exponent a and coefficient k e b Joseph M Mahaffy h jmahaffysdsuedu i Lecture

Exponent a and coefficient k e b joseph m mahaffy h

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exponent, a , and coefficient, k = e b Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (51/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Allometric Models 4 Define a MatLab function for the sum of square errors between the logarithm of the length ( ltdfish ) and weight ( wtdfish ) data and the logarithmic model ln( W ) = a ln( L ) + ln( k ) 1 function [ k , a ] = powerfit ( ldata , wdata ) 2 % Power law f i t f o r model W = k * Lˆa 3 % Uses l i n e a r l e a s t squares f i t to logarithms of data 4 Y = log ( wdata ) ; % Logarithm of W - data 5 X = log ( ldata ) ; % Logarithm of L - data 6 p = p o l y f i t (X,Y, 1 ) ; % Linear f i t to X and Y with p = [ slope , i n t e r c e p t ] 7 a = p (1) ; % Value of exponent 8 k = exp (p (2) ) ; % Value of leading c o e f f i c i e n t 9 end Apply this MatLab function to obtain k = 0 . 015049 and a = 2 . 8591, giving a best allometric model W ( L ) = 0 . 015049 L 2 . 8591 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (52/64)
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Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Allometric Models 5 Nonlinear Least Squares Fit: W ( L ) = kL a This uses the nonlinear best fit to the length and weight data using a MatLab program almost identical to the one used for the time and length data for the von Bertalanffy model Create a sum of square errors function and use MatLab’s fminsearch function Produces best fitting model with smallest sum of square error J 2 = 2 . 8683 × 10 6 given by W ( L ) = 0 . 068695 L 2 . 5052 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (53/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Allometric Models 6 Dimensional Analysis for W ( L ) = kL a Two previous models indicate a 3 Similarity argument Lake Trout look similar at most ages Increasing length scales the width and height similarly or V L 3 Since weight is proportional to volume, W L 3 Create MatLab program to find best k to the model W ( L ) = kL 3 Program finds best fitting model as W ( L ) = 0 . 00799791 L 3 Joseph M. Mahaffy, h jmahaff[email protected] i Lecture Notes – Linear Differential Equations — (54/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Graph of Allometric Models Graph of Allometric Models: Shows data and 3 Allometric Models, which are all very close to each other (similar least sum of square errors) 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 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (55/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Mercury Accumulation in Lake Trout 1 Mercury Accumulation in Lake Trout Mercury (Hg) accumulates in fish from feeding and water passing over the gills
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