02-Growth - GROWTH IN LENGTH: a model for the growth of an...

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GROWTH IN LENGTH: a model for the growth of an individual Consider the following differential equation: What are the dimensions for parameters J and K? dL dt L' = J K L - = As was the case with the differential equation relating N' with N, once again there is a linear relationship between L' and L. In this new differential equation J is the intercept of the line and K is the slope. The Differential Equation. dL/dt L Solution to the Differential Equation. t J K J 0 0 The graph on the left defines the slope for each point of the curve on the right. Analytical Solution to dL/dt = J - K L As was the case for the differential equation N' t ( ) M - N t ( ) = , we can use an algebraic approach to drive an analytical solution for the differential equation for growth-in-length. L is the dependent variable , t is the independent variable . dL dt K - L J K - = dL L J K - K - dt = Separate the variables by putting L on one side and t on the other. FW431/531 Copyright 2008 by David B. Sampson Growth - Page 11
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Integrate both sides to "undo" the differentiation. Remember the Rule L 1 L J K - d t K - d = F x ( ) a x u f u ( ) d = <==> x F x ( ) d d f x ( ) = If we differentiate F(x), the function that the integral is equal to, we should recover the function f(x) that is under the integral sign. Start with the right hand side: t K - d K - t C + = The derivative of [-K·t + C] is -K dt . Then do the left hand side: dX /(X + a) = ln(X + a) + constant L 1 L J K - d ln L J K - C' + = The derivative of [ln(X+a) + C'] is dX/(X+a) + C' . The handout "Review of Some Mathematics" has tables of integrals that you can use. Wikipedia has extensive lists. Finding integrals can be hard. Now combine the two sides: ln L J K - K - t C'' + = C'' combines the earlier arbitraty constants, C''= C - C' L t ( ) J K C''' exp K - t ( ) + = Exponentiate both sides to get the general solution , C''' = exp(C''). Define a new parameter L inf = J / K. and substitute it for J / K. L t ( ) L inf C''' exp K - t ( ) + = Specify the initial conditions, L=0 at t=t 0 , and solve for the arbitrary constant C'''. L t 0 ( 29 0 = L inf C''' exp K - t 0 ( 29 + = ==> L inf - C''' exp K - t 0 ( 29 = C''' L inf - exp K - t 0 ( 29 = ==> C''' L inf - exp K t 0 ( 29 = The particular solution is L t ( ) L inf L inf exp K t 0 ( 29 exp K - t ( ) - = L t ( ) L inf 1 exp K - t t 0 - ( 29 - = FW431/531 Copyright 2008 by David B. Sampson Growth - Page 12
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This equation in fisheries science is usually known as the von Bertalanffy growth equation and it is widely used as a model for the growth in length of individual fish. However, see Knight (1968) and Roff (1980) on the Supplemental Reading list for dissenting opinions about the utility of the von Bertalanffy equation. Ricker (1979) on the Recommended Reading
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02-Growth - GROWTH IN LENGTH: a model for the growth of an...

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