α
1
= 0
.
0079798, so the weight of a fish
satisfies the following composite function:
W
(
t
) = 6295
.
4
1

e

0
.
14553
t
3
,
where
t
is the age of the fish, giving a sum of square errors of
J
= 1
.
3168
×
10
7
. A second method
is to parallel the fitting of the length vs age for the von Bertalanffy equation with two parameters,
W
(
t
) =
α
2
1

e

β
2
3
. With a nonlinear least squares fit of this model to the weight vs age data
(almost identical to
sumsq
vonBert
Mscript, so omitted), the composite function for weight of a
fish satisfies:
W
(
t
) = 5677
.
67
1

e

0
.
16960
t
3
,
with a sum of square errors of
J
= 1
.
2049
×
10
7
. These two composite functions are graphed to
show this fit of the weight of the fish as it ages.
(The graphing script is similar to the one for
the von Bertalanffy model, so is omitted here.)
The graphs are very similar, and there is not a
significant difference in the sum of square errors. The calculations will use the model generalized
model derived from the von Bertalanffy equation
W
(
t
) =
W
*
1

e

bt
3
,
with
W
*
= 6295
.
4 or 5677.67 and
b
= 0
.
14553 or 0.16960, corresponding to the von Bertalanffy or
nonlinear least squares fits, respectively.
0
2
4
6
8
10
12
14
16
18
20
0
1000
2000
3000
4000
5000
6000
Age (Years)
Weight (g)
Weight of Lake Trout
Cubic von Bertalanffy
Nonlinear Fit
Data
The lecture notes discuss how mercury (Hg) accumulates in the body from feeding and water
passing over the gills. Since fish are cold blooded animals, their energy expenditure (balanced by
food and O
2
intake) should be roughly proportional to the weight of the fish. (Alternate models
might consider
Kleiber’s law
or more mercury laden food sources with aging.) The lecture notes
point out that Hg doesn’t tend to leave the body after entering.
This suggests the rate of Hg
entering the body of a fish should be proportional to the weight of the fish, giving the differential
equation:
dH
dt
=
κW
(
t
) =
κW
*
1

e

bt
3
,
where
H
(
t
) is the amount of Hg in the fish. This differential equation is readily solved with details
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 Fall '08
 staff