Introduction Separation of Variables Modified Malthusian Growth Model

Introduction separation of variables modified

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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Modified Malthusian Growth Model 1 Modified Malthusian Growth Model Consider the model dP dt = ( a t + b ) P with P (0) = P 0 This equation is linear and separable Z dP P = Z ( a t + b ) dt Integrating ln( P ( t )) = a t 2 2 + b t + C Exponentiating P ( t ) = e a t 2 2 + b t + C Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (19/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Modified Malthusian Growth Model 2 Modified Malthusian Growth Model: With P (0) = e C = P 0 , the model can be written P ( t ) = P 0 e a t 2 2 + b t This model has 3 unknowns, P 0 , a , and b We fit the census data in 1790 and 1990 of 3.93 million and 248.7 million Choose the third data value from the census in 1890, where the population is 62.95 million Again take t to be the years after 1790, then P 0 = 3 . 93 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (20/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Modified Malthusian Growth Model 2 Population Model for U. S. The nonautonomous model is P ( t ) = 3 . 93 e a t 2 2 + b t Use the census data in 1890 and 1990 to find a and b The model gives P (100) = 62 . 95 = 3 . 93 e 5000 a +100 b P (200) = 248 . 7 = 3 . 93 e 20000 a +200 b Taking logarithms, we have the linear equations 5000 a + 100 b = ln ( 62 . 95 3 . 93 ) = 2 . 7737 20 , 000 a + 200 b = ln ( 248 . 7 3 . 93 ) = 4 . 1476 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (21/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Modified Malthusian Growth Model 3 Population Model for U. S. Solving the linear equations 5000 a + 100 b = ln ( 62 . 95 3 . 93 ) = 2 . 7737 20 , 000 a + 200 b = ln ( 248 . 7 3 . 93 ) = 4 . 1476 Multiply the first equation by - 2 and add to the second 10 , 000 a = - 2(2 . 7737) + 4 . 1476 = - 1 . 3998 Thus, a = - 0 . 00013998, which is substituted into the first equation It follows that 100 b = 5000(0 . 00013998) + 2 . 7737 = 3 . 473 Solution is a = - 0 . 00013998 and b = 0 . 03473 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (22/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Modified Malthusian Growth Model 4 Population Models for U. S. The Malthusian growth model fitting the census data at 1790 and 1990 is P ( t ) = 3 . 93 e 0 . 02074 t The nonautonomous model fitting the census data at 1790, 1890, and 1990 is P ( t ) = 3 . 93 e 0 . 03474 t - 0 . 00006999 t 2 Model 1900 2000 2010 U. S. Census Data 76.21 281.4 308.7 Malthusian Growth 38.48 306.1 376.7 Nonautonomous 76.95 264.4 277.0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (23/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S.
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