5 10 15 t 06 08 1 12 14 16 18 2 Length m von Bertalanffy Model of

# 5 10 15 t 06 08 1 12 14 16 18 2 length m von

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5 10 15 t (yrs) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Length (m) von Bertalanffy Model of Marlin The graph shows that the rate of growth of the fish is very fast in the early years and then slows down, approaching an asymptote, as the fish ages. The model quite clearly matches the data very well. The graph shows that the maximum length the fish is the asymptotic limit, which is about 1.89 m. c. The Matlab script in Part a shows how to obtain the allometric model by finding the linear least squares best fit to the logarithms of the data.

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d. The graph of the best fitting allometric model for the Striped Marlin with the data set is seen below, using the model: W ( L ) = 8 . 2861 L 3 . 6166 . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Length (m) 0 10 20 30 40 50 60 70 80 90 100 Weight (kg) Allometric Model of Marlin The graph and the model show that the allometric model roughly follows a cubic relationship between the length and the weight, which is expected based on dimensional analysis (between length and volume). The model quite clearly matches the data very well with a little more variation than seen in the previous graph. With the asymptotic limit of the length of the Striped Marlin from Part a, this model would indicate that the Striped Marlin has a limit in weight of approximately 82.8 kg. e. The composite function simply combines the von Bertalanffy model with the allometric model, yielding: W ( t ) = a ( L (1 - e - bt ) ) k . f. The functions above are combined in a composite function to give W ( t ) for the Striped Marlin, W ( t ) = 82 . 7983 1 - e - 0 . 5764 t 3 . 6166 . The growth function satisfies the derivative of W ( t ) with W 0 ( t ) = 172 . 6020184 1 - e - 0 . 5764 t 2 . 6166 e - 0 . 5764 t . 0 5 10 15 Age (yr) 0 10 20 30 40 50 60 70 80 90 100 Weight (kg) Weight of Marlin 0 5 10 15 Age (yr) 0 5 10 15 20 25 Growth Rate (kg/yr) Growth of Marlin W ( t ) W 0 ( t )
The graph of W ( t ) shows the increase in weight as the fish ages with the increase accelerating for the first 2-3 years then slowing down as it approaches a maximum weight for large time. The point of inflection for the first graph matches the maximum of the growth curve on the right. The growth curve shows the increasing growth rate until approximately the age of 2.25 before growth slows to almost zero for older Marlin. This maximum growth shows the 2 year old Marlin putting on almost 20 kg/yr. 19. The graphs and discussions for this problem are found with the written solutions to this problem. Here the techniques for solving the three different differential equations are shown. a. We consider the linear DE with initial condition (IVP): dA dt + kA = be - qt , A (0) = 0 and k 6 = q. It is clear that the integrating factor is μ ( t ) = e kt , so d dt e kt A = be ( k - q ) t , so e kt A ( t ) = be ( k - q ) t k - q + C. This is readily solved to give: A ( t ) = b k - q e - qt + Ce - kt . The initial condition gives C = - b k - q , so A ( t ) = b k - q e - qt - e - kt .

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