intake should be roughly proportional to the weight of the fish Mostly Hg stays

Intake should be roughly proportional to the weight

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intake) should be roughly proportional to the weight of the fish Mostly, Hg stays in the body once it enters The rate of Hg entering the body of a fish should be proportional to the weight of the fish Resulting differential equation: dH dt = κW ( t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (56/64)
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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 Mercury Accumulation in Lake Trout 2 Mercury Accumulation Model: We select the cubic weight model as the simplest of similar models dH dt = κW * ( 1 - e - bt ) 3 This is a time-varying only DE Solve by integration or H ( t ) = κW * Z ( 1 - e - bt ) 3 dt with H (0) = 0 Integrate by expanding cubic expression H ( t ) = κW * Z ( 1 - 3 e - bt + 3 e - 2 bt - e - 3 bt ) dt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (57/64)
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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 Mercury Accumulation in Lake Trout 4 Integrating the DE: H ( t ) = κW * Z ( 1 - 3 e - bt + 3 e - 2 bt - e - 3 bt ) dt Gives H ( t ) = κW * t + 3 e - bt b - 3 e - 2 bt 2 b + e - 3 bt 3 b + C The initial condition H (0) = 0 gives C = κW * - 3 b + 3 2 b - 1 3 b = - 11 κW * 6 b The solution for Hg accumulation is H ( t ) = κW * 6 b ( 6 bt + 18 e - bt - 9 e - 2 bt + 2 e - 3 bt - 11 ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (58/64)
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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 Mercury Accumulation in Lake Trout 5 Lake Trout Data for Hg concentration vs Age (Lake Superior, 1997) age Hg age Hg age Hg age Hg (yr) (ppm) (yr) (ppm) (yr) (ppm) (yr) (ppm) 6 0.17 8 0.2 9 0.15 13 0.53 7 0.17 8 0.14 9 0.4 14 0.39 7 0.18 8 0.2 10 0.34 15 0.33 7 0.1 9 0.13 11 0.39 18 0.52 8 0.19 9 0.46 13 0.39 The solution of the DE, H ( t ), gives the total amount of Hg in Lake Trout Find the concentration of Hg in Lake Trout , which satisfies c ( t ) = H ( t ) W ( t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (59/64)
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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 Mercury Accumulation in Lake Trout 6 Concentration of Hg in Lake Trout requires Weight vs Age For integration we assume a weight model of the form W ( t ) = W * ( 1 - e - bt ) 3 The 2 parameters W * and b are fit to time and weight data on Lake Trout We can use the parameters from the von Bertalanffy fit with b = 0 . 14553 giving W * = 6295 . 4 We can fit both parameters to the time weight data, giving b = 0 . 16960 giving W * = 5677 . 67 The von Bertalanffy model fits existing length data best Fitting both matches weight/time data best Similar graphs and least square errors Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (60/64)
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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 Modeling Mercury in Fish 7 Weight vs Age of Lake Trout: Two models presented above are graphed 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
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