Information Technology Reference
In-Depth Information
2.1.1
The Cultivation of Rabbits
We are entering the business of rabbit farming. Let us assume that we place a number
of rabbits on an isolated island with a perfect environment for them. How will the
number of rabbits grow? We cannot solve this problem just based on mathematical
reasoning. Some fundamental data have to be provided. But by using mathematical
models, we can figure out exactly what we need to know in order to make realistic
predictions, and we can figure out some interesting facts about the growth by doing
some very simple assumptions.
2.1.2
The Simplest Possible Case
Let r
D
r.t/ be the number of rabbits on the island at time t: We assume that at
time t
D
0; the number is given by r
0
; so
r.0/
D
r
0
:
(2.1)
That is all we know initially. That is the basic state. Next we consider the change.
Let t > 0 be a small period of time. If the change of rabbits per time is given by
f.t/; we have
r.t
C
t/
r.t/
t
f.t/:
(2.2)
Next we assume that the number of rabbits is large and that it can be modeled as a
continuous function of time. This is, of course, wrong since the number of rabbits
has to change discontinuously when one single rabbit is born. But when the number
is large, a continuous r can be a fair approximation. If we assume that the number
of rabbits is continuous and differentiable, we obtain the model
r
0
.t /
D
f.t/
(2.3)
by letting t go to zero.
3
If we assume that (2.1)and(2.3) hold, we get
r.t/
D
r.0/
C
Z
t
0
f.s/ds
(2.4)
3
Recall the definition of the derivative from calculus:
y.x
C
x/
y.x/
y
0
.x/
D
lim
x
:
x
!
0
