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{SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 1 " " }{TEXT 258 20 "Predato
r-prey models" }{TEXT -1 1 "\n" }{TEXT 260 79 "by Prof. Matt Miller, D
epartment of Mathematics, University of South Carolina \n" }{TEXT 259 
6 "email:" }{TEXT 261 19 " miller@math.sc.edu" }{TEXT -1 2 " \n" }}
{PARA 269 "" 0 "" {TEXT 257 1 "\n" }{TEXT -1 122 "We use Maple's DEtoo
ls to study solutions of the Lotka-Volterra system and its refinements
 as described in section 4.1 of " }{TEXT 256 19 "Calculus in Context" 
}{TEXT -1 82 " and chapter 6 of E-K.  As you play with the models, kee
p these questions in mind:" }}{PARA 0 "" 0 "" {TEXT -1 822 "\n1.  What
 is the long term behavior of the system?\n2.  In the case of oscillat
ions, what is the period (time interval from peak to peak or trough to
 trough), and what is the amplitude?\n3.  How does changing the initia
l conditions affect your answers to qustions 1 and 2?  \n4.  Does the \+
system have any steady states (equilibria)?  Do these appear to be sta
ble or unstable?\n5.  If there are steady states, are they in any way \+
related to the long term behavior?\n\nWe first consider problem 6 on p
age 190 of CIC; see also E-K pp 218-220, equations (7ab).  Here h repr
esents the hare population and u represents 60 times the lynx populati
on (since the lynx population is numerically much smaller than the har
e population, we scale it up to fit on the same graph).  We are going \+
to work with three different initial conditions. " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 37 "restart: with(plots):  with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "rate_eqn1:= diff(h(t),t)=(0
.1)*h-(0.005)*h*(1/60)*u; rate_eqn2:=diff(u(t),t)=(0.00004)*h*u-(0.04)
*u;\nvars:= [h(t), u(t)];  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*rate
_eqn1G/-%%diffG6$-%\"hG6#%\"tGF,,&F*$\"\"\"!\"\"*&F*F/%\"uGF/$!+MLLL$)
!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*rate_eqn2G/-%%diffG6$-%\"uG6
#%\"tGF,,&*&%\"hG\"\"\"F*F0$\"\"%!\"&F*$!\"%!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%varsG7$-%\"hG6#%\"tG-%\"uGF(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 110 "init1:=[h(0)=2000,u(0)=600]; init2:=[h(0)=200
0,u(0)=1200]; init3:=[h(0)=2000, u(0)=3000]; \ndomain := 0 .. 320;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init1G7$/-%\"hG6#\"\"!\"%+?/-%\"uGF
)\"$+'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init2G7$/-%\"hG6#\"\"!\"%
+?/-%\"uGF)\"%+7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init3G7$/-%\"hG
6#\"\"!\"%+?/-%\"uGF)\"%+I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'domai
nG;\"\"!\"$?$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "\n We plot the \+
hare and lynx populations jointly against time using the first of the \+
given initial conditions.   You should repeat this with the other init
ial conditions.  Get a feeling for the accuracy of the computations by
 changing the step size, and for the long term behavior by changing th
e time interval.  Keep a record of the results with questions 1-5 in m
ind!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "L:= DEplot(\{rate_
eqn1, rate_eqn2\}, vars, domain,\{init1 \}, stepsize=0.5, scene=[t, u]
, arrows=NONE):\nH:= DEplot(\{rate_eqn1, rate_eqn2\}, vars, domain,\{i
nit1 \}, stepsize=0.5, scene=[t, h], arrows=NONE,linecolor=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( \{L,H\} , title = `
Hares and 60 * Lynxes vs. time` );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 341 "Next we plot the hare and lynx populations against one a
nother in what is called a PHASE PORTRAIT.  We do this for three diffe
rent initial conditions.  ***Can you identify which curve goes with wh
ich initial condition?  How is the independent variable t showing up i
n these pictures?  (Hint: try it again with time interval t = 0 .. 20.
) ***" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "DEplot(\{rate_eqn
1, rate_eqn2\}, vars, t= 0 .. 160, \{init1, init2, init3,[h(0)=3000,u(
0)=50]\}, stepsize=0.5, scene=[h,u], title=`Hares vs. 60 * Lynxes for \+
t = 0 .. 160`, arrows=NONE,linecolor=[blue,green,black,red]);" }}
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\"1]e&p)>&[q\"F07$Fbdhl7$$\"1)HZK2!RazF3$\"1U&p\\J\"yt;F07$Fhdhl7$$\"1
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7$$\"1yd<JBqRvF3$\"142m3Lk$e\"F07$Fjehl7$$\"1n2NQ)RYU(F3$\"1$*e;@%4Yb
\"F07$F`fhl7$$\"1WdG;t5?tF3$\"1W%>=Lug_\"F07$Fffhl7$$\"1ku%z$zbDsF3$\"
11k!z>L!)\\\"F07$F\\ghl7$$\"1p(4W(\\\\SrF3$\"1+.o54[q9F07$Fbghl7$$\"1y
dcUwYkqF3$\"1'zg<87MW\"F07$Fhghl7$$\"1qI$4Pnq*pF3$\"1-#))eK@oT\"F07$F^
hhl7$$\"1q&H1WCz$pF3$\"1z\\S!z-2R\"F07$Fdhhl7$$\"1p`Lj\\q')oF3$\"1K]z
\"o]]O\"F07$Fjhhl7$$\"1W#eA;3J%oF3$\"1QkdR!f)R8F07$F`ihl7$$\"1u#\\e#R'
o!oF3$\"1#47y!=7:8F07$Ffihl7$$\"1uSK21txnF3$\"1'*\\ZaG$3H\"F07$F\\jhl7
$$\"1\\I)f/$\\bnF3$\"1G-))))f)pE\"F07$Fbjhl7$$\"1BN$*)yg*RnF3$\"1VOPx
\\dV7F07$Fhjhl7$$\"1Pa8%\\m4t'F3$\"1\"yZub$f?7F07$F^[il7$$\"1MTv(\\k$G
nF3$\"1()zV]a.)>\"F07$Fd[il7$$\"1[\"Q1`H?t'F3$\"1)>OHP%*e<\"F07$Fj[il7
$$\"1>JPqb&=u'F3$\"1[f/ZS;a6F07$F`\\il7$$\"1]!=o\"[vdnF3$\"1b(*e4#QG8
\"F07$Ff\\il7$$\"1\\wK!yc'znF3$\"1\\*o.i5>6\"F07$F\\]il7$$\"1!QAIZ2v!o
F3$\"1=hcp]P\"4\"F07$Fb]il7$$\"1!>-Xpo7%oF3$\"1i9P%QD72\"F0-F\\dr6&F^d
rF_drF*F*FbdrFfdr-%%VIEWG6$;$!+\"Gie&[!\"($\"+#>55-\"!\"&;$!+rx)zu#Fe^
il$\"+*\\YMf'!\"'-%+AXESLABELSG6$Q\"h6\"Q\"uFc_il-%&TITLEG6#%GHares~vs
.~60~*~Lynxes~for~t~=~0~..~160G" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "What is
 the significance of the next calculation?  (Hint: try using these val
ues of h and u as initial conditions.)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "equil:= solve( \{rhs(rate_eqn1), rhs(rate_eqn2)\}, \{
h , u \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equilG6$<$/%\"uG\"\"!
/%\"hGF)<$/F+$\"%+5F)/F($\"+++++7!\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 92 "\n***Discuss the answers to questions 1-5 above in light \+
of your examination of the model.***" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 489 " Next we study solutions of th
e Lotka-Volterra system of CIC pp 184-186; see also E-K pp 220-222 equ
ations (8ab).  In this model the prey is assumed to grow logistically \+
in the absence of any predators.  Can you see how the rate equations h
ave been changed from the original L-V model to incorporate this assum
ption?  This time h represents the hare (rabbit) population and u repr
esents 100 times the lynx (fox) population.   We are going to work wit
h three different initial conditions.  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 141 "rate_eq1:= diff(h(t),t) = (0.1)*h-.00001*h^2-(0.005)
*h*(1/100)*u ;\nrate_eq2:= diff(u(t),t) = (0.00004)*h*u-(0.04)*u  ;\nv
ars:= [h(t), u(t)];  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rate_eq1G/
-%%diffG6$-%\"hG6#%\"tGF,,(F*$\"\"\"!\"\"*$)F*\"\"#\"\"\"$F0!\"&*&F*F/
%\"uGF/$!+++++]!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rate_eq2G/-%%
diffG6$-%\"uG6#%\"tGF,,&*&%\"hG\"\"\"F*F0$\"\"%!\"&F*$!\"%!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG7$-%\"hG6#%\"tG-%\"uGF(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "init1:=[h(0)=2000,u(0)=500]
; init2:=[h(0)=2000,u(0)=1000]; init3:=[h(0)=2000,u(0)=5000]; \ndomain
:= 0 .. 380;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init1G7$/-%\"hG6#\"
\"!\"%+?/-%\"uGF)\"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init2G7$/
-%\"hG6#\"\"!\"%+?/-%\"uGF)\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&init3G7$/-%\"hG6#\"\"!\"%+?/-%\"uGF)\"%+]" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'domainG;\"\"!\"$!Q" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 237 "\n First we plot the hare and lynx populations jointly against
 time using the first of the given initial conditions.   As above you \+
should repeat this with the other initial conditions, different time i
ntervals, different step sizes, etc." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 212 "L:= DEplot([rate_eq1, rate_eq2], vars, domain,\{init1 \},stepsi
ze=0.5, scene=[t, u], arrows=NONE):\nH:= DEplot([ rate_eq1, rate_eq2],
 vars, domain, \{init1 \}, stepsize=0.5, scene=[t, h], arrows=NONE, li
necolor=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( \+
\{L,H\}, title = `Hares and 100 * Lynxes vs. time` );" }}{PARA 13 "" 
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0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 366 "\nNext we plot the hare and
 lynx populations against one another in what is called a PHASE PORTRA
IT.  We do this for two different initial conditions.  Can you identif
y which curve goes with which initial condition?  How is the independe
nt variable t showing up in these pictures?  (Hint: try it again with \+
time interval t = 0 .. 20, or use the option arrows = SLIM.)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 176 "DEplot([rate_eq1, rate_eq2], vars, domai
n,\{init2, init3 \}, stepsize=0.5, scene=[h,u], title = `Hares vs. 100
 * Lynxes for t = 0 .. 320`, arrows= SLIM,linecolor=[blue,black] ) ;" 
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Fdc_o7$$\"1*f()fD/d!oFb`]m$\"1R:Mibs/>Fci[m7$Fjc_o7$$\"1@k(pc-)*z'Fb`]
m$\"16jG2Se#*=Fci[m-Fbf]n6&Fdf]nFgh[mFgh[mFgh[mFgf]nF[g]n-%%VIEWG6$;$!
+f7$H])F*$\"+Te+[KFis;$\"+%z3@!>F-$\"+#*)H)3cFis-%+AXESLABELSG6$Q\"h6
\"Q\"uFhe_o-%&TITLEG6#%HHares~vs.~100~*~Lynxes~for~t~=~0~..~320G" 1 2 
0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 56 "equil:= solve( \{rhs(rate_eq1), rhs(rate_eq2)
\}, \{h , u\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equilG6%<$/%\"uG
$\"\"!F*/%\"hGF)<$/F,$\"&++\"F*F'<$/F,$\"%+5F*/F($\"%+=F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 95 "\nWhat is the significance of this last c
alculation?  ***Answer questions 1-5 for this model.***" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 494 "Finally \+
we study solutions of the May system (problem 5 on page 189 of CIC; se
e also E-K page 265 problem 32 where the predators are whales and the \+
prey are krill).   Here x is measured in units of \"hectohares\" (i.e.
, the number of hares in units of 100) and y is the number of lynxes. \+
 Choose a variety of initial conditions, time intervals, stepsizes, an
d so forth.  ***Are there any QUALITATIVE similarities and/or differen
ces that you notice between the May model and the two L-V models?***" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eq1:=diff( x(t), t) = 0.6
 * x *(1 -(x / 10)) - 0.5 * x * y /(x + 1);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "eq2:=diff( y(t), t) = 0.1 * y * ( 1 - y / (2 * x) );
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:= [x(t), y(t)];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$-%\"xG6#%\"tGF,,&*&F*\"\"
\",&F/F/F*#!\"\"\"#5F/$\"\"'F2*&*&F*\"\"\"%\"yGF/F8,&F*F/F/F/!\"\"$!\"
&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/-%%diffG6$-%\"yG6#%\"tG
F,,$*&F*\"\"\",&F/F/*&F*\"\"\"%\"xG!\"\"#!\"\"\"\"#F/$F/F6" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%varsG7$-%\"xG6#%\"tG-%\"yGF(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "init1:=[0, 10, 10]; init2:=[0, 10, \+
15]; init3:=[0, 10, 5]; domain:= 0 .. 120;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&init1G7%\"\"!\"#5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&init2G7%\"\"!\"#5\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init
3G7%\"\"!\"#5\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'domainG;\"\"!
\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "X:= DEplot([eq1, \+
eq2], vars, domain, \{init1 \}, stepsize=0.5, scene=[t,x], linecolor=b
lue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Y:= DEplot([eq1, eq2], var
s, domain, \{init1 \}, stepsize=0.5, scene=[t,y], linecolor=green):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display( \{X, Y\}, title =
 `May model: Rabbits/100 and Foxes vs. time` );" }}{PARA 13 "" 1 "" 
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"" {TEXT -1 286 "\nWhat is the significance of the last calculation?  \+
*** In Biology 301 you may have heard the term \"limit cycle\".  Can y
ou explain where this term comes from?  Predict what will happen if yo
u use initial conditions close to, but not at, the equilibrium values.
  Test your prediction!***" }}{PARA 0 "" 0 "" {TEXT -1 816 "\n***At la
st it's time to judge the models.  What sort of field measurements wou
ld you want to have in order to choose one over another?  Are there an
y purely mathematical features of the predictions of the models that m
ight help?  (One can critique the models on the basis of unreasonable \+
assumptions that might go into their construction--this is a separate \+
matter.)  For example, you have surely observed that each of the three
 models (L-V with unbounded prey growth, L-V with bounded prey growth,
 and May) exhibits a different possibility for long term behavior.  Wh
y is this considered to be such an important aspect of the model?  Wha
t about sensitivity to initial conditions--how does long term behaviou
r depend on initial conditions, and what does this mean in terms of ac
tual observation of populations?***  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "11 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }