Math 2280-1 and 2250-1Monday September 8, 2008Examples 4,5 from %2.1 of the text, pages 83-84. The Belgian demographer P.F. Verhulst introduced the logistic model around 1840, as a tool for studying human population growth. Our text demonstrates its superiority to the simple exponential growth model, and also illustrates why mathematical modelers must always exercise care, by comparing the two models to actual U.S. population data. here are actual U.S. populations from 1800-1990, see e.g. the table on page 84:> restart: #clear Maple memory> pops:= [[1800,5.3],[1810,7.2],[1820,9.6],[1830,12.9], [1840,17.1],[1850,23.2],[1860,31.4],[1870,38.6], [1880,50.2],[1890,63.0],[1900,76.2],[1910,92.2], [1920,106.0],[1930,123.2],[1940,132.2],[1950,151.3], [1960,179.3],[1970,203.3],[1980,225.6],[1990,248.7]]: Unlike Verhulst, the book uses data from 1800, 1850 and 1900 to get constants in our two models. We let t=0 correspond to 1800. Exponential Model: For the exponential growth model =( )P t P0e( )r t we use the 1800 and 1900 data to get values for P0 and r :> P0:=5.308;solve(P0*exp(r*100)=76.212,r); := P0 5.3080.02664303814> P1:=t->5.308*exp(.02664*t); #exponential model := P1 →t 5.308 e( )0.02664 tLogistic Model: We get P0 from 1800, and use the 1850 and 1900 data to find k and M :> P2:=t->M*P0/(P0+(M-P0)*exp(-M*k*t)); #logisitic function, with our P0 := P2 →tMP0+P0 ( )−M P0 e( )−M k t> solve({P2(50)=23.192, P2(100)=76.212},{M,k});{ },=k 0.0001677157274 =M 188.1208275> M:=188.1208275;k:=.1677157274e-3;P2(t); #should be our logistic model function, #equation (11) page 82. := M 188.1208275 := k 0.0001677157274998.5453524+5.308 182.8128275 e( )−0.03155082142 tNow compare the two models with the real data, and discuss:> with(plots):plot1:=plot(P1(t-1800),t=1800..1950,color=black, linestyle=3): #this linestyle gives dashes for the exponential curveplot2:=plot(P2(t-1800),t=1800..2000,color=black):plot3:=pointplot(pops,symbol=cross):display({plot1,plot2,plot3},title=‘U.S. population dataand models‘);Warning, the name changecoords has been redefinedU.S. population dataand models0501001502002501800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000t> The exponential model takes no account of the fact that the U.S. has only finite resources. Any ideas on why the logistic model begins to fail (with our parameters) around
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