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 Population of Italy 4 Solution (cont): The nonautonomous Malthusian growth model is dP dt = ( a t + b ) P with P (0) = 47 . 1 Separating variables Z dP P = Z ( a t + b ) dt Thus, ln( P ( t )) = at 2 2 + bt + c Exponentiating P ( t ) = e at 2 2 + bt + c = e c e at 2 2 + bt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (29/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 5 Solution (cont): The initial condition gives P (0) = 47 . 1 = e c The solution can be written P ( t ) = 47 . 1 e at 2 2 + bt The logarithmic form satisfies at 2 2 + bt = ln( P ( t )) - ln(47 . 1) The data from 1970 and 1990 give 200 a + 20 b = ln(53 . 7) - ln(47 . 1) = 0 . 13114 800 a + 40 b = ln(56 . 8) - ln(47 . 1) = 0 . 18726 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (30/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 6 Solution (cont): The equations in a and b are linear equations Multiply the first equation by - 2 and add it to the second - 2(200 a + 20 b ) = - 2(0 . 13114) 800 a + 40 b = 0 . 18726 400 a = - 0 . 07502 It follows that a = - 0 . 00018755 From either equation above b = 0 . 0084325 The solution becomes P ( t ) = 47 . 1 e 0 . 0084325 t - 0 . 00009378 t 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (31/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 7 Solution (cont): The two models are given by P ( t ) = 47 . 1 e 0 . 004682 t and P ( t ) = 47 . 1 e 0 . 0084325 t - 0 . 00009378 t 2 Below is a Table comparing the models at 1960 and 2000 Model 1960 % Error 2000 % Error Italy Census Data 50.2 - 57.6 - Malthusian 49.4 - 1 . 7% 59.5 3.3% Nonautonomous 50.8 1.1% 56.8 - 1 . 4% Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (32/41)
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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 8 Graphs of Population Models for Italy 1950 1960 1970 1980 1990 2000 48 50 52 54 56 58 Year Population (in millions) Models of Italian Population Malthusian Nonautonomous Census Data Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (33/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 9 Solution (cont): The nonautonomous model is dP dt = (0 . 0084325 - 0 . 00018755 t ) P ( t ) The population growth slows to zero, so the population levels off, when dP dt = 0 This occurs when 0 . 0084325 - 0 . 00018755 t = 0 or t = 44 . 96 years The nonautonomous Malthusian growth model predicts that Italy’s population leveled off in 1995 (45 years after 1950) Data indicates that 2000 was the peak of Italy’s population, so the model does reasonably well Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (34/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S.
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