Information Technology Reference
In-Depth Information
Analytical Solution
Model (2.7) is a differential equation. If we write this equation and the associated
initial equation together, we have the following initial value problem:
r
0
.t /
D
ar
.t /;
(2.8)
r.0/
D
r
0
:
The amazing thing is that these two simple equations determine r uniquely for all
time. So if we are able to estimate the initial number of rabbits r
0
and the growth
rate a, we get an estimate for the number of rabbits in the future. Since
dr
dt
D
ar
;
we have
6
1
r
dr
D
a
dt
;
and then integration
Z
1
r
dr
D
Z
a
dt
gives
ln.r/
D
at
C
c
(2.9)
where c is a constant of integration. By setting t
D
0; we have
c
D
ln.r
0
/;
where we have used the initial condition r.0/
D
r
0
: It now follows from (2.9)that
ln.r.t //
ln.r
0
/
D
at
;
6
It is perfectly ok to multiply
dr
dt
1
r
dr
a
dt
and then integrate. But if you
are unfamiliar with that way of manipulating differentials, you may also observe directly that since
1
D
ar
by
dt
to obtain
D
r
r
0
.t /
D
a, we can integrate in time to get
Z
1
r
r
0
.t /
dt
Z
a
dt
:
D
r
0
.t /
dt
and thus the left-hand side is
Z
1
r
dr
;
If we now substitute r
D
r.t/,wehave
dr
D
and we arrive at
ln.r /
D
at
C
const:
