{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 151 "Here we co
nsider the diffusive Malthus equation with homogeneous Dirichlet bound
ary condition. From our calculation, the critical growth rate is a=Pi^
2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart:with(PDEtools);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 80 "First we try a=5, t
hus it is below critical rate, then the population collapses." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eq:=diff(u(t,x),t)=diff(u(t,x),x,x)
+5*u(t,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "IBC:=\{u(0,x)
=sin(Pi*x), u(t,0)=0, u(t,1)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pds:=pd
solve(eq, IBC, numeric, spacestep=1/40);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "pds:-animate(t=2,frames=100);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT -1 78 "Next we try a=15, thta is over the cr
itical rate, and the population explodes." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "eq1:=diff(u(t,x),t)=diff(u(t,x),x,x)+15*u(t,x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "IBC:=\{u(0,x)=sin(Pi*x), u(t
,0)=0, u(t,1)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "pds:=pdsolve(eq1, IBC, numer
ic, spacestep=1/40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "pds
:-animate(t=2,frames=100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{MARK "6 0 1" 78 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }