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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 17 "Examining models " }}
{PARA 18 "" 0 "" {TEXT -1 3 "and" }}{PARA 18 "" 0 "" {TEXT -1 20 "nume
rical solutions." }}{PARA 19 "" 0 "" {TEXT -1 24 "Math 374    Winter  \+
1999" }}{PARA 19 "" 0 "" {TEXT -1 11 "Jay Treiman" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 46 "One of the most important uses of computers in" }}
{PARA 0 "" 0 "" {TEXT -1 49 "differential equations is graphically vie
wing the" }}{PARA 0 "" 0 "" {TEXT -1 52 "behavior of a differential eq
uation.  With this tool" }}{PARA 0 "" 0 "" {TEXT -1 44 "one can often \+
tell if a model is reasonable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "Load the " }{HYPERLNK 17 "DEtools" 2 "Deto
ols" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT 
-1 23 " packages for plotting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart;\nwith(DEtools):with(plots)
:" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "The logistic population mod
el" }}{PARA 3 "" 0 "" {TEXT -1 16 "with harvesting." }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 21 "The logistic equation" }}{PARA 0 "" 0 "" {TEXT 
-1 51 "The basic logistic population differential equation" }}{PARA 0 
"" 0 "" {TEXT -1 2 "is" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+
 " }{XPPEDIT 18 0 "diff(P,t) = k*P*(M-P);" "6#/-%%diffG6$%\"PG%\"tG*(%
\"kG\"\"\"F'F+,&%\"MGF+F'!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 14 "Here the term " }{XPPEDIT 18 0 "kMP;" "6#%$kMPG" }{TEXT 
-1 34 " on the right side represents the " }}{PARA 0 "" 0 "" {TEXT -1 
49 "reproduction of the population.  The other term, " }{XPPEDIT 18 0 
"-k*P^2;" "6#,$*&%\"kG\"\"\"*$)%\"PG\"\"#F&F&!\"\"" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 51 "represents a limit on the number of membe
rs of the " }}{PARA 0 "" 0 "" {TEXT -1 56 "population.  It can be inte
rpreted as being proportional" }}{PARA 0 "" 0 "" {TEXT -1 52 "to the n
umber of interactions between members of the" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "population.  This is a crowding term." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Leaving the constants in we get
 the following solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "de1 := diff(P(t),t)=k*M*P(t)-k*P(t)
^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "infolevel[dsolve] :=
 3:\ngen_sol1 := dsolve(de1,P(t));\ninfolevel[dsolve] := 0:" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "One can substit
ute values for " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 6 ", and " }{TEXT -1 26 "_C1 an
d plot the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 69 "Sol_1 := subs(\{_C1=1,M=20,k=.01\},rhs(ge
n_sol1));\nplot(Sol_1,t=0..50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 42 "To see the overall behavior one plots the
 " }}{PARA 0 "" 0 "" {TEXT -1 34 "slope field and several solutions." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 159 "de2 := subs(\{M=20,k=.01\},de1);\nDEplot(de2,P(t),t=0..50,[[P(0
)=10],[P(0)=1],\n   [P(0)=30],[P(0)=50]],P=0..50,color=black,\n   line
color=[red,blue,maroon,brown]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "soln_de2 := dsolve(\{de2,P(0
)=100\},P(t));" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Harvesting" }
}{PARA 0 "" 0 "" {TEXT -1 49 "The addition of harvesting will change t
he model." }}{PARA 0 "" 0 "" {TEXT -1 53 "One subtracts a constant ter
m to represent a constant" }}{PARA 0 "" 0 "" {TEXT -1 58 "rate of harv
est.  This could be a constant yearly harvest " }}{PARA 0 "" 0 "" 
{TEXT -1 59 "of deer.  Many wildlife managers try to keep a herd of de
er" }}{PARA 0 "" 0 "" {TEXT -1 48 "near a constant population using th
is technique." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "Her the constant is 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "de3 := lhs(de2) = rhs(de2)-
1.;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "H
ave Maple try to solve the differential equation." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(de3,P
(t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 
"Now one can plot the slope field and the solutions " }}{PARA 0 "" 0 "
" {TEXT -1 39 "with the same initial values as before." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "DEplot
(de3,P(t),t=0..50,[[P(0)=10],[P(0)=1],\n   [P(0)=30],[P(0)=50]],P=0..5
0,color=black,\n   linecolor=[red,blue,maroon,brown]);" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "It is clear that the
 behavior of the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 12 
"has changed." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Solve a logistic growth population
 model with" }}{PARA 0 "" 0 "" {TEXT -1 45 "harvesting where the P(0)=
100, P(10)=250, and" }}{PARA 0 "" 0 "" {TEXT -1 44 "P(15)=300.  What i
s the limiting population?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Exp
lain how the term  " }{XPPEDIT 18 0 "k*P^2;" "6#*&%\"kG\"\"\"*$)%\"PG
\"\"#F%F%" }{TEXT -1 18 " can represent the" }}{PARA 0 "" 0 "" {TEXT 
-1 57 "number of interactions between members of the population?" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "How do the solutions to these diff
eential equations differ " }}{PARA 0 "" 0 "" {TEXT -1 24 "from exponen
tial growth?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "How do the soluti
ons to these differential equations" }}{PARA 0 "" 0 "" {TEXT -1 52 "di
ffer from what you expect to happen.  Can you give" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "a plausible reason for the differences?" }}}}}{MARK "4" 
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