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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 14 "Nonlinear.mws " }
{TEXT -1 61 "   This Maple worksheet is designed to illustrate two pap
ers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "
May, RM 1974.  Biological populations with nonoverlapping generations:
 stable points, stable cycles and chaos.  Science 186: 645-647. " }}
{PARA 0 "" 0 "" {TEXT -1 126 "May, RM & GF Oster. 1976.  Bifurcations \+
and dynamic complexity in simple ecological models.  American Naturali
st 110: 573-599." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 310 "May examined the behavior of a difference equatio
n version of the logistic equation, and found that it exhibited very c
omplex dynamics, depending upon the value of the growth rate `r'.  He \+
found monotonic damping to an equilibrium, damped oscillations, stable
 oscillations, and chaos.  The equation he used was" }}{PARA 260 "" 0 
"" {XPPEDIT 18 0 "N[t+1] = N[t]*exp(r*(1-N[t])" "6#/&%\"NG6#,&%\"tG\"
\"\"F)F)*&&F%6#F(F)-%$expG6#*&%\"rGF),&F)F)&F%6#F(!\"\"F)F)" }}{PARA 
0 "" 0 "" {TEXT -1 196 "This equation can never have negative populati
on sizes, and has a carrying capacity of 1, like the dimensionless log
istic that we developed earlier in the course (look what happens when \+
N(t) = 1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with
(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "timeplot:=NULL:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for r from 1 to 3.0 by 0.2 do" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "   n[0]:=0.8;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "   for t from 0 to 30 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "     n[t+1]:=n[t]*exp(r*(1-n[t]));" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "     od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 " ti
meplot:=timeplot,plot([ [[x,n[x]]$x=0..30], [x,r,x= -3..-1]], color=[r
ed, blue], thickness=2, xtickmarks=4, ytickmarks=4, title=cat(`N vs t \+
for r=`, convert(r, string)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
26 "The following graph is an " }{TEXT 258 9 "animation" }{TEXT -1 
406 " of the dynamics of the population at different values of `r'.  T
he short blue line on the left side of the graph represents `r', and t
he red trajectory is the population size plotted as a function of time
.  Note that the population shows logistic behavior at low values of r
, shows damped oscillations, stable 2-cycles, stable 4-cycles, 8-cycle
s and then chaos.  The behavior is predicted by the plots of " }
{XPPEDIT 18 0 "N[t+8]" "6#&%\"NG6#,&%\"tG\"\"\"\"\")F(" }{TEXT -1 2 ",
 " }{XPPEDIT 18 0 "N[t+4]" "6#&%\"NG6#,&%\"tG\"\"\"\"\"%F(" }{TEXT -1 
6 ", and " }{XPPEDIT 18 0 "N[t+2]" "6#&%\"NG6#,&%\"tG\"\"\"\"\"#F(" }
{TEXT -1 4 " vs " }{XPPEDIT 18 0 "N[t]" "6#&%\"NG6#%\"tG" }{TEXT -1 
149 " above.  Compare this graph to the figures in May 1974.  Then cha
nge the initial condition from n[0]:= 0.01 to n[0]=0.81; what changes \+
do you obseve?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 407 "To set the animation in play, click on the graph, and th
en use the play and stop buttons on the toolbar.  The ->| button does \+
single frame advance, the << and >> decrease and increase the speed of
 playback.  The  -> and <- buttons change the direction of playback.  \+
The  short purple line at the left side of the graph is the value of r
 (read on vertical axis scale).  Continuous loop play is also possible
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display([timeplot],ins
equence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 447 "T
he bifurcation plot shows the stable points as a function of r.  When \+
r is less than 2, there is a single stable point.  When r reaches 2, t
he single stable point bifurcates into 2 stable points with a stable p
eriod-2 cycle.  When r reaches 2.5 the two point cycle bifurcates into
 a stable period-4 cycle with 4 stable points.  This is Figure 8a of M
ay and Oster, who provide an explanation for the changes in behavior o
f the system as r changes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "bifurcplot:=NULL:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for r from
 1 to 3.0 by 0.01 do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "   n[0]:=0
.99;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "   for t from 0 to 70 do;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "     n[t+1]:=n[t]*exp(r*(1-n[t]))
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "     od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 " bifurcplot:=bifurcplot, plot( [ [r,n[x]]$x =40..70 ]
, color=black, style=point, symbol=POINT):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "disp
lay([bifurcplot]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "We
 will consider equation 1 of May and Oster, which is the same equation
 used  by May 1974." }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "N[t+1]" "6#&%
\"NG6#,&%\"tG\"\"\"F(F(" }{TEXT -1 5 " = F(" }{XPPEDIT 18 0 "N[t]" "6#
&%\"NG6#%\"tG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
}{PARA 259 "" 0 "" {TEXT -1 7 "F(N) = " }{XPPEDIT 18 0 "N[t] *exp(r*(1
-N[t]))" "6#*&&%\"NG6#%\"tG\"\"\"-%$expG6#*&%\"rGF(,&F(F(&F%6#F'!\"\"F
(F(" }}{PARA 0 "" 0 "" {TEXT -1 193 "This is analogous to the logistic
 equation, but is better behaved, because it cannot predict negative p
opulation sizes.  By deciding on the value of the growth rate `r', we \+
can plot a graph of " }{XPPEDIT 18 0 "N[t+1]" "6#&%\"NG6#,&%\"tG\"\"\"
F(F(" }{TEXT -1 8 " versus " }{XPPEDIT 18 0 "N[t]" "6#&%\"NG6#%\"tG" }
{TEXT -1 109 ", and see the shape of the function F(N). The red line i
s the function F(N), and the black line is a plot of " }{XPPEDIT 18 0 
"N[t+1]" "6#&%\"NG6#,&%\"tG\"\"\"F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "N[t]" "6#&%\"NG6#%\"tG" }{TEXT -1 53 ".  This is figure 2a of May a
nd Oster, using r = 0.5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 
"plot([n*exp(0.5*(1-n)),n],n=0..2,color=[red,black]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 456 "The intersections of F(N) wit
h the line N(t+1)=N(t) are the equilibria or fixed points,  N*.  The s
tability of the fixed points is easily determined because in this situ
ation, the eigenvalue of F at a fixed point N* is the slope of F at th
e intersection with the 45 degree line.  If the eigenvalue is less tha
n 1 (slope  <1), the fixed point  is stable (attractor), if the eigenv
alue is greater than 1 (slope > 1),  the fixed  point is unstable (rep
ulsor)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 472 "The analogous procedure can be used to determine the rel
ation between N(t+2) and N(t), or between N(t+4) and N(t).  In these c
ases, if N(t+4)=N(t),  there is a stable period-4 oscillation, and if \+
N(t+2) = N(t), there is a stable period-2 oscillation.  The stability \+
of these fixed points is determined in the same way as described above
.  The population at time t+4 is found by recursively applying the fun
ction F . (Maple uses %1 to abbreviate a repeated chunk of stuff.)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F:= n -> n*exp(0.5*(1-n)); \+
t:= 't';  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "N[t+1]:= F(N[
t]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N[t+2]:= F(N[t+1]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N[t+3]:= F(N[t+2]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N[t+4]:= F(N[t+3]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 699 "N
ext we illustrate the May & Oster cobweb plot method to find bifurcati
ons.  Note that as the growth rate r increases, the slope of function \+
F(N) changes as does function F(F(N)).  There are conditions where a s
ingle fixed point becomes 3 fixed points: an unstable point between tw
o stable points with oscillation period 2.  When this occurs, there is
 a `bifurcation' in the population process, and the population switche
s from a single stable population  to a stable oscillation between the
 two new fixed points.  This process repeats as each of the  stable po
ints bifurcates into 2 stable  points with oscillation period 4, and r
epeats again, forming now 8 stable points with oscillation period 8." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p2_list:=NULL: p_list:=NU
LL: p4_list:=NULL: p8_list:=NULL:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "for r from 1.0 to 3.1 by 0.15 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "F:= n -> n*exp(r*(1-n)); #  modified logistic model " }}{PARA 0 
"" 0 "" {TEXT -1 51 "Look at time 2 in the future to find 2 point cycl
es" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "n1:= F(n): n2:= F(n1):  n3:= \+
F(n2):" }}{PARA 0 "" 0 "" {TEXT -1 51 "Look at time 4 in the future to
 find 4 point cycles" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n4:= F(n3):
  n5:= F(n4):  n6:= F(n5): n7:= F(n6):" }}{PARA 0 "" 0 "" {TEXT -1 52 
"Look at time 8 in the future to find 8 point cycles." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "n8:= F(n7):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
114 "p2_list:= p2_list, plot([[n,n2,n=0..2], [n,n,n=0..2], [n,r,n=-0.3
..-0.15]], color=[red,black,plum], ytickmarks=4);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 132 "p4_list:=p4_list, plot([[n,n4,n=0..2], [n,n,n=0..2
], [n,n2,n=0..2],[n,r,n=-0.3..-0.15]], color=[blue,black,red,plum], yt
ickmarks=4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "p8_list:=p8_list,p
lot([[n,n8,n=0..3],[n,n4,n=0..3],[n,n,n=0..3],[n,r,n=-0.3..-0.15]], co
lor=[sienna,blue,black,plum], ytickmarks=4);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 257 10 "animations" }
{TEXT -1 147 " display the fixed points of the logistic process and th
eir stabilities. The two-point cycle graph is figure 6 in May and Oste
r.  This graph plots " }{XPPEDIT 18 0 "N[t+2]" "6#&%\"NG6#,&%\"tG\"\"
\"\"\"#F(" }{TEXT -1 4 " vs " }{XPPEDIT 18 0 "N[t]" "6#&%\"NG6#%\"tG" 
}{TEXT -1 40 " in red.  The function is determined by " }{XPPEDIT 18 
0 "N[t+2]" "6#&%\"NG6#,&%\"tG\"\"\"\"\"#F(" }{TEXT -1 4 "= F(" }
{XPPEDIT 18 0 "N[t+1]" "6#&%\"NG6#,&%\"tG\"\"\"F(F(" }{TEXT -1 8 ") = \+
F(F(" }{XPPEDIT 18 0 "N[t]" "6#&%\"NG6#%\"tG" }{TEXT -1 226 ")).  Note
 that the single stable point bifurcates into a stable 2-point cycle w
hen r >  2, as noted in the legend of Fig 6 in  May and Oster.  The pa
rameter r is indicated by a short plum line to the left of the vertica
l axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "display([p2_list
],insequence=true,title=`2 point cycles`);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 17 "This graph plots " }{XPPEDIT 18 0 "N[t+4]" "6#&%
\"NG6#,&%\"tG\"\"\"\"\"%F(" }{TEXT -1 5 " vs  " }{XPPEDIT 18 0 "N[t]" 
"6#&%\"NG6#%\"tG" }{TEXT -1 21 " in blue and replots " }{XPPEDIT 18 0 
"N[t+2]" "6#&%\"NG6#,&%\"tG\"\"\"\"\"#F(" }{TEXT -1 4 " vs " }
{XPPEDIT 18 0 "N[t] " "6#&%\"NG6#%\"tG" }{TEXT -1 99 "  in red.   Noti
ce how the 2-cycle stable points bifurcate into 4-cycle stable points \+
when r = 2.6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "display([p
4_list],insequence=true,title=`4 point cycles`);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "This graph plots " }{XPPEDIT 18 0 "N[t+8]
" "6#&%\"NG6#,&%\"tG\"\"\"\"\")F(" }{TEXT -1 4 " vs " }{XPPEDIT 18 0 "
N[t]" "6#&%\"NG6#%\"tG" }{TEXT -1 23 " in sienna and replots " }
{XPPEDIT 18 0 "N[t+4]" "6#&%\"NG6#,&%\"tG\"\"\"\"\"%F(" }{TEXT -1 5 " \+
vs  " }{XPPEDIT 18 0 "N[t]" "6#&%\"NG6#%\"tG" }{TEXT -1 224 " in blue.
  Note the number of crossing points of the mapping function and the 4
5 degree line.  Each crossing point represents a fixed point that is e
ither stable or unstable, depending upon the slope of the mapping func
tion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "display([p8_list],
insequence=true,title=`8 point cycles`);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 601 "Finally we use the F(N) vs N and N vs N plots to \+
produce a sort of phase plane equivalent for the plot of N vs t.  On t
he vetical axis we have N(t+1) and on the horizontal N(t).  The first \+
point plotted is (N[0], F(N[0])). The next value N[1] = F(N[0]) can be
 found graphically by extending the horizontal line at level F(N[0]) a
cross to the 45 degree line. Using this position, which is N[1], on th
e horizontal axis, we move up (or down) to the curve to find N[2] = F(
N[1]). From here we again move across to the line, then up or down to \+
the curve, and so on.  Try the following simple example first." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "F:= n -> n*exp(r*(1-n));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "n[0]:= 0.1; r:= 1.8;  itera
tions:= 12;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plotlist:= N
ULL:  x:= n[0]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for j from 1 to
 iterations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y:= F(x):  plotli
st:= plotlist, [x, y], [y, y]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x:
= y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "P:= plot([plotlist], color=black, thickness=2): C:= p
lot( \{F(n), n\}, n = 0 .. 2.0):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "display( \{P, C\} );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 190 "Change n[0], and see what happens.  Record the re
sults of your experiment!  Next change the value of r and see what hap
pens for the same sequence of values of n[0] that you used previously.
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }
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