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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 39 "Phase Portrait for a Co
mpetitive System" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart
:\nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Define the differential equatio
n using parameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=2
: b:=1: c:= 1: d:= 3: e:=1: f:=2: g:=2:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 104 "CompDE := [diff(x(t),t) = x(t)*(a-b*x(t)-c*y(t)),\n          \+
     diff(y(t),t) = y(t)*(d-e*x(t)-f*y(t))];" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 120 "Plot the phaseportrait using Runge-Kutta method;  the \+
initial conditions are given in the first line and can be changed." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "IC:=[[x(0)=0.1,y(0)=0.1],[
x(0)=2,y(0)=0.1],[x(0)=0.1,y(0)=1.5],\n[x(0)=3,y(0)=3],[x(0)= 2,y(0)=3
.0],[x(0)= 3,y(0)=2.0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "phasep
ortrait(CompDE, [x(t),y(t)], t=0..7,  \nIC,linecolour=BLUE, x=0..4, y=
0..4, stepsize=0.1,\narrows=NONE, method=classical[rk4], title=`Compet
itive System`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "You could try changing the paramet
ers in the equation to get different examles. " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Plot x
 versus t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=2: b:=1: c:
= 1: d:= 3: e:=1: f:=2: g:=2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "C
ompDE := [diff(x(t),t) = x(t)*(a-b*x(t)-c*y(t)),\n               diff(
y(t),t) = y(t)*(d-e*x(t)-f*y(t)),\n               diff(T(t),t)=1];" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "IC:=[[x(0)=0.1,y(0)=0.1,T(
0)=0],[x(0)=2,y(0)=0.1,T(0)=0],\n    [x(0)=0.1,y(0)=1.5,T(0)=0],[x(0)=
3,y(0)=3,T(0)=0],\n    [x(0)= 2,y(0)=3.0,T(0)=0],[x(0)= 3,y(0)=2.0,T(0
)=0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "DEplot(CompDE, [x(t),y(t
),T(t)], t=0..7,  \nIC,linecolour=BLUE, x=0..4, y=0..4, stepsize=0.1,
\narrows=NONE, method=classical[rk4],scene=[T,x], \ntitle=`Competitive
 System`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Plot the vector fie
ld for   x' = x(2-x-y),  y' = y(3-x-2y)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "fieldplot([x*(2-x-y),y*(3-x-2*y)],  x = 0..4, y=0..4)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 71 "Plot the solutions with several initial conditions u
sing Euler method: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "IC:
=[[x(0)=0.1,y(0)=0.1,T(0)=0],[x(0)=2,y(0)=0.1,T(0)=0],\n   [x(0)=0.1,y
(0)=1.5,T(0)=0],[x(0)=3,y(0)=3,T(0)=0]];\nphaseportrait(CompDE, [x(t),
y(t),T(t)], t=0..7,  \nIC,linecolour=BLUE, x=0..4, y=0..4,\narrows=NON
E,method=classical[foreuler],scene=[T,x], \ntitle=`Competitive System`
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Notice the problem with th
e solution overshooting and crossing another solution.  This is caused
 by using the Euler method and does not happen for the true solutions.
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 9 
}{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }