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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 39 
"The Logistic Model of Population Growth" }}{PARA 19 "" 0 "" {TEXT -1 
61 "by Daniel Schwalbe, Macalester College, schwalbe@macalstr.edu" }}
{PARA 259 "" 0 "" {TEXT 256 9 "Abstract:" }{TEXT -1 95 " In this works
heet we study the logistic model of population growth, (dP/dt) = aP(t)
 - bP(t)^2." }}{PARA 259 "" 0 "" {TEXT 259 28 "Application Areas/Subje
cts: " }{TEXT 257 0 "" }{TEXT -1 51 " Biology/Medicine, Plotting, Diff
erential equations" }}{PARA 259 "" 0 "" {TEXT 258 9 "Keywords:" }
{TEXT -1 63 " differential equations, ODEs, DEs, population, direction
 plots" }}{PARA 259 "" 0 "" {TEXT 260 14 "Prerequisites:" }{TEXT -1 
16 "  Share/ODE code" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 " Introduction" }}{PARA 0 "" 
0 "" {TEXT -1 538 "In this worksheet we study the logistic model of po
pulation growth, (dP/dt) = aP(t) - bP(t)^2.  a and b are known as the \+
vital coefficients of the population growth. In the exponential growth
 model (dP/dt) = kP(t), we can find a value for k if we are given the \+
population at two different times. We use Maple to show that in the lo
gistics model we can calculate a and b if we are given the population \+
at three different values which are equally spaced.  We use the U.S. p
opulation from 1790 to 1990 as a test case for the resulting model." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " The US Population" }}
{PARA 0 "" 0 "" {TEXT -1 106 "We first make a table of the actual popu
lation of U.S. taken from a publication of the U.S. census bureau." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 340 "population:=table([1790=392
9214,\n1800=5308483, 1810=7239881,\n1820=9638453, 1830=12866020,\n1840
=17069453, 1850=23191876,\n1860=31433321, 1870=39818449,\n1880=5015578
3, 1890=62947714,\n1900=75994575, 1910=91972266,\n1920=105710620, 1930
=122775046,\n1940=131669275, 1950=151325798,\n1960=179323175, 1970=203
302031,\n1980=226545805, 1990=248709873 ]):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 238 "We then define a function which represents the right h
and side of the differential equation. Before doing this, we assume th
at a and b are positive real numbers.  Maple will need this informatio
n to compute a limit later in the worksheet." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "assume(a>0); assume(b>0);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "logeq := unapply(a*P - b*P^2,t,P);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&logeqGR6$%\"tG%\"PG6\"6$%)operatorG%&arrowGF),&
*&%#a|irG\"\"\"9%F0F0*&%#b|irGF0)F1\"\"#\"\"\"!\"\"F)F)F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 467 "Before using Maple to analyze this equat
ion we read in a file of procedures which are designed to analyze diff
erential equations. The file is located in the share library. If you h
ave an old version of the share library you may need to update it befo
re some of  the commands in this worksheet will work.  The following c
ommand will read in the file from the share library.  The ODE package \+
contains a help file and is also documented in the Maple V Flight Manu
al [1]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(share): with(ODE);
" }}{PARA 6 "" 1 "" {TEXT -1 19 "Share Library:  ODE" }}{PARA 6 "" 1 "
" {TEXT -1 24 "Author: Daniel Schwalbe." }}{PARA 6 "" 1 "" {TEXT -1 
79 "Description:  package for solving ODE's numerically and plottingth
eir solutions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'Q/directionfield6\"
Q)impeulerF%Q+rungekuttaF%Q-rungekuttahfF%Q*phaseplotF%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 111 "We can use Maple to look at a direction \+
field for particular values of a and b, here we choose a=2 and b = 1/2
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "directionfield(subs(a=2,b=1/2,
eval(logeq)),0..8,0..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6jx-%%VIEWG6$;$!+++++]!#5$\"+++++&)!\"*;F'$\"+++++lF,-%'C
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F7F87$$\"1++++++D6!#:F8F<-F16$7$7$$\"1++++++v8FRF87$$\"1++++++D;FRF8F<
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7$$\"1++++++DEFRF8F<-F16$7$7$$\"1++++++vGFRF87$$\"1++++++DJFRF8F<-F16$
7$7$$\"1++++++vLFRF87$$\"1++++++DOFRF8F<-F16$7$7$$\"1++++++vQFRF87$$\"
1++++++DTFRF8F<-F16$7$7$$\"1++++++vVFRF87$$\"1++++++DYFRF8F<-F16$7$7$$
\"1++++++v[FRF87$$\"1++++++D^FRF8F<-F16$7$7$$\"1++++++v`FRF87$$\"1++++
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RF8F<-F16$7$7$$\"1++++++vtFRF87$$\"1++++++DwFRF8F<-F16$7$7$$\"1++++++v
yFRF87$$\"1++++++D\")FRF8F<-F16$7$7$F5$\"1+++++D1RF77$F:$\"1+++++v$4'F
7F<-F16$7$7$FDF_v7$FGFbvF<-F16$7$7$FMF_v7$FPFbvF<-F16$7$7$FWF_v7$FZFbv
F<-F16$7$7$FjnF_v7$F]oFbvF<-F16$7$7$FcoF_v7$FfoFbvF<-F16$7$7$F\\pF_v7$
F_pFbvF<-F16$7$7$FepF_v7$FhpFbvF<-F16$7$7$F^qF_v7$FaqFbvF<-F16$7$7$Fgq
F_v7$FjqFbvF<-F16$7$7$F`rF_v7$FcrFbvF<-F16$7$7$FirF_v7$F\\sFbvF<-F16$7
$7$FbsF_v7$FesFbvF<-F16$7$7$F[tF_v7$F^tFbvF<-F16$7$7$FdtF_v7$FgtFbvF<-
F16$7$7$F]uF_v7$F`uFbvF<-F16$7$7$FfuF_v7$FiuFbvF<-F16$7$7$$!1LLLLLLL$)
!#<FM7$$\"1LLLLLLL$)Fj[lFPF<-F16$7$7$$\"1nmmmmmmTF7FM7$$\"1MLLLLLLeF7F
PF<-F16$7$7$$\"1mmmmmmm\"*F7FM7$$\"1LLLLLL$3\"FRFPF<-F16$7$7$$\"1nmmmm
m;9FRFM7$$\"1LLLLLL$e\"FRFPF<-F16$7$7$$\"1nmmmmm;>FRFM7$$\"1LLLLLL$3#F
RFPF<-F16$7$7$$\"1nmmmmm;CFRFM7$$\"1LLLLLL$e#FRFPF<-F16$7$7$$\"1nmmmmm
;HFRFM7$$\"1LLLLLL$3$FRFPF<-F16$7$7$$\"1nmmmmm;MFRFM7$$\"1LLLLLL$e$FRF
PF<-F16$7$7$$\"1nmmmmm;RFRFM7$$\"1LLLLLL$3%FRFPF<-F16$7$7$$\"1nmmmmm;W
FRFM7$$\"1LLLLLL$e%FRFPF<-F16$7$7$$\"1nmmmmm;\\FRFM7$$\"1LLLLLL$3&FRFP
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RFM7$$\"1LLLLLL$3'FRFPF<-F16$7$7$$\"1nmmmmm;kFRFM7$$\"1LLLLLL$e'FRFPF<
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M7$$\"1LLLLLL$e(FRFPF<-F16$7$7$$\"1nmmmmm;zFRFM7$$\"1MLLLLL$3)FRFPF<-F
16$7$7$$!1nmmmmmmmFj[lFW7$$\"1nmmmmmmmFj[lFZF<-F16$7$7$$\"1LLLLLLLVF7F
W7$$\"1nmmmmmmcF7FZF<-F16$7$7$$\"1LLLLLLL$*F7FW7$$\"1nmmmmmm5FRFZF<-F1
6$7$7$$\"1LLLLLLL9FRFW7$$\"1nmmmmmm:FRFZF<-F16$7$7$$\"1LLLLLLL>FRFW7$$
\"1nmmmmmm?FRFZF<-F16$7$7$$\"1LLLLLLLCFRFW7$$\"1nmmmmmmDFRFZF<-F16$7$7
$$\"1LLLLLLLHFRFW7$$\"1nmmmmmmIFRFZF<-F16$7$7$$\"1LLLLLLLMFRFW7$$\"1nm
mmmmmNFRFZF<-F16$7$7$$\"1LLLLLLLRFRFW7$$\"1mmmmmmmSFRFZF<-F16$7$7$$\"1
MLLLLLLWFRFW7$$\"1mmmmmmmXFRFZF<-F16$7$7$$\"1MLLLLLL\\FRFW7$$\"1mmmmmm
m]FRFZF<-F16$7$7$$\"1MLLLLLLaFRFW7$$\"1mmmmmmmbFRFZF<-F16$7$7$$\"1MLLL
LLLfFRFW7$$\"1mmmmmmmgFRFZF<-F16$7$7$$\"1MLLLLLLkFRFW7$$\"1mmmmmmmlFRF
ZF<-F16$7$7$$\"1MLLLLLLpFRFW7$$\"1mmmmmmmqFRFZF<-F16$7$7$$\"1MLLLLLLuF
RFW7$$\"1mmmmmmmvFRFZF<-F16$7$7$$\"1MLLLLLLzFRFW7$$\"1mmmmmmm!)FRFZF<-
F16$7$7$$!1++++++]iFj[lFjn7$$FHFj[lF]oF<-F16$7$7$$FhqF7Fjn7$$F]sF7F]oF
<-F16$7$7$$\"1++++++v$*F7Fjn7$$\"1+++++]i5FRF]oF<-F16$7$7$$\"1+++++]P9
FRFjn7$$\"1+++++]i:FRF]oF<-F16$7$7$$\"1+++++]P>FRFjn7$$\"1+++++]i?FRF]
oF<-F16$7$7$$\"1+++++]PCFRFjn7$$\"1+++++]iDFRF]oF<-F16$7$7$$\"1+++++]P
HFRFjn7$$\"1+++++]iIFRF]oF<-F16$7$7$$\"1+++++]PMFRFjn7$$\"1+++++]iNFRF
]oF<-F16$7$7$$\"1+++++]PRFRFjn7$$\"1+++++]iSFRF]oF<-F16$7$7$$\"1+++++]
PWFRFjn7$$\"1+++++]iXFRF]oF<-F16$7$7$$\"1+++++]P\\FRFjn7$$\"1+++++]i]F
RF]oF<-F16$7$7$$\"1+++++]PaFRFjn7$$\"1+++++]ibFRF]oF<-F16$7$7$$\"1++++
+]PfFRFjn7$$\"1+++++]igFRF]oF<-F16$7$7$$\"1+++++]PkFRFjn7$$\"1+++++]il
FRF]oF<-F16$7$7$$\"1+++++]PpFRFjn7$$\"1+++++]iqFRF]oF<-F16$7$7$$\"1+++
++]PuFRFjn7$$\"1+++++]ivFRF]oF<-F16$7$7$$\"1+++++]PzFRFjn7$$\"1+++++]i
!)FRF]oF<-F16$7$7$FbelFco7$FeelFfoF<-F16$7$7$F[flFco7$F^flFfoF<-F16$7$
7$FdflFco7$FgflFfoF<-F16$7$7$F]glFco7$F`glFfoF<-F16$7$7$FfglFco7$FiglF
foF<-F16$7$7$F_hlFco7$FbhlFfoF<-F16$7$7$FhhlFco7$F[ilFfoF<-F16$7$7$Fai
lFco7$FdilFfoF<-F16$7$7$FjilFco7$F]jlFfoF<-F16$7$7$FcjlFco7$FfjlFfoF<-
F16$7$7$F\\[mFco7$F_[mFfoF<-F16$7$7$Fe[mFco7$Fh[mFfoF<-F16$7$7$F^\\mFc
o7$Fa\\mFfoF<-F16$7$7$Fg\\mFco7$Fj\\mFfoF<-F16$7$7$F`]mFco7$Fc]mFfoF<-
F16$7$7$Fi]mFco7$F\\^mFfoF<-F16$7$7$Fb^mFco7$Fe^mFfoF<-F16$7$7$Fh[lF\\
p7$F\\\\lF_pF<-F16$7$7$Fb\\lF\\p7$Fe\\lF_pF<-F16$7$7$F[]lF\\p7$F^]lF_p
F<-F16$7$7$Fd]lF\\p7$Fg]lF_pF<-F16$7$7$F]^lF\\p7$F`^lF_pF<-F16$7$7$Ff^
lF\\p7$Fi^lF_pF<-F16$7$7$F__lF\\p7$Fb_lF_pF<-F16$7$7$Fh_lF\\p7$F[`lF_p
F<-F16$7$7$Fa`lF\\p7$Fd`lF_pF<-F16$7$7$Fj`lF\\p7$F]alF_pF<-F16$7$7$Fca
lF\\p7$FfalF_pF<-F16$7$7$F\\blF\\p7$F_blF_pF<-F16$7$7$FeblF\\p7$FhblF_
pF<-F16$7$7$F^clF\\p7$FaclF_pF<-F16$7$7$FgclF\\p7$FjclF_pF<-F16$7$7$F`
dlF\\p7$FcdlF_pF<-F16$7$7$FidlF\\p7$F\\elF_pF<-F16$7$7$F5$\"1++++]i!R$
FR7$F:$\"1++++]P4OFRF<-F16$7$7$FDF[cn7$FGF^cnF<-F16$7$7$FMF[cn7$FPF^cn
F<-F16$7$7$FWF[cn7$FZF^cnF<-F16$7$7$FjnF[cn7$F]oF^cnF<-F16$7$7$FcoF[cn
7$FfoF^cnF<-F16$7$7$F\\pF[cn7$F_pF^cnF<-F16$7$7$FepF[cn7$FhpF^cnF<-F16
$7$7$F^qF[cn7$FaqF^cnF<-F16$7$7$FgqF[cn7$FjqF^cnF<-F16$7$7$F`rF[cn7$Fc
rF^cnF<-F16$7$7$FirF[cn7$F\\sF^cnF<-F16$7$7$FbsF[cn7$FesF^cnF<-F16$7$7
$F[tF[cn7$F^tF^cnF<-F16$7$7$FdtF[cn7$FgtF^cnF<-F16$7$7$F]uF[cn7$F`uF^c
nF<-F16$7$7$FfuF[cn7$FiuF^cnF<-F16$7$7$F5$\"\"%F87$F:FdhnF<-F16$7$7$FD
Fdhn7$FGFdhnF<-F16$7$7$FMFdhn7$FPFdhnF<-F16$7$7$FWFdhn7$FZFdhnF<-F16$7
$7$FjnFdhn7$F]oFdhnF<-F16$7$7$FcoFdhn7$FfoFdhnF<-F16$7$7$F\\pFdhn7$F_p
FdhnF<-F16$7$7$FepFdhn7$FhpFdhnF<-F16$7$7$F^qFdhn7$FaqFdhnF<-F16$7$7$F
gqFdhn7$FjqFdhnF<-F16$7$7$F`rFdhn7$FcrFdhnF<-F16$7$7$FirFdhn7$F\\sFdhn
F<-F16$7$7$FbsFdhn7$FesFdhnF<-F16$7$7$F[tFdhn7$F^tFdhnF<-F16$7$7$FdtFd
hn7$FgtFdhnF<-F16$7$7$F]uFdhn7$F`uFdhnF<-F16$7$7$FfuFdhn7$FiuFdhnF<-F1
6$7$7$$\"166666666F7Fgq7$$!166666666F7FjqF<-F16$7$7$$\"17666666hF7Fgq7
$$\"1*)))))))))))))QF7FjqF<-F16$7$7$$F\\^oFRFgq7$$\"1))))))))))))))))F
7FjqF<-F16$7$7$$\"16666666;FRFgq7$$\"1*))))))))))))Q\"FRFjqF<-F16$7$7$
$\"16666666@FRFgq7$$\"1*)))))))))))))=FRFjqF<-F16$7$7$$\"16666666EFRFg
q7$$\"1*))))))))))))Q#FRFjqF<-F16$7$7$$\"16666666JFRFgq7$$\"1*))))))))
)))))GFRFjqF<-F16$7$7$$\"16666666OFRFgq7$$\"1*))))))))))))Q$FRFjqF<-F1
6$7$7$$\"16666666TFRFgq7$$Fh^oFRFjqF<-F16$7$7$$\"16666666YFRFgq7$$\"1*
))))))))))))Q%FRFjqF<-F16$7$7$$\"16666666^FRFgq7$$\"1*)))))))))))))[FR
FjqF<-F16$7$7$$\"16666666cFRFgq7$$\"1*))))))))))))Q&FRFjqF<-F16$7$7$$
\"16666666hFRFgq7$$\"1*)))))))))))))eFRFjqF<-F16$7$7$$\"16666666mFRFgq
7$$\"1*))))))))))))Q'FRFjqF<-F16$7$7$$\"16666666rFRFgq7$$\"1*)))))))))
))))oFRFjqF<-F16$7$7$$\"16666666wFRFgq7$$\"1*))))))))))))Q(FRFjqF<-F16
$7$7$$\"16666666\")FRFgq7$$\"1*)))))))))))))yFRFjqF<-F16$7$7$$\"1+++++
++]Fj[lF`r7$$!1+++++++]Fj[lFcrF<-F16$7$7$$\"1+++++++bF7F`r7$$\"1++++++
+XF7FcrF<-F16$7$7$$\"1++++++]5FRF`r7$$\"1+++++++&*F7FcrF<-F16$7$7$$\"1
++++++]:FRF`r7$$\"1++++++]9FRFcrF<-F16$7$7$$\"1++++++]?FRF`r7$$\"1++++
++]>FRFcrF<-F16$7$7$$\"1++++++]DFRF`r7$$\"1++++++]CFRFcrF<-F16$7$7$$\"
1++++++]IFRF`r7$$\"1++++++]HFRFcrF<-F16$7$7$$\"1++++++]NFRF`r7$$\"1+++
+++]MFRFcrF<-F16$7$7$$\"1++++++]SFRF`r7$$\"1++++++]RFRFcrF<-F16$7$7$$
\"1++++++]XFRF`r7$$\"1++++++]WFRFcrF<-F16$7$7$$\"1++++++]]FRF`r7$$\"1+
+++++]\\FRFcrF<-F16$7$7$$\"1++++++]bFRF`r7$$\"1++++++]aFRFcrF<-F16$7$7
$$\"1++++++]gFRF`r7$$\"1++++++]fFRFcrF<-F16$7$7$$\"1++++++]lFRF`r7$$\"
1++++++]kFRFcrF<-F16$7$7$$\"1++++++]qFRF`r7$$\"1++++++]pFRFcrF<-F16$7$
7$$\"1++++++]vFRF`r7$$\"1++++++]uFRFcrF<-F16$7$7$$\"1,+++++]!)FRF`r7$$
\"1++++++]zFRFcrF<-F16$7$7$$\"1IIIIIIIIFj[lFir7$$!1IIIIIIIIFj[lF\\sF<-
F16$7$7$$\"1.......`F7Fir7$$\"1(ppppppp%F7F\\sF<-F16$7$7$$\"1IIIIIII5F
RFir7$$\"1(ppppppp*F7F\\sF<-F16$7$7$$\"1IIIIIII:FRFir7$$\"1qpppppp9FRF
\\sF<-F16$7$7$$\"1IIIIIII?FRFir7$$\"1qpppppp>FRF\\sF<-F16$7$7$$\"1IIII
IIIDFRFir7$$\"1qppppppCFRF\\sF<-F16$7$7$$F\\apFRFir7$$\"1qppppppHFRF\\
sF<-F16$7$7$$\"1IIIIIIINFRFir7$$\"1qppppppMFRF\\sF<-F16$7$7$$\"1IIIIII
ISFRFir7$$\"1qppppppRFRF\\sF<-F16$7$7$$\"1IIIIIIIXFRFir7$$\"1qppppppWF
RF\\sF<-F16$7$7$$\"1IIIIIII]FRFir7$$\"1qpppppp\\FRF\\sF<-F16$7$7$$\"1I
IIIIIIbFRFir7$$\"1qppppppaFRF\\sF<-F16$7$7$$\"1IIIIIIIgFRFir7$$\"1qppp
pppfFRF\\sF<-F16$7$7$$\"1IIIIIIIlFRFir7$$\"1qppppppkFRF\\sF<-F16$7$7$$
\"1IIIIIIIqFRFir7$$\"1qpppppppFRF\\sF<-F16$7$7$$\"1IIIIIIIvFRFir7$$\"1
qppppppuFRF\\sF<-F16$7$7$$\"1JIIIIII!)FRFir7$$\"1qppppppzFRF\\sF<-F16$
7$7$$Fa^lFj[lFbs7$$!1LLLLLL$3#Fj[lFesF<-F16$7$7$$\"1MLLLLL3_F7Fbs7$$\"
1nmmmmm\"z%F7FesF<-F16$7$7$$\"1LLLLL$3-\"FRFbs7$$\"1mmmmmm\"z*F7FesF<-
F16$7$7$$\"1LLLLL$3_\"FRFbs7$$\"1nmmmm;z9FRFesF<-F16$7$7$$\"1LLLLL$3-#
FRFbs7$$\"1nmmmm;z>FRFesF<-F16$7$7$$\"1LLLLL$3_#FRFbs7$$\"1nmmmm;zCFRF
esF<-F16$7$7$$\"1LLLLL$3-$FRFbs7$$\"1nmmmm;zHFRFesF<-F16$7$7$$\"1LLLLL
$3_$FRFbs7$$\"1nmmmm;zMFRFesF<-F16$7$7$$\"1LLLLL$3-%FRFbs7$$\"1nmmmm;z
RFRFesF<-F16$7$7$$\"1LLLLL$3_%FRFbs7$$\"1nmmmm;zWFRFesF<-F16$7$7$$\"1L
LLLL$3-&FRFbs7$$\"1nmmmm;z\\FRFesF<-F16$7$7$$\"1LLLLL$3_&FRFbs7$$\"1nm
mmm;zaFRFesF<-F16$7$7$$\"1LLLLL$3-'FRFbs7$$\"1nmmmm;zfFRFesF<-F16$7$7$
$\"1LLLLL$3_'FRFbs7$$\"1nmmmm;zkFRFesF<-F16$7$7$$\"1LLLLL$3-(FRFbs7$$
\"1nmmmm;zpFRFesF<-F16$7$7$$\"1LLLLL$3_(FRFbs7$$\"1nmmmm;zuFRFesF<-F16
$7$7$$\"1MLLLL$3-)FRFbs7$$\"1nmmmm;zzFRFesF<" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
184 "We can see from the direction field that the model predicts that \+
the population levels off at 4. Can you verify that in general the mod
el predicts the population leveling off at a/b ? " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Maple is then used to sol
ve the equation symbolically." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "lo
gsol := dsolve(\{diff(P(t),t)=logeq(t,P(t)), P('t0')='P0'\}, P(t));\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'logsolG/-%\"PG6#%\"tG*&%#a|irG
\"\"\",&%#b|irG\"\"\"*&*(-%$expG6#,$*&F+F/F)F/!\"\"F/-F36#*&%#t0GF/F+F
,F/,&F+F7*&%#P0GF/F.F/F/F/F,F>!\"\"F7F?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 76 "We next verify that this solution approaches a/b for t ap
proaching infinity." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "limit(rhs(lo
gsol),t=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#a|irG\"\"\"%
#b|irG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "What we are after
 next is a way to find the values of a and b based on the known popula
tion values.  To motivate the steps we must take to do this we first r
ecall how we find the growth rate for the simpler case of exponential \+
growth." }}{PARA 0 "" 0 "" {TEXT -1 161 "In the exponential growth mod
el, (dP)/(dt) = kP(t), we can find the growth rate k, by using two act
ual values from the population, P(t0)=P0 and P(t0+delta) = P1." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "expsol:=dsolve(\{diff(P(t),t)=k*P(t
),P('t0')='P0'\},P(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'expsolG/
-%\"PG6#%\"tG*&*&%#P0G\"\"\"-%$expG6#*&%\"kG\"\"\"F)F-F-F3-F/6#*&F2F-%
#t0GF-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(subs(P
(t)='P1',t=t0+'delta',expsol),\{k\});" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#<#/%\"kG*&-%#lnG6#*&%#P1G\"\"\"%#P0G!\"\"F,%&deltaGF." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 180 "For the case of the logistics population
 model we need three equally spaced values,P(t0), P(t0+delta), P(t0+2*
delta) and we get two equations to solve for the two unknowns a and b.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "eq:=\{subs(P(t)=P1,t=t0+delta,l
ogsol), subs(P(t)=P2,P0=P1,t0=t0+delta, t=t0+2*delta,logsol)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG<$/%#P2G*&%#a|irG\"\"\",&%#b|irG
\"\"\"*&*(-%$expG6#,$*&F)F-,&%#t0GF-%&deltaG\"\"#F-!\"\"F--F16#*&F)F*,
&F6F-F7F-F-F-,&F)F9*&%#P1GF-F,F-F-F-F*F@!\"\"F9FA/F@*&F)F*,&F,F-*&*(-F
16#,$F<F9F--F16#*&F6F-F)F*F-,&F)F9*&%#P0GF-F,F*F-F-F*FOFAF9FA" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "absol:=solve(eq,\{a,b\});" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&absolG<$/%#b|irG*&*&-%#lnG6#*&*&%#
P0G\"\"\",&%#P1GF0%#P2G!\"\"F0\"\"\"*&F3\"\"\",&F2F4F/F0\"\"\"!\"\"F0,
&*&F/F5F3F0F0*$)F2\"\"#F5F4F0F5*(%&deltaG\"\"\",(*&F3F5F2F0F0F<!\"#*&F
2F5F/F5F0\"\"\"F2\"\"\"F:/%#a|irG,$*&F*F5FAF:F4" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 131 "To find a and b for the U.S. population, we use thr
ee values from the table of U.S. populations for the years 1790, 1850,
 and 1910." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "absol1 := evalf(subs
(P0=population[1790],P1=population[1850], P2=population[1910], delta=6
0, t0 = 1790, absol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'absol1G<$
/%#a|irG$\"+U$HQ8$!#6/%#b|irG$\"+Q]Q)e\"!#>" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 149 "We then use these values of a and b to plot the popula
tion predicted by the logistics model.  We also include a plot of the \+
actual population values." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "plot(
\{[seq([1790+i*10,population[1790+i*10]],i=0..20)], subs(op(absol1),t0
=1790,P0=population[1790],rhs(logsol))\}, t=1790..2100,0..3*10^8);" }}
{PARA 13 "" 1 "" {GLPLOT2D 368 189 189 {PLOTDATA 2 "6&-%'CURVESG6$7S7$
$\"%!z\"\"\"!$\"1$y[m+9#HR!\"*7$$\"1n;z<rv'z\"!#7$\"1`vSpe?L[F-7$$\"1$
3_)\\kj-=F1$\"18ltySP#y&F-7$$\"1nTlt$[#4=F1$\"1\\,,kw+mqF-7$$\"1n\"zb7
/f\"=F1$\"1wbvp#*4L')F-7$$\"1$3xQCGD#=F1$\"1]ujOF'=0\"!\")7$$\"1<a[\\'
p'G=F1$\"1\\%)z`H$3E\"FH7$$\"1](o#>(G]$=F1$\"1m'omJ%\\<:FH7$$\"1<aLy_g
T=F1$\"1[jW!Q\\D$=FH7$$\"1]PHY2;[=F1$\"1W8c)p+Q?#FH7$$\"1L$e3&Q!\\&=F1
$\"1p\"f!HRk_EFH7$$\"1n\"H5=V3'=F1$\"101))>&y(4JFH7$$\"1+]2g%Hv'=F1$\"
1f,v^ny)p$FH7$$\"1+]i$>VU(=F1$\"13)*)[8]HP%FH7$$\"1+]#*>Jr!)=F1$\"10k6
E/r,^FH7$$\"1<a$GV)e')=F1$\"1%oLpG/%GeFH7$$\"1L$e\"[Zd$*=F1$\"1#e%=Txu
lnFH7$$\"1LLjtI\\**=F1$\"1g\\O@4_7wFH7$$\"1](=J\\xj!>F1$\"1h'=h'QdU')F
H7$$\"1LL)3PrC\">F1$\"1h5xqNHy&*FH7$$\"1](=1Kd\">>F1$\"1ZR')ff-h5!\"(7
$$\"1]i&e#R_D>F1$\"1H#yHECy:\"F^r7$$\"1L3_5o;K>F1$\"1Qo3ISHb7F^r7$$\"1
$3FB0n#Q>F1$\"1g'RCT=,M\"F^r7$$\"1n\"Hs)p%[%>F1$\"10&[l91`U\"F^r7$$\"1
<aL!p\"o^>F1$\"1(HBV)***e]\"F^r7$$\"1]7[>8jd>F1$\"1UomWY3p:F^r7$$\"1nT
&>3dS'>F1$\"1b%*o\")=%*H;F^r7$$\"1++NBbpq>F1$\"1$pwi,l\\o\"F^r7$$\"1+v
8V**=x>F1$\"1'*Rh='4:t\"F^r7$$\"1](=#GOZ$)>F1$\"1!>;SfL-x\"F^r7$$\"1+D
;*e]/*>F1$\"17O)y]$p1=F^r7$$\"1LL)))p>n*>F1$\"1vhB'HmU$=F^r7$$\"1+]i6L
T.?F1$\"1U!Q0!e**e=F^r7$$\"1<aj>(y%4?F1$\"19j\\mGxx=F^r7$$\"1+]n7)4h,#
F1$\"1+quB!G\\*=F^r7$$\"1$3_W:\\B-#F1$\"1'\\g%HOL3>F^r7$$\"1]PRk5()G?F
1$\"1];$=.g*>>F^r7$$\"1L$eMUZ_.#F1$\"15)*Gp%[$H>F^r7$$\"1](o/)G#>/#F1$
\"1q#)R5OVP>F^r7$$\"1nm'*Q@N[?F1$\"120!R!=#Q%>F^r7$$\"1n\"zV)p#\\0#F1$
\"1iZ4*3$=\\>F^r7$$\"1$3_nQZ91#F1$\"146Zk^``>F^r7$$\"1++Nf*Qu1#F1$\"1_
\"ouJMo&>F^r7$$\"1LepxfIu?F1$\"1t\">N2P*f>F^r7$$\"1nmw?zW!3#F1$\"1)fc(
)z0A'>F^r7$$\"1](=J^'*p3#F1$\"1&)[$)eK>k>F^r7$$\"1]7e^VE$4#F1$\"1[2C%*
>vl>F^r7$$\"%+@F*$\"1_K`n17n>F^r-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-F$6$7
77$F($\"(9#HRF*7$$\"%+=F*$\"($[3`F*7$$\"%5=F*$\"(\"))RsF*7$$\"%?=F*$\"
(`%Q'*F*7$$\"%I=F*$\")?g'G\"F*7$$\"%S=F*$\")`%pq\"F*7$$\"%]=F*$\")w=>B
F*7$$\"%g=F*$\")@LVJF*7$$\"%q=F*$\")\\%=)RF*7$$\"%!)=F*$\")$yb,&F*7$$
\"%!*=F*$\")9x%H'F*7$$\"%+>F*$\")vX*f(F*7$$\"%5>F*$\")mA(>*F*7$$\"%?>F
*$\"*?1r0\"F*7$$\"%I>F*$\"*Y]xA\"F*7$$\"%S>F*$\"*v#p;8F*7$$\"%]>F*$\"*
)zD8:F*7$$\"%g>F*$\"*vJKz\"F*7$$\"%q>F*$\"*J?I.#F*7$$\"%!)>F*$\"*0eaE#
F*7$$\"%!*>F*$\"*t)4([#F*-F\\[l6&F^[lF*F_[lF*-%+AXESLABELSG6$Q\"t6\"%!
G-%%VIEWG6$;F(Fgz;F*$\"*++++$F*" 1 2 0 1 0 2 9 1 4 2 1.000000 
44.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
21 " Summary & References" }}{PARA 0 "" 0 "" {TEXT -1 318 "Notice that
 the population closely agree with the model up to around 1945 meaning
 that the model accurately predicted the population for about 35 years
 past 1910.  What happened in 1945 is that World War II and the result
ing population boom resulted in the values of the vital population coe
fficients a and b changing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
0 {PARA 5 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 
181 "[1] W. Ellis, E. Lodi, E. Johnson, and D. Schwalbe.  Maple V Flig
ht Manual: Tutorials for Calculus, Linear Algebra and Differential Equ
ations, Books/Cole Publishing Company, (1991)." }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "8 0 0" 8 }{VIEWOPTS 1 1 0 1 1 
1803 }