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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 256 11 "test.mws   " }{TEXT -1 21 " Predator-prey models
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: with(DEtools):
 with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq1:=diff
( V(t), t) = a* V *(1 -(V/ K)) - b* H * V /(V+ e);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "eq2:=diff( H(t), t) = c * V *H/(e+V) -d* H ;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:= [V(t), H(t)];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "equil:= solve( \{rhs(eq1), rhs(eq2)
\}, \{V,H\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "\nHere is the al
ternate method for computing the coefficient matrix using the jacobian
 command." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "J:= jacobian([rhs(eq1)
,rhs(eq2)], [V,H]); # computes the partials and forms the matrix" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "A:= subs(\{V = d*e/(c-d), H
 = -a*e*c*(K*d-K*c+d*e)/b/K/(c^2-2*c*d+d^2)\}, evalm(J));  # same as b
efore" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B:= eigenvals(A); \+
 h:=solve(rhs(eq1), H);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
params:=a=0.5, b=.001, c=.5, d=.4, K=500, e=100;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "subs(params, \{B[1], B[2]\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "bar:= subs(params, equil[3]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Vbar:=subs(bar, V);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "P1:= plot(subs(params,h), V=
0..500, thickness=3,color=black, axesfont=[COURIER,BOLD,12]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P2:= plot([[Vbar,0], [Vbar, \+
90000]], style=line, thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "plots[display](\{P1,P2\});" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "para
ms:=a=0.5, b=.001, c=.5, d=.4, K=500, e=25;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "subs(params, \{B[1], B[2]\});" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "bar:= subs(params, equil[3]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Vbar:=subs(bar, V);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "P1:= plot(subs(params,h), V=0..500,
 thickness=3,color=black, axesfont=[COURIER,BOLD,12]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P2:= plot([[Vbar,0], [Vbar, 75000]]
, style=line, thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "plots[display](\{P1,P2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{MARK "24 0" 1 }
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