Information Technology Reference
In-Depth Information
carrying capacity
of the island. This number is the maximum number of rabbits
that the island can supply with food, space, and so forth. We introduce the so called
logistic model
r
0
.t /
D
ar
.t /
1
r.t/
R
;
(2.19)
where a>0is the growth rate and R is the carrying capacity. We assume that R
is a very big number and, in particular, we assume that r
0
<< R: So for t close to
zero, we have
r.t/
R
0;
and thus
r
0
.t /
ar
.t /;
which means that the logistic model and the exponential model give similar predic-
tions for small values of t: But as t increases, the two models provide very different
predictions. By looking at (2.19), you can notice that the right hand side is positive
at time t
D
0: In fact, since r
0
<R;we have
1
>0:
r
R
r
0
.0/
D
ar
0
This means that r is increasing. And we observe that r will continue to increase as
long as
1
>0;
r.t/
R
that is, as long as
r.t/ < R:
Thus, as time increases, r will increase, but as it approaches R; the growth rate will
become smaller and smaller, and if we reach r.t/
D
R at, say, t
D
t
, we will have
r
0
.t
/
D
0:
This means that the logistic model predicts that the number of rabbits will steadily
increase but never exceed the carrying capacity. Note, however, that we have not
proved that we will actually reach the carrying capacity; more on this later.
Exceeding the Carrying Capacity
Suppose that we place lots of rabbits on the island initially, such that
r
0
>R;
that is, the initial number of rabbits exceeds the carrying capacity of the island. Then
it follows from (2.19)that
