Information Technology Reference
In-Depth Information
from the fundamental theorem of Calculus.
4
By the methods introduced in the pre-
vious chapter, we know that r
D
r.t/ can be computed as accurately as we want for
any reasonable function f:
Equation (2.3) is a mathematical model of rabbit growth, but is it useful? No, not
really. It simply states that if you know the number at t
D
0 and each day you add
the net number of new rabbits, you will have r each time. This is somewhat trivial.
Let us turn to something slightly more advanced.
2.1.3
Exponential Growth
How does the number of rabbits really change? What characterizes the growth? It
is reasonable to assume that safe sex is not a big issue among rabbits, and it is also
fairly well known that they do
it
very often. We find it reasonable to assume that
the rate of change is proportional to the number of rabbits. More specifically, we
assume that
r.t
C
t/
r.t/
t
D
ar
.t /;
(2.5)
where a is a positive constant. Another way to put this is to state that the relative
change is constant:
r.t
C
t/
r.t/
r.t/t
D
a:
(2.6)
In order to phrase this as a differential equation, we assume that r is differentiable
and let t go to zero. This gives the exponential growth equation
5
r
0
.t /
D
ar
.t /:
(2.7)
Let us assume that the growth rate a is a given positive constant. Of course, in a
real-life application, this number has to be measured and we will return to this issue
later. But let us just assume that it is given.
4
The fundamental theorem of Calculus states that if f
D
f.x/ is continuous and F
0
.t /
D
f.t/,
then
R
a
f.t/dt
F.a/.
5
We can state this slightly more precisely: Let b
D
F.b/
D
b.t/ be births and d
D
d.t/ be deaths. Then
r
0
.t /
D
b.t/
d.t/:
Now we assume that the births are proportional to the population, i.e., b.t/
D
ˇr.t/ and similarly
d.t/
D
ır.t/. Thus,
r
0
.t /
D
.ˇ
ı/r.t/;
which is (2.7) with a
D
ˇ
ı.
