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{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
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" 0 "" {TEXT -1 69 "   This worksheet analyzes a variation of the pred
ator-prey model for" }}{PARA 0 "" 0 "" {TEXT -1 72 "   both the discre
te and continuous cases. We look at specific values at" }}{PARA 0 "" 
0 "" {TEXT -1 11 "   the end." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
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" {MPLTEXT 1 0 21 "F := a*m-b*m*w-c*m^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "G :=  -d*w^2+b*m*w;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "Ff := unapply(F,m,w);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "Gf := unapply(G,m,w);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 64 "  Get the derivatives for the entries of the stabi
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" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
Fm := diff(F,m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Fw := d
iff(F,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Gm := diff(G,m
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Gw := diff(G,w);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "Fmf := unapply(Fm,m,w);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "Fwf := unapply(Fw,m,w);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "Gmf := unapply(Gm,m,w);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "Gwf := unapply(Gw,m,w);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 38 "  Determine the equilibrium solutions." }}{PARA 
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}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "alpha := (solve(\{Ff(x,y)
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ta := (solve(\{F,G\},\{m,w\}));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "alpha;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "u[1] := 0; u[2] := a/c; u[3]
 := d*a/(b^2+c*d); #d/b; #d*a/(b^2+c*d);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "v[1] := 0; v[2] := 0; v[3] := b*a/(b^2+c*d); #1/b^2*(
b*a-c*d); #b*a/(b^2+c*d);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
53 "   Do the stability analysis for the continuous case." }}{PARA 0 "
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for j from 1 by 1 to 3 do M[
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{TEXT -1 28 "   Set up the discrete case." }}{PARA 0 "" 0 "" {TEXT -1 
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "  Get the derivatives.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "fm := diff(f,m);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "fw := diff(f,w);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "gm := diff(g,m);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "gw := diff(g,w);" }}}{EXCHG {PARA 0 "> " 0 "" 
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Print out the equilibriums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
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" 0 "" {TEXT -1 34 "   Input some values for examples." }}{PARA 0 "" 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a := 0.025; b := 0.003; c :=
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;u[2];u[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v[1];v[2];v[
3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "m[0] := 2; w[0] := 5;" }}}{EXCHG {PARA 0 "> \+
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{PARA 0 "" 0 "" {TEXT -1 35 "   Do the continuous example first." }}
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0 141 "predprey := dsolve(\{diff(x(t),t)=Ff(x(t),y(t)),diff(y(t),t)=Gf
(x(t),y(t)),x(0)=m[0],y(0)=w[0]\},\{x(t),y(t)\},type=numeric,method=ta
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "tym := 400; nump := 500;" }}}
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0..tym,numpoints=nump);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "
odeplot(predprey,[t,y(t)],t=0..tym,numpoints=nump);" }}}{EXCHG {PARA 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "trace(M[1]); det(M[1]);" }}}
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" 0 "" {TEXT -1 27 "  Now do the discrete case." }}{PARA 0 "" 0 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for j from 0 by 1 to 100*tym do" }}
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