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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart;  with(DEtoo
ls): with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "COMPETITION \+
MODELS" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq1:= diff(P(t),t) = r1*P
(t)-k1*(P(t))^2 -a1*P(t)*Q(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq2:= diff(Q(t),t) = r2
*Q(t)-k2*(Q(t))^2 -a2*P(t)*Q(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "vars:= [P(t),Q(t)] ; # \+
what depends on what?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "r1:=1.0; r2:=.75; k1:=1.0; k
2:=1.0; a1:=1.0; a2:=.5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "init1:= [P(0)=1.0,Q(0)=1.0]; # initial values" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "DEplot([diff(P(t),t)=r1*P(t
)-k1*(P(t))^2 -a1*P(t)*Q(t),diff(Q(t),t)=r2*Q(t)-k2*(Q(t))^2 -a2*P(t)*
Q(t)], vars,t=0..50,number=2, [init1], stepsize = 0.1, scene = [t, P],
 arrows = NONE,title=`time plot for P`);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 208 "DEplot([diff(P(t),t)=r1*P(t)-k1*(P(t))^2 -a1*P(t)*Q(
t),diff(Q(t),t)=r2*Q(t)-k2*(Q(t))^2 -a2*P(t)*Q(t)], vars,t=0..50,numbe
r=2, [init1], stepsize = 0.1, scene = [t, Q], arrows = NONE, title=`ti
me plot for Q`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "DEplot([diff(P
(t),t)=r1*P(t)-k1*(P(t))^2 -a1*P(t)*Q(t),diff(Q(t),t)=r2*Q(t)-k2*(Q(t)
)^2 -a2*P(t)*Q(t)], vars,t=0..50,number=2, [init1], stepsize = 0.1, sc
ene = [P, Q], arrows = NONE,title=`phase plane for P vs Q`);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "DEplot([diff(P(t),t)=r1*P(t
)-k1*(P(t))^2 -a1*P(t)*Q(t),diff(Q(t),t)=r2*Q(t)-k2*(Q(t))^2 -a2*P(t)*
Q(t)], vars, t=0..100,P=0..1,Q=0..1,arrows=LARGE,title=`competition mo
del direction field`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "DEplot([
diff(P(t),t)=r1*P(t)-k1*(P(t))^2 -a1*P(t)*Q(t),diff(Q(t),t)=r2*Q(t)-k2
*(Q(t))^2 -a2*P(t)*Q(t)], vars, t=0..10,[[P(0)=.2,Q(0)=.1],[P(0)=.8,Q(
0)=.8]],stepsize=.01,title=`competition model direction field with tra
jectories`);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "" "6#%#%?G" }}
{PARA 0 "" 0 "" {TEXT -1 40 "PREDATOR PREY EQUATIONS AND PHASE PLANES
" }}{PARA 0 "" 0 "" {TEXT -1 25 "PLUS FANCY MAPLE GRAPHICS" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 147 "DEplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(
t),t)=.3*y(t)*(x(t)-1)],\n[x(t),y(t)],t=-2..2,x=-1..2,y=-1..2,arrows=L
ARGE,\ntitle=`Lotka-Volterra model`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 169 "DEplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),t)=.3*y(t)*(x(t)-1
)],\n[x(t),y(t)],t=-7..7,[[x(0)=1.2,y(0)=1.2],[x(0)=1,y(0)=.7]],stepsi
ze=.2,\ntitle=`Lotka-Volterra model`);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 202 "DEplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),t)=.3*y(t)*(x(t)
-1)], [x(t),y(t)],t=0..50,number=2, [[x(0)=1.2,y(0)=1.2]], stepsize = \+
0.1, scene = [t, x], arrows = NONE, title=`predator  -prey eqns-prey`)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "DEplot([diff(x(t),t)=x(t)*(1-
y(t)),diff(y(t),t)=.3*y(t)*(x(t)-1)], [x(t),y(t)],t=0..50,number=2, [[
x(0)=1.2,y(0)=1.2]], stepsize = 0.1, scene = [t, y], arrows = NONE, ti
tle=`predator  -prey eqns-predator`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "DEplot([diff(x(t),t)=x(t)
*(1-y(t)),diff(y(t),t)=.3*y(t)*(x(t)-1)], [x(t),y(t)],t=0..50,number=2
, [[x(0)=1.2,y(0)=1.2]], stepsize = 0.1, scene = [x, y], arrows = NONE
, title=`predator  -prey eqns-phase portrait`);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 443 "DEplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),
t)=.3*y(t)*(x(t)-1)],\n[x(t),y(t)],t=-2..2,x=-1..2,y=-1..2,arrows=LARG
E,\ntitle=`Lotka-Volterra model`,color=[.3*y(t)*(x(t)-1),x(t)*(1-y(t))
,.1]);\nDEplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),t)=.3*y(t)*(x(t)-
1)],\n[x(t),y(t)],t=-7..7,[[x(0)=1.2,y(0)=1.2],[x(0)=1,y(0)=.7]],steps
ize=.2,\ntitle=`Lotka-Volterra model`,color=[.3*y(t)*(x(t)-1),x(t)*(1-
y(t)),.1],\nlinecolor=t/2,arrows=MEDIUM,method=rkf45);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "THI
S IS A SIMPLE LINEAR 3-DIMENSIONAL SYSTEM. One can generate" }}{PARA 
0 "" 0 "" {TEXT -1 59 "phase portraits between any two of the variable
s at a time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "DEplot([D(
x)(t)=y(t)-z(t),D(y)(t)=z(t)-x(t),D(z)(t)=x(t)-y(t)*2],\n[x(t),y(t),z(
t)],t=-2..2,[[x(0)=1,y(0)=0,z(0)=2]],stepsize=.05,\nscene=[z(t),x(t)],
linecolour=sin(t*Pi/2),method=classical[foreuler]);\n" }}}}{MARK "14" 
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