Information Technology Reference
In-Depth Information
0.08
0.06
0.04
0.02
0
0
20
40
60
80
100
Generation
n
Figure 4.8.
The growth of the logistic equation to saturation for a parameter value given by
λ
=
1
.
1sothat
Y
sat
=
0
.
09
.
Thesolutionto(
4.76
) is readily obtained through the transformation
Z
=
1
/
Y
, which
yields the linear equation
d
dt
(
−
)
=−
(
−
),
Z
1
k
Z
1
(4.77)
which immediately integrates to
log
Z
(
)
−
t
1
=−
.
kt
(4.78)
(
)
−
Z
0
1
A little algebra gives the solution in terms of the original variable
e
kt
Y
(
0
)
Y
(
t
)
=
)
,
(4.79)
e
kt
1
+
(
−
1
)
Y
(
0
where we see that asymptotically the saturation level is unity. This asymptotic value
clearly shows that the logistic map is not simply a discretization of the continuous
logistic equation; the relation between them is more subtle. Moreover, the early-time
growth is exponential since, when
t
1
/
k
, the second term in the denominator can be
neglected to obtain
e
kt
(
)
≈
(
)
Y
t
Y
0
(4.80)
in accordance with the unbounded growth feared by Malthus and many of his contem-
poraries. Note, however, that this exponential behavior is indicative of the growth only
at very early times.
A great deal has been written about how nonlinear maps become unstable in certain
regimes of the control parameters, but we do not go into that literature here. Our intent
is to show how an equation as benign looking as (
4.75
) can give rise to a wide variety of
dynamical behaviors. In Figure
4.9
the iterations of this equation are graphed for four
values of the control parameter
9 the population grows,
oscillates about its asymptotic level and eventually is attracted to the saturation level
0.655. In Figure
4.9
(b) the control parameter is
λ
. In Figure
4.9
(a) with
λ
=
2
.
This value gives rise to a bifur-
cation in the dynamics and the solution oscillates between two values, asymptotically
producing a 2-cycle. Note that a bifurcation is a qualitative change in the dynamics.
λ
=
3
.
2
.
