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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 74 "Worksheet on Population Dynamics. Logistic Growth and  Cr
itical Threshold." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 127 "First Model has to do with \+
logistic growth. The saturation level is k = 100000 and the intrinsic \+
growth rate is given by r = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "PD1:= diff(P(t),t) =  (1 - P
(t)/100000)*P(t) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DEplo
t(PD1, P(t), t=0..5, P=0..120000, arrows=thin);" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 83 " We can clearly see an \"asymptotically stable\" e
quilibrium solution, P(t) = 100000." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "DEplot(PD1, P(t), t=0..5, \+
P=-60000..120000, arrows=thin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 93 "Enlarging the y axis range, we can
 also see another \"unstable\" equilibrium solution P(t) = 0." }}
{PARA 256 "" 0 "" {TEXT -1 65 "Next, we will see several solutions for
 different initial values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "DEplot(PD1, P(t), t=0..5, P=0..180
000,\{[0,120000],[0,80000],[0,60000],[0,40000],[0,20000],[0,160000]\},
 arrows=thin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 59 "This new model corresponds to a critical threshold p
roblem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "PD2:= diff(P(t),t) =  -(1 - P(t)/20000)*P(t) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "DEplot(PD2, P(t), t=0..5, P
=0..100000,\{[0,40000],[0,30000],[0,22000],[0,20000],[0,18000], [0,150
00]\}, arrows=thin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 132 "Is clear from this model  that there exists an
 unstable equilibrium solution P(t) = 20000 and an asymptotically stab
le one P (t)= 0." }}{PARA 0 "" 0 "" {TEXT -1 180 "Our next model combi
nes both properties studied above. It means a logistic growth combined
 with a critical threshold, which seems to be the population dynamics \+
for certain species." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "PD3:= diff(P(t),t) =  -(1 - P(t)/20000)*(
1-P(t)/100000)*P(t) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "D
Eplot(PD3, P(t), t=0..5, P=0..140000,\{[0,60000],[0,80000],[0,120000],
[0,140000],[0,40000],[0,30000],[0,22000],[0,20000],[0,18000], [0,15000
]\}, arrows=thin, stepsize=0.1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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