{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" 
-1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 
0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "
" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier
" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 
2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3
" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 260 1 {CSTYLE 
"" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "R3 Font 5" -1 261 1 {CSTYLE "" -1 -1 "Helvetica
" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "R3 Font 6" -1 262 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 
0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font
 7" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 264 1 {CSTYLE 
"" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 9" -1 265 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "R3 Font 10" -1 266 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 
0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font
 11" -1 267 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 
0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 256 15 "volterra.mws   " }{TEXT -1 22 " Predator-prey mod
els\n" }{TEXT 258 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 974 "\n1.  What is
 the long term behavior of the system?\n2.  In the case of oscillation
s, what is the period (time interval from peak to peak or trough to tr
ough), and what is the amplitude?\n3.  How does changing the initial c
onditions affect your answers to qustions 1 and 2?  \n4.  Does the sys
tem have any steady states (equilibria)?  Do these appear to be stable
 or unstable?\n5.  If there are steady states, are they in any way rel
ated to the long term behavior?\n\n We are investigating the Volterra \+
model (equations 5.1 and 5.2 in May).  In the May article, the prey is
 represented by N and the predator by P.  For our example, we consider
 the prey to be arctic hares and the predator to be Canadian lynx.  He
re h represents the hare population and u represents 60 times the lynx
 population (since the lynx population is numerically much smaller tha
n the hare population, we scale it up to fit on the same graph).  We a
re 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)];  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 110 "init1:=[h(0)=2000,u(0)=600]; init2:=[h(0)=2000,u(0)=
1200]; init3:=[h(0)=2000, u(0)=3000]; \ndomain := 0 .. 320;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "\n We plot the hare and lynx popu
lations jointly against time using the first of the given initial cond
itions.   You should repeat this with the other initial conditions.  G
et a feeling for the accuracy of the computations by changing the step
 size, and for the long term behavior by changing the time interval.  \+
Keep a record of the results with questions 1-5 in mind!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "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,\{init1 \}, stepsi
ze=0.5, scene=[t, h], arrows=NONE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "display( \{L,H\} , title = `Hares and 60 * Lynxes vs.
 time` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "\nWhich graph is which?  You may want to
 inset options such as linecolor= or thickness= to distinguish them.  \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
341 "Next we plot the hare and lynx populations against one another in
 what is called a PHASE PORTRAIT.  We do this for three different init
ial conditions.  ***Can you identify which curve goes with which initi
al condition?  How is the independent variable t showing up in these p
ictures?  (Hint: try it again with time interval t = 0 .. 20.) ***" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "DEplot(\{rate_eqn1, rate_e
qn2\}, vars, t= 0 .. 160, \{init1, init2, init3\}, stepsize=0.5, scene
=[h,u], title=`Hares vs. 60 * Lynxes for t = 0 .. 160`, arrows=NONE);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
115 "What is the significance of the next calculation?  (Hint: try usi
ng these values of h and u as initial conditions.)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 59 "equil:= solve( \{rhs(rate_eqn1), rhs(rate_e
qn2)\}, \{h , u \});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "\n***Disc
uss 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 -1 430 " Next we study solutions of the Lotka-Volterra syst
em.  In this model the prey is assumed to grow logistically in the abs
ence of any predators.  Can you see how the rate equations have been c
hanged from the original L-V model to incorporate this assumption?  Th
is time h represents the hare (rabbit) population and u represents 100
 times the lynx (fox) population.   We are going to work with three di
fferent 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  ;\nvars:= [h(t)
, u(t)];  " }}}{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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "
\n First we plot the hare and lynx populations jointly against time us
ing the first of the given initial conditions.   As above you should r
epeat this with the other initial conditions, different time intervals
, different step sizes, etc." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "L:
= DEplot([rate_eq1, rate_eq2], vars, domain,\{init1 \},stepsize=0.5, s
cene=[t, u], arrows=NONE):\nH:= DEplot([ rate_eq1, rate_eq2], vars, do
main, \{init1 \}, stepsize=0.5, scene=[t, h], arrows=NONE):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( \{L,H\}, title = `H
ares and 100 * Lynxes vs. time` );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 366 "\nNext we plot the hare and lynx populations against one anoth
er in what is called a PHASE PORTRAIT.  We do this for two different i
nitial conditions.  Can you identify which curve goes with which initi
al condition?  How is the independent variable t showing up in these p
ictures?  (Hint: try it again with time interval t = 0 .. 20, or use t
he option arrows = SLIM.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "DEplo
t([rate_eq1, rate_eq2], vars, domain,\{init2, init3 \}, stepsize=0.5, \+
scene=[h,u], title = `Hares vs. 100 * Lynxes for t = 0 .. 320`, arrows
= NONE) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "equil:= solve(
 \{rhs(rate_eq1), rhs(rate_eq2)\}, \{h , u\});" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 95 "\nWhat is the significance of this last calculation?
  ***Answer questions 1-5 for this model.***" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 399 "Finally we study solut
ions of the May system (equations 5.6 and 5.7).   Here x is measured i
n 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, and so forth.  ***Are there any QUALITATI
VE similarities and/or differences that you notice between the May mod
el 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)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "init1:=[0
, 10, 10]; init2:=[0, 10, 15]; init3:=[0, 10, 5]; domain:= 0 .. 120;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "X:= DEplot([eq1, eq2], va
rs, domain, \{init1 \}, stepsize=0.5, scene=[t,x], arrows=NONE):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Y:= DEplot([eq1, eq2], vars, domain
, \{init1 \}, stepsize=0.5, scene=[t,y], arrows=NONE):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display( \{X, Y\}, title = `May mod
el: Rabbits/100 and Foxes vs. time` );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 132 "DEplot([eq1, eq2], vars, domain, \{init1, init2, ini
t3\}, stepsize=0.5, scene=[x,y], title = `May model phase portrait`, a
rrows=NONE);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "equil:= sol
ve( \{rhs(eq1), rhs(eq2)\}, \{x, y\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 287 "\nWhat is the significance of the last calculation?  ***
 In Biology 765  you may have heard the term \"limit cycle\".  Can you
 explain where this term comes from?  Predict what will happen if you \+
use initial conditions close to, but not at, the equilibrium values.  \+
Test your prediction!***" }}{PARA 0 "" 0 "" {TEXT -1 816 "\n***At last
 it's time to judge the models.  What sort of field measurements would
 you want to have in order to choose one over another?  Are there any \+
purely mathematical features of the predictions of the models that mig
ht help?  (One can critique the models on the basis of unreasonable as
sumptions that might go into their construction--this is a separate ma
tter.)  For example, you have surely observed that each of the three m
odels (L-V with unbounded prey growth, L-V with bounded prey growth, a
nd May) exhibits a different possibility for long term behavior.  Why \+
is this considered to be such an important aspect of the model?  What \+
about sensitivity to initial conditions--how does long term behaviour \+
depend on initial conditions, and what does this mean in terms of actu
al observation of populations?***  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }