Information Technology Reference
In-Depth Information
If we let
d.t/
D
r.t/
q.t/;
we find that d solves the problem
d
0
.t /
D
ad
.t /;
(2.16)
d.0/
D
d
0
D
r
0
q
0
:
It follows that
d.t/
D
d
0
e
at
;
so
r.t/
q.t/
D
.r
0
q
0
/e
at
;
or
j
r.t/
q.t/
j D j
r
0
q
0
j
e
at
: (2.17)
This equation states that the difference is magnified by a factor of e
at
. To under-
stand what this means, we divide the left-hand side of (2.17)byr.t/ and the
right-hand side by r
0
e
at
: This is fine since
r.t/
D
r
0
e
at
:
We fi n d t h a t
j
r.t/
q.t/
j
r.t/
j
r
0
q
0
j
r
0
D
;
(2.18)
which means that the relative difference between the two solutions remains constant.
We also note from (2.18)thatifweletq
0
tend toward r
0
,thenq.t/ will also tend
toward r.t/: Since the relative difference between these two solutions does not blow
up, we refer to this problem as
stable
with respect to changes in the initial data.
More generally, we refer to a problem as stable with respect to perturbations
in one parameter if such changes do not lead to very different solutions. With the
rabbit counting in mind, we hope you appreciate the need for some sort of stability.
Usually, the parameters that we use in such equations are based on some sort of
measurements, and measurements often involve errors.
2.1.4
Logistic Growth
As mentioned above, model (
2.8
) is an interesting model for the growth of rabbits
on an island, but it is not perfect. Mathematical models are not perfect; we can
never incorporate every tiny effect in a model, and therefore it will always remain
a
model
. However, there is one particular feature that is unrealistic about model
(
2.8
), and that is that the number of rabbits goes to infinity as time increases. This is
unrealistic, because each rabbit needs sufficient food and some space to stay alive.
Therefore, we need to refine the model with an additional term. Let R denote the
