IterativeMaps.nb
The following simple "iterative" rule provides a model of how
the population of rabbits changes each year for a given
amount of clover as the number of foxes responds to the
number of rabbits (more rabbits = more food for foxes = more
foxes= fewer rabbits next year).
à Let x be the population of rabbits divided by 1000. Multiply x by 1000 to
get the population of rabbits in any given year.
à Let c be a measure of the clover available annually to the rabbits. If there
were no foxes, a possible model for the number of rabbits next year in
terms of the number of rabbits this year might be
(x next yr) = c times (x this yr)
More clover means more rabbits the next year (assuming constant
breeding habits),
à But many rabbits this year means lots of food for foxes which will then
thrive and reproduce. So we need a term in our formula which causes
the number of rabbits to decrease when there are too many of them.
Assume the following rule for determining the number of rabbits the next
year if the number this year is known:
(x next yr) = c times (x this yr)
times [1- (x this yr)]
Nextx@x_D := c x H1 - xL
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IterativeMaps.nb
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à This is called the "prey-predator" model. It is a grossly oversimplified
"computer simulation model" of the rabbit population (there are more
complicated ones). The fraction of rabbits, x, will be seen to vary
between 0 and 1. (The number of rabbits will vary between 0 and one
times 100.) When x is close to 1 many foxes appear to kill them off, so
there are fewer rabbits the next year(the number (1-x) is small).
à We will keep c in a range which guarantees that x will be a number
between 0 and 1. In the example below, c will start at 2.9 and several
higher values will be chosen up to a maximum of 4.
à Here is what happens when we start with 900 rabbits, and c = 2.9, after 48
years:
x = 0.9; c = 2.9; ListPlot@NestList@Nextx, x, 48DD
0.7
0.68
0.66
0.64
0.62
10
-Graphics-
20
30
40
50
IterativeMaps.nb
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x = 0.9; c = 2.9; ListPlot@NestList@Nextx, x, 48D, PlotJoined ® TrueD
0.7
0.68
0.66
0.64
0.62
10
20
30
40
50
-Graphics-
à Note that the number of rabbits tends after many years to become fixed
annually at 65. This represents a kind of ecological balance between
population increase due to a good clover supply and population
decrease due to foxes.
à Here is what happens when we increase c slightly (more clover and more
foxes):
x = 0.9; c = 3.2; ListPlot@NestList@Nextx, x, 48DD
0.9
0.8
0.7
0.6
0.5
0.4
10
-Graphics-
20
30
40
50
IterativeMaps.nb
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x = 0.9; c = 3.2; ListPlot@NestList@Nextx, x, 48D, PlotJoined ® TrueD
0.9
0.8
0.7
0.6
0.5
0.4
10
20
30
40
50
à In this case, the population of rabbits eventually oscillates from year to
year between 51 and 80.
à Now let's increase c slightly, again.
c = 3.52; ListPlot@NestList@Nextx, x, 48D, PlotJoined ® TrueD
0.9
0.8
0.7
0.6
0.5
0.4
10
-Graphics-
20
30
40
50
IterativeMaps.nb
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à In this case, the population of rabbits comes to oscillate between 4
different values, repeating every 4th year.
à Here is what happens when c = 4:
c = 4; ListPlot@NestList@Nextx, x, 96DD
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0.8
0.6
0.4
0.2
20
40
60
80
-Graphics-
à Here the population is chaotic and virtually unpredictable from year to
year!
Next we study what what happens when we
repeatedly apply a rule which determines TWO
numbers
Next@8x_, y_<D := 81 - c x2 + y, b x<
b = 0.3; c = 1.4;
IterativeMaps.nb
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ListPlot@NestList@Next, 80.5, 0.2<, 2000DD
0.4
0.2
-1
-0.5
0.5
-0.2
-0.4
… Graphics …
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