Graphics Programs Reference
In-Depth Information
u = 3.4; X = itseq(f, Xinit, 20, u); plot(X)
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0
5
10
15
20
25
In the first computation, we have used our iterative program to compute
the population density for 20 time intervals, assuming a logistic growth
constant
u
=
0
.
5 and an initial population density of 50%. The population
seems to be dying out. In the remaining examples, we kept the initial
population density at 50%; the only thing we varied was the logistic growth
constant. In the second example, with a growth constant
u
=
1, once again
the population is dying out — although more slowly. In the third example,
witha growthconstant of 1.5 the population seems to be stabilizing at
33.3...%. Finally, in the last example, with a constant of 3.4 the population
seems to oscillate between densities of approximately 45% and 84%.
These examples illustrate the remarkable features of the logistic
population dynamics model. This model has been studied for more than 150
years, with its origins lying in an analysis by the Belgian mathematician
Pierre Verhulst. Here are some of the facts associated with this model. We
will corroborate some of them with MATLAB. In particular, we shall use
bar
as well as
plot
to display some of the data.
(1) The LogisticConstant Cannot Be Larger Than 4
For the model to work, the output at any point must be between 0 and 1. But
the parabola
ux
(1
−
x
)
,
for 0
≤
x
≤
1, has its maximum height when
x
=
1
/
2,
where its value is
u
/
4. To keep that number between 0 and 1, we must
restrict
u
to be at most 4. Here is what happens if
u
is bigger than 4:
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