{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 " " {TEXT -1 27 "Wednesday September 3, 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 48 "Examples 2,3 from %2.1 of the \+ text, pages 79-80." }}{PARA 0 "" 0 "" {TEXT -1 427 " The Belgian d emographer P.F. Verhulst introduced the logistic model around 1840, as a tool for studying human population growth. Our text demonstrates i ts superiority to the simple exponential growth model, and also illust rates why mathematical modelers must always exercise care, by comparin g 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 80:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart: #clear Maple memory" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "pops:= [[1800,5.3],[1810,7. 2],[1820,9.6],[1830,12.9],\n [1840,17.1],[1850,23.2],[1860,31.4] ,[1870,38.6],\n [1880,50.2],[1890,63.0],[1900,76.2],[1910,92.2], \n [1920,106.0],[1930,123.2],[1940,132.2],[1950,151.3],\n \+ [1960,179.3],[1970,203.3],[1980,225.6],[1990,248.7]]:" }}}{PARA 0 "" 0 "" {TEXT -1 134 " Unlike Verhulst, the book uses data from 1800, 1850 and 1900 to get constants in our two models. We let t=0 correspo nd to 1800. " }}{PARA 0 "" 0 "" {TEXT 257 18 "Exponential Model:" } {TEXT -1 34 " For the exponential growth model " }{XPPEDIT 18 0 "P(t) \+ = P[0]*exp(r*t);" "6#/-%\"PG6#%\"tG*&&F%6#\"\"!\"\"\"-%$expG6#*&%\"rGF ,F'F,F," }{TEXT -1 49 " we use the 1800 and 1900 data to get values fo r " }{XPPEDIT 18 0 "P[0];" "6#&%\"PG6#\"\"!" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 2 " :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "P0:=5.308;\nsolve(P0*exp(r*100)=76.212,r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P0G$\"%3`!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9QIkE!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P1:=t->5.308*exp(.02664*t); #exponential model" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#P1Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&$\"%3`! \"$\"\"\"-%$expG6#,$*&$\"%kE!\"&F19$F1F1F1F1F(F(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 258 15 "Logistic Model:" }{TEXT -1 8 " We get \+ " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 52 " from 1800, a nd use the 1850 and 1900 data to find " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 2 " :" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "P2:=t->M*P0/(P0+(M-P0)*exp(- M*k*t));\n #logisitic function, with our P0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2Gf*6#%\"tG6\"6$%)operatorG%&arrowGF(*(%\"MG\"\"\"% #P0GF.,&F/F.*&,&F-F.F/!\"\"F.-%$expG6#,$*(F-F.%\"kGF.9$F.F3F.F.F3F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "solve(\{P2(50)=23.192, P2(100)=76.212\},\{M,k\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"M G$\"+v#37)=!\"(/%\"kG$\"+us:x;!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "M:=188.1208275;\nk:=.1677157274e-3;\nP2(t); #should be our logistic model function,\n #equation (8) page 79." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"+v#37)=!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+us:x;!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&$\"+CNX&)**!\"(\"\"\",&$\"%3`!\"$F(*&$\"+v#G\"G=F'F(-%$expG6#,$*&$ \"+U@3bJ!#6F(%\"tGF(!\"\"F(F(F9F(" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 59 "Now compare the two models with the real data, and discuss:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "with(plots):\nplot1:=plot(P 1(t-1800),t=1800..1950,color=black, linestyle=3):\n #this linestyle g ives dashes for the exponential curve\nplot2:=plot(P2(t-1800),t=1800.. 2000,color=black):\nplot3:=pointplot(pops,symbol=cross):\ndisplay(\{pl ot1,plot2,plot3\},title=`U.S. population data\nand models`);" }}{PARA 13 "" 1 "" {GLPLOT2D 568 362 362 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"%+= \"\"!$\"3$)***********zI&!#<7$$\"33+]7t&pK!=!#9$\"3)z\"GyAp1\"z&F-7$$ \"34](=7T9h!=F1$\"3$)oIi70,ZiF-7$$\"3.+v=HPJ4=F1$\"3!=!oCr-z-oF-7$$\"3 -+DJaU`7=F1$\"3NL;eoE?7uF-7$$\"3/]P%GZRd\"=F1$\"3V;:yN`\"H2)F-7$$\"3&* \\(=276(==F1$\"3e?g.!*G)zt)F-7$$\"3+](o**3)y@=F1$\"3@unAGTT%[*F-7$$\"3 &*\\(ofHq\\#=F1$\"3!>d2?)4NK5!#;7$$\"3'*\\Pf'HU\"G=F1$\"3W\"FW7$$\"3)***\\iNGwS=F1$\"3lvXtt &>Bd\"FW7$$\"3(***\\7XM*Q%=F1$\"3xW-E%[t!40)H&=F1$\"3T< 7k48=x@FW7$$\"32](=-p6j&=F1$\"3AXRsz6\")=F1$\"3c](\\OzBkg%FW7$$\"3/+voo6A%)=F1$\"3, vY\"f+AU+&FW7$$\"3,++vF1$\"3y&R]=*\\4CwFW7$$\"33+]i0XE.>F1$\"3)*3CE bs86$)FW7$$\"3&*\\(o/Q*>1>F1$\"3nk[\"G%=-()*)FW7$$\"3#***\\(Q(zS4>F1$ \"3l/5Z$f\"**)y*FW7$$\"3'*\\(=-,FC\">F1$\"3widmN=)31\"!#:7$$\"3*)\\P4t Fe:>F1$\"3y\"G\")*ef#R:\"F^w7$$\"3*)**\\73\"o'=>F1$\"3g:0W1yx_7F^w7$$ \"3$*\\(oz;)*=#>F1$\"31-Q4,=Nl8F^w7$$\"30++]*44]#>F1$\"3B@]E3nK$[\"F^w 7$$\"35+DJw/>G>F1$\"3'**e%*\\XAXh\"F^w7$$\"3,](=(4bMJ>F1$\"3Z_4'>B*3c< F^w7$$\"3*****\\xlWU$>F1$\"3[jKG2G4(*=F^w7$$\"33+Dc3ucP>F1$\"3UfqR\\jn s?F^w7$$\"3/++];$R0%>F1$\"3;jx;2VWVAF^w7$$\"3'*\\(=-*zqV>F1$\"3]T.tMR/ TCF^w7$$\"3)*\\7G:3uY>F1$\"3sT?K\"*HXYEF^w7$$\"%]>F*$\"3$)Q3Ay1]')GF^w -%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"$-F$6$7S7$F($\"3B]l_z***zI& F-7$$\"3OLL$3VfV!=F1$\"3VC-%[vQa1'F-7$$\"3cm;H[D:3=F1$\"3kl=iS)Q'3oF-7 $$\"3ALLe0$=C\"=F1$\"3'*R3I;P7\\xF-7$$\"3HLL3RBr;=F1$\"3ogkVB9q?))F-7$ $\"3cm;zjf)4#=F1$\"3E0OHV0h-5FW7$$\"3ML$e4;[\\#=F1$\"3)e:XUY7\"G6FW7$$ \"3++]i'y]!H=F1$\"3@n(*zLcXt7FW7$$\"3UL$ezs$HL=F1$\"3#>.`:!p)=W\"FW7$$ \"3-+]7iI_P=F1$\"3l'4VC8&))H;FW7$$\"3wmm;_M(=%=F1$\"3u0w`t%[h%=FW7$$\" 3FLL3y_qX=F1$\"33_L7-yWd?FW7$$\"3$*****\\1!>+&=F1$\"3XX%Rkc>/K#FW7$$\" 3(*****\\Z/Na=F1$\"3v*QUnl?Ih#FW7$$\"3/++]$fC&e=F1$\"35SX!eZ'FW7$ $\"3qm;zihl&*=F1$\"3+/nwSNx3qFW7$$\"3SLL3#G,***=F1$\"3aqe0=M32wFW7$$\" 3PL$ezw5V!>F1$\"3?Oef%40\\C)FW7$$\"35+]PQ#\\\"3>F1$\"3aH%p*HJK4))FW7$$ \"3JLLe\"*[H7>F1$\"3s;p`U*pOU*FW7$$\"3-+++dxd;>F1$\"3#zgug;8e+\"F^w7$$ \"31++D0xw?>F1$\"3C1l1'=:t1\"F^w7$$\"33+]i&p@[#>F1$\"3s6jo-[zD6F^w7$$ \"3+++vgHKH>F1$\"3O&HJ%*R2!*=\"F^w7$$\"3smmmZvOL>F1$\"3E_,47(QQC\"F^w7 $$\"3'*****\\2goP>F1$\"3c2PQn\"R**H\"F^w7$$\"3EL$eR<*fT>F1$\"33$4'R7vK [8F^w7$$\"30++])Hxe%>F1$\"3/kFp=8M)R\"F^w7$$\"3om;H!o-*\\>F1$\"3W)yqxB -DW\"F^w7$$\"3$***\\7k.6a>F1$\"3_9=^4Bd&[\"F^w7$$\"3vmm;WTAe>F1$\"3cV \"QaJ,Y_\"F^w7$$\"3)***\\i!*3`i>F1$\"39^'[$QqAi:F^w7$$\"3DLLL*zym'>F1$ \"3Y)o%fN8T&f\"F^w7$$\"3JLL3N1#4(>F1$\"3A1f[M;OE;F^w7$$\"3vm;HYt7v>F1$ \"3T$o2))[:Ul\"F^w7$$\"39+++xG**y>F1$\"3&4#Q(yCbun\"F^w7$$\"3immT6KU$) >F1$\"3\"HFgX(e\\,F1$\"3cfzkJ`\"3s\"F^w7$$\"3%*** \\i`1h\"*>F1$\"3Zb4SI@KRF1$\"3M2efr(y^v\"F^w7$$ \"%+?F*$\"3.3NU\"ou.x\"F^wF[[l-%'POINTSG677$F($\"#`!\"\"7$$\"%5=F*$\"# sF_[m7$$\"%?=F*$\"#'*F_[m7$$\"%I=F*$\"$H\"F_[m7$$\"%S=F*$\"$r\"F_[m7$$ \"%]=F*$\"$K#F_[m7$$\"%g=F*$\"$9$F_[m7$$\"%q=F*$\"$'QF_[m7$$\"%!)=F*$ \"$-&F_[m7$$\"%!*=F*$\"$I'F_[m7$$\"%+>F*$\"$i(F_[m7$$\"%5>F*$\"$A*F_[m 7$$\"%?>F*$\"%g5F_[m7$$\"%I>F*$\"%K7F_[m7$$\"%S>F*$\"%A8F_[m7$Fgz$\"%8 :F_[m7$$\"%g>F*$\"%$z\"F_[m7$$\"%q>F*$\"%L?F_[m7$$\"%!)>F*$\"%cAF_[m7$ $\"%!*>F*$\"%([#F_[m-%'SYMBOLG6#%&CROSSG-%+AXESLABELSG6%Q\"t6\"Q!Feam- %%FONTG6#%(DEFAULTG-%&TITLEG6#%@U.S.~population~data|+and~modelsG-%%VI EWG6$;F(FejlFjam" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 178 "Th e exponential model takes no account of the fact that the U.S. has onl y finite resources. Any ideas on why the logistic model begins to fai l (with our parameters) around 1950?" }}}{MARK "16 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }