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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 139 "      Stability of Syst
ems of Equations = file:  prey.mws\n\n   Predator Prey Example directi
on fields, DEplot3d and DEplot with scene\n\nName:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "restart;\nwith(linalg):\nwith(DEtools):" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 53 "Define a list containing the right side
 of the system" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "r := 1: a := 1: m
:= 2: b:= 1:\nf := [r*x-a*x*y,-m*y+b*x*y];" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT -1 22 "Locate the rest points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "soln:=[solve(\{f[1]=0,f[2]=0\},\{x,y\})];" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 29 "Calculate the Jacobian matrix" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "jacf := jacobian(f,[x,y]);" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT -1 54 "Find the community matrices at each of the rest poin
ts" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a1 := subs(soln[1],evalm(jacf
));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a2 := subs(soln[2],e
valm(jacf));" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 49 "Compute the eig
envalues of the community matrices" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "eigenvals(a1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigen
vals(a2);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 64 "What can you concl
ude about the stability of the rest points?\n\n\n" }}{PARA 256 "" 0 "
" {TEXT -1 44 "Direction Field - must specify the constants" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "r := 1: a := 1: m:= 2: b:= 1:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "prey := diff(x(t),t)=r*x(t)-a*x(t)*
y(t), diff(y(t),t)=-m*y(t)+b*x(t)*y(t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "DEplot(\{prey\},[x(t),y(t)],t=0..20,x=0..4,y=0..2,arr
ows=SLIM);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 40 "Numerical Solutio
ns\n  3 dimensional plot" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "DEplot3
d(\{prey\},[x(t),y(t)],t=0..10,\{[0,1,1]\},color=black,\nstepsize=0.1)
;" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 12 "scene =[t,y]" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 83 "DEplot(\{prey\},[x(t),y(t)],t=0..10,\{[0,1,1]
\},color=black,\nstepsize=0.1,scene=[t,y]);" }}}{EXCHG {PARA 256 "" 0 
"" {TEXT -1 13 "scene = [x,y]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "DE
plot(\{prey\},[x(t),y(t)],t=0..10,\{[0,1,1]\},color=black,\nstepsize=0
.1,scene=[x,y]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "scene = [t,
x]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "DEplot(\{prey\},[x(t),y(t)],t
=0..10,\{[0,1,1]\},color=black,\nstepsize=0.1,scene=[t,x]);" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 71 "Investigate the solution of line
ar problem near the interior rest point" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 \+
0" 101 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }