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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 61 "In this worksheet, we show how to get numerical solutions to " 
}{TEXT 256 20 "systems of equations" }{TEXT -1 183 " and show how to p
lot the different graphs that might be of interest.  This is an exampl
e of competing species (x and y are populations of animals that compet
e for the same resources)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "de1:=diff(x(t),t)=2*x(t)*(1-(1/2)*x(t))-x(t)*y(t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "de2:=diff(y(t),t)=3*y(t)*(
1-(1/3)*y(t))-2*x(t)*y(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
134 "DEplot([de1,de2],[x(t),y(t)],t=-1..6,x=0..4,y=0..4,[[x(0)=0.3,y(0
)=1],[x(0)=2,y(0)=1],[x(0)=2,y(0)=3]],stepsize=0.05,linecolor=black);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "A:=DEplot([de1,de2],[x
(t),y(t)],t=0..15,x=0..4,y=0..4,[[x(0)=2,y(0)=2.5]],stepsize=0.05,scen
e=[t,x],linecolor=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
115 "B:=DEplot([de1,de2],[x(t),y(t)],t=0..15,x=0..4,y=0..4,[[x(0)=2,y(
0)=2.5]],stepsize=0.05,scene=[t,y],linecolor=red):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "display(\{A,B\},title=\"Competing Species\");" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Solve for equilibrium solutions (d
x/dt=0 and dy/dt=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn
s:=\{2*x*(1-(1/2)*x)-x*y=0,3*y*(1-(1/3)*y)-2*x*y=0\};" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(eqns,\{x,y\});" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 279 "The firs
t equilibrium corresponds to both populations dying.  The second corre
sponds to population x taking control, pop. y dying off.  The opposite
 is true for the third case, and in the final case, we have \"Utopia\"
.  Locate these points in the direction field (our first plot)." }}
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