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Math 636 - Mathematical Modeling Discrete Modeling 1 Discrete Modeling – Population of the United States U Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Variation in Growth Rate Variation in Growth Rate Autonomous Models Autonomous Models Outline Math 636 - Mathematical Modeling Discrete Modeling 1 Discrete Modeling { Population of the United States U. S. Population Malthusian Growth Model Programming Malthusian Growth Joseph M. Mahaffy, 2 Variation in Growth Rate [email protected] General Discrete Dynamical Population Model Linear Growth Rate U. S. Population Model Nonautonomous Malthusian Growth Model Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center 3 Autonomous Models San Diego State University Logistic Growth Model San Diego, CA 92182-7720 Beverton-Holt Model http://jmahaffy.sdsu.edu Analysis of Autonomous Models Fall 2018 Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (1/34) Joseph M. Mahaffy, [email protected] (2/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Malthusian Growth Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Programming Malthusian Growth Programming Malthusian Growth Autonomous Models Autonomous Models United States Census Census Data Census Data United States Census 1790 3,929,214 1870 39,818,449 1950 150,697,361 Constitution requires census every 10 years 1800 5,308,483 1880 50,189,209 1960 179,323,175 Census used for budgeting federal payments and representation 1810 7,239,881 1890 62,947,714 1970 203,302,031 in Congress 1820 9,638,453 1900 76,212,168 1980 226,545,805 1830 12,866,020 1910 92,228,496 1990 248,709,873 Process can be politically charged 1840 17,069,453 1920 106,021,537 2000 281,421,906 Accurately predicting demographic data are important for 1850 23,191,876 1930 122,775,046 2010 308,745,538 planning communities in the future 1860 31,443,321 1940 132,164,569 Calculations for the future populations uses sophisticated History of the Census Bureau shows that these numbers are collected mathematical models in various ways, including information only from the heads of Models are constantly improved and revised households, the Bureau depending on only 183 clerks for over 30M people, failure to count slaves, and other irregularities such as recent undercounting minorities. Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (3/34) Joseph M. Mahaffy, [email protected] (4/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Malthusian Growth Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Programming Malthusian Growth Programming Malthusian Growth Autonomous Models Autonomous Models Growth Rate of U. S. Malthusian Growth Model Growth Rate in Early U. S. Malthusian Growth Model The growth rate for the decade of 1790-1800 Simplest growth model uses a constant rate of growth, r Population in 1800 5; 308; 483 = = 1:351 Start with the population in 1790, P0 Population in 1790 3; 292; 214 Population in the next decade is current population plus the The growth rate for the decade of 1800-1810 population times the average growth rate Population in 1810 7; 239; 881 = = 1:364 Pn+1 = Pn + rPn = (1 + r)Pn Population in 1800 5; 308; 483 Sequence of predicted populations based solely on population The growth rate for the decade of 1810-1820 from preceding population Population in 1820 9; 638; 453 = = 1:331 The Malthusian Growth Model is easily solved to give: Population in 1810 7; 239; 881 n Pn = P0(1 + r) : These growth rates are relatively constant, averaging 0.349/decade, and are typical of agrarian societies. Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (5/34) Joseph M. Mahaffy, [email protected] (6/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Malthusian Growth Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Programming Malthusian Growth Programming Malthusian Growth Autonomous Models Autonomous Models Malthusian Growth Model Malthusian Growth Model Table for U. S. Population (early years), assuming a constant Graph of the Malthusian Growth Model and Census Data for growth rate of 0.349 the U. S. U. S. Population 45 Year Census Model P = 1:349P % Error Malthusian Model n+1 n 40 Census Data 1790 3; 929; 214 3; 929; 214 35 ) 6 30 1800 5; 308; 483 5; 300; 510 −0:15 10 × 1810 7; 239; 881 7; 150; 388 −1:24 25 20 1820 9; 638; 453 9; 645; 873 0:08 15 Population ( 1830 12; 866; 020 13; 012; 282 1:14 10 1840 17; 069; 453 17; 553; 569 2:84 5 0 1790 1800 1810 1820 1830 1840 1850 1860 1870 1850 23; 191; 876 23; 679; 765 2:10 Year 1860 31; 433; 321 31; 944; 002 1:59 Error remains small until 1870, as U. S. is primarily an Agrarian 1870 39; 818; 449 43; 092; 459 8:22 society with a constant growth rate around 35% The Malthusian Growth Model tracks early U. S. population well Civil War dramatically increased the death rate, but more until the Civil war. significantly, society moved into the industrial revolution Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (7/34) Joseph M. Mahaffy, [email protected] (8/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Malthusian Growth Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Programming Malthusian Growth Programming Malthusian Growth Autonomous Models Autonomous Models Programming Discrete Models Malthusian Growth Model Programming Discrete Models: Discrete dynamical models are one of the easiest models to program: xn+1 = f(xn) 1 clear 2 xlab='Year'; Spreadsheet software 3 ylab='Population($ ntimes 10ˆ6$)'; Excel is designed for discrete dynamical models 4 mytitle='U.S. Population'; The pull-down or updating feature automatically uses the value in 5 pcensus = [3.929 5.308 7.240 9.638 12.866 17.069 ... the cell above to insert into the \updating function" in the new cell 23.192 31.433 39.818]; MatLab or other programming languages 6 tt = [1790]; pmal = [3.929]; Discrete dynamical models are iterated processes, which naturally 7 fori=1:8 program with for loops 8 t= tt(i) + 10; In MatLab if we want to iterate the model above N times creating a 9 mal= pmal(i) + 0.349 *pmal(i); solution vector x from a given function f(x), we write: 10 tt=[tt,t]; 11 pmal=[pmal,mal]; 12 formatSpec='Year is%d, Population is %8.3f ... 1 x=[x0]; M n'; 2 fori = 1:N n 13 fprintf(formatSpec,t,mal) 3 x0=f(x0); 14 end 4 x=[x,x0]; 5 end Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (9/34) Joseph M. Mahaffy, [email protected] (10/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United StatesGeneral Discrete Dynamical Population Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Linear Growth Rate U. S. Population Model Programming Malthusian Growth Autonomous Models Autonomous Models Nonautonomous Malthusian Growth Model Malthusian Growth Model Analysis of the Growth Rate Variation in Growth Rate 15 plot(tt,pmal,'r-','LineWidth',1.5); 16 hold on Assume this Malthusian growth model were extended 17 plot(tt,pcensus,'o','MarkerSize',6); In 1920, model predicts 192,365,343 (population in 1960s), 18 grid 19 legend('Malthusian Model','Census ... which is 82% too high Data','Location','Northwest'); In 1970, model predicts 859,382,645, which is 323% too high 20 Growth rate of U. S. has generally decreased over the 21 ax = [1790 1870 0 50]; Census history 22 fontlabs='Times New Roman'; 23 xlabel(xlab,'FontSize',14,'FontName',fontlabs,'interpreter','latex'); Average growth rate over census history is 22.3% 24 ylabel(ylab,'FontSize',14,'FontName',fontlabs,'interpreter','latex'); In 1800, 1900, and 2000, the growth rates were 36.4%, 25 title(mytitle,'FontSize',16,'FontName','Times New ... 21.0%, and 9.71%, showing the general decline Roman','interpreter','latex'); The Great Depression gave lowest growth rate of 7.2% 26 set(gca,'FontSize',12); Obvious time variation in growth rate 27 print-depsc uspopmal.eps Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (11/34) Joseph M. Mahaffy, [email protected] (12/34) Discrete Modeling { Population of the United StatesGeneral Discrete Dynamical Population Model Discrete Modeling { Population of the United StatesGeneral Discrete Dynamical Population Model Variation in Growth Rate Linear Growth Rate U. S. Population Model Variation in Growth Rate Linear Growth Rate U. S. Population Model Autonomous Models Nonautonomous Malthusian Growth Model Autonomous Models Nonautonomous Malthusian Growth Model Modeling the Growth Rate General Discrete Dynamical Population Model Modeling the Growth Rate General Discrete Dynamical Population Models Malthusian growth model is too simplistic except early population growth with plenty of resources If the population at the next discrete time period depends only on the current population, then given initial population, P , Crowding and limited resources are usually next dominant term 0 in modeling (logistic growth) Pn+1 = f(Pn); Crowding/squalor leads to declining birth rates and increased death rates for some function f(P ) Improved health decreases death rates and significantly This first order difference equation is autonomous drops birth rates(greatest birth rate decline comes from Malthusian growth uses a linear growth with educating women) f(P ) = P + rP , while logistic growth uses a quadratic Clearly, human populations have important contributions from growth with f(P ) = P + rP (1 − P=M) time varying historical, societal, and technological changes Many different nonlinear forms of f(P ) are used with (temporal variation) different interpretations and mathematical behaviors Animal population models usually ignore these time-varying Animal populations often use these models factors (autonomous) Discrete Modeling U.
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