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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 21 "More Realistic Models" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The cell
 below defines the model used as an example with the Java applet in th
is module.  " }{TEXT 267 15 "Evaluate it now" }{TEXT -1 80 ".  You wil
l be able to work with similar models by modifying the definition of  \+
" }{TEXT 268 6 "pts[0]" }{TEXT -1 4 ", .." }{TEXT 269 8 ". pts[20" }
{TEXT -1 58 "].  These are the values of the population multiplier at \+
 " }{TEXT 270 5 "p = 0" }{TEXT -1 2 ", " }{TEXT 271 6 "p = 50" }{TEXT 
-1 4 ", .." }{TEXT 272 9 ". p = 100" }{TEXT -1 2 "0." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 639 "pts[ 0] \+
:= 0.8:\npts[ 1] := 0.8:\npts[ 2] := 0.8:\npts[ 3] := 0.8:\npts[ 4] :=
 1.0:\npts[ 5] := 1.2:\npts[ 6] := 1.2:\npts[ 7] := 1.2:\npts[ 8] := 1
.2:\npts[ 9] := 1.2:\npts[10] := 1.2:\npts[11] := 1.2:\npts[12] := 1.2
:\npts[13] := 1.2:\npts[14] := 1.2:\npts[15] := 1.0:\npts[16] := 0.8:
\npts[17] := 0.6:\npts[18] := 0.4:\npts[19] := 0.2:\npts[20] := 0.0:\n
\nmult := proc(p)\n        local j, t, pout;\n        j := trunc(p/50)
:\n        t := (p - 50 * j)/50:\n        if (j = 20) then\n          \+
 pout := pts[20]:\n        else\n           pout := (1 - t) * pts[j] +
 t * pts[j + 1]:\n        fi;\npout;\nend:\n\nplot(mult(p),p=0..1000);
\n\nf := p -> a * p * mult(p):\na := 1:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 39 "The cell below defines \+
a procedure  -- " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 262 6 "cobweb" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 89 "  -- that can be used to draw cobweb diagrams.\nThis pr
ocedure requires three arguments.\n\n" }}{PARA 0 "" 0 "" {TEXT 263 4 "
fcn " }}{PARA 0 "" 0 "" {TEXT -1 100 " -- the function that describes \+
how each term of the sequence is computed from the preceding term.\n\n
" }}{PARA 0 "" 0 "" {TEXT 264 5 "first" }}{PARA 0 "" 0 "" {TEXT -1 50 
" -- the value of the first term of the sequence.\n\n" }}{PARA 0 "" 0 
"" {TEXT 265 1 "n" }}{PARA 0 "" 0 "" {TEXT -1 76 " -- the number of te
rms of the sequence to be shown in the cobweb diagram \n\n" }}{PARA 0 
"" 0 "" {TEXT 266 27 "Evaluate the cell below now" }}{PARA 0 "" 0 "" 
{TEXT -1 4 ".  \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1488 "with(plots):\nwith(plottools):\n\nPopModel
 :=\n     proc(fcn, first, n) \n        if n <= 1 then first else fcn(
PopModel(fcn, first, n-1)) fi:\n     end:\n\ncobweb := \n     proc(fcn
, first, n, top)\n     local blk, rd, blu, frm, cobweblist, j;\n\n    \+
 cobweblist := \{line([first, 0], [first, fcn(first)])\}:\n\n     for \+
j from 3 to n do\n        cobweblist := cobweblist union\n            \+
\{line([PopModel(fcn, first, j-2), PopModel(fcn, first, j-1)],\n      \+
            [PopModel(fcn, first, j-1), PopModel(fcn, first, j-1)]), \+
\n             line([PopModel(fcn, first, j-1), PopModel(fcn, first, j
-1)],\n                  [PopModel(fcn, first, j-1), PopModel(fcn, fir
st, j)])\}\n     od:\n\n     frm := display(\{line([0, 0], [0, top]),
\n                     line([0, top], [top, top]),\n                  \+
   line([top, top], [top, 0]),\n                     line([top, 0], [0
, 0])\},\n                     color = BLACK,\n                     ax
es = NONE,\n                     scaling = CONSTRAINED):\n\n     blk :
= plot(f(p),\n                    p = 0..top,\n                    y =
 0..top,\n                    color = BLACK,\n                    scal
ing = CONSTRAINED):\n\n     rd  := display(cobweblist,\n              \+
      color = RED,\n                    axes = NONE,\n                \+
    scaling = CONSTRAINED):\n\n     blu := display(\{line([0, 0], [top
, top])\},\n                    color = BLUE,\n                    axe
s = NONE,\n                    scaling = CONSTRAINED):\n\n     display
([blk, rd, blu, frm]);   \n  end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The following cell illustrates ho
w this new procedure can used along with other Maple procedures to inv
estigate long term behavior.  " }{TEXT 261 16 "Evaluate it now." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 300 "  cobweb(f, 50, 30, 1000);\n\n  pop := proc(n) option remember;
\n         if n = 1 then 50 else f(pop(n-1)) fi\n         end:\n\n  fo
r k from 1 to 30 \n      do printf(`%4d %10.4f`, k, pop(k));\n      pr
int();\n      od;\n\n  plot([seq([n, pop(n)], n = 1 .. 30)], \n       \+
 year = 1 .. 30, title = `Population`);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 }