Population of Italy 2 Population Models Modified Malthusian Growth Model 5

Population of italy 2 population models modified

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Population of Italy 2 Population Models Modified Malthusian Growth Model 5 Population Models for U. S. The models use limited data for prediction For 1900 The Malthusian growth model is too low by 49.5% The nonautonomous growth model is too high by 0.97% The nonautonomous growth model fits quite well For 2000 and 2010 The Malthusian growth model is too high by 8.8% and 22% The nonautonomous growth model is too low by 6.0% and 10.3% Neither model fits the census data very well The nonautonomous though fitting better misses the recent higher growth from immigration Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equations — (24/41)
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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 6 Graphs of Population Models for U. S. 1800 1850 1900 1950 2000 50 100 150 200 250 300 Year Population (in millions) U. S. Population Malthusian Nonautonomous Census Data Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equations — (25/41) Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 1 Population of Italy: For the last few decades, Italy has had its growth rate decline to where the country does not even have enough births (or immigration) to replace the number of deaths in the country The population of Italy was 47.1 million in 1950, 53.7 million in 1970, and 56.8 million in 1990 Use the data in 1950 and 1990 to find a Malthusian growth model for Italy’s population Consider the nonautonomous Malthusian growth model given by the differential equation dP dt = ( a t + b ) P with P (0) = 47 . 1 with t in years after 1950 Solve this differential equation Find the constants a and b from the data Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equations — (26/41) Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 2 Population of Italy (cont): If the population of Italy was 50.2 million in 1960 and 57.6 million in 2000, then use each of these models to estimate the populations and determine the error between the models and the actual census values Graph the solutions of the two models and the data points from 1950 to 2000 Find when Italy’s population levels off and begins to decline according to the nonautonomous Malthusian growth model Solution: The Malthusian growth model satisfies dP dt = rP with P (0) = 47 . 1 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equations — (27/41) Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Italy 3 Solution (cont): The solution of the Malthusian growth model is P ( t ) = 47 . 1 e rt In 1990 the population was 56.8 million, so P (40) = 47 . 1 e 40 r = 56 . 8 Thus, e 40 r = 56 . 8 47 . 1 or r = 1 40 ln ( 56 . 8 47 . 1 ) = 0 . 004682 The Malthusian growth model for Italy is P ( t ) = 47 . 1 e 0 . 004682 t Joseph M. Mahaffy, h [email protected] i
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