The best allometric model where the exponent is assumed to be a 3 is W 0

The best allometric model where the exponent is

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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 M-script, 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.
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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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