Information Technology Reference
In-Depth Information
2.2.6
Numerical Solution of the Logistic Equation
The Explicit Scheme
Certainly, the methods introduced to solve the equations above can also be used to
solve the logistic equation. Recall that this initial value problem is given by
r
0
.t /
D
ar
.t /
1
r.t/
R
;
(2.40)
r.0/
D
r
0;
(2.41)
where a>0is the growth rate and R is the carrying capacity. An explicit scheme
for this model is given by
r
nC1
r
n
t
r
R
/;
D
ar
n
.1
or
r
nC1
D
r
n
C
ar
n
t.1
r
R
/:
(2.42)
Properties
In our discussions of the differential equation (
2.40
) above, we made the following
observations just based on the form of the equation:
If R>>r
0
; then for small t ,wehaver
0
.t /
ar
.t /, and thus approximately
exponential growth.
-
R and r
0
.t /
-
If 0<r
0
<R;then the solution satisfies r
0
6
r.t/
6
>
0 for all
time.
r
0
and r
0
.t /
-
If r
0
>R;then the solution satisfies R
6
r.t/
6
6
0 for all time.
We can make exactly the same observations for the numerical solution generated
by the scheme (2.42). In order to do so, we have to assume that the time step is not
too big. More precisely, we assume that
t < 1=a:
(2.43)
Numerical solutions generated by the explicit scheme (2.42) have the following
properties:
If R>>r
0
; we have r
nC1
r
n
C
at r
n
; for small values of n: This is the
explicit scheme for the exponential growth model; see (2.28).
-
-
Assume that 0<r
0
<R:If, for some value of n; we have that r
n
6
R,then
r
R
/
.1
>
0;
