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The only value of the parameter for which the population does not have these extreme
tendencies asymptotically is
1, for which, since the population reproduces itself
exactly in each generation, we obtain the unstable situation
λ
=
lim
n
→∞
Q
n
=
Q
0
if
λ
=
1
.
(4.72)
In this way the initial population remains unchanged from one generation to the next,
unless
λ
deviates slightly from unity, in which case the population diverges to zero or
infinity.
Verhulst, himself a scientist, put forth a theory that mediated the pessimistic view of
the cleric Malthus. He noted that the growth of real populations is not unbounded, since
such factors as the availability of food, shelter and sanitary conditions all restrict, or at
least influence, the growth of populations. In particular, he assumed the net birth rate to
decrease with increasing population in a linear way and made the replacement
λ
→
λ(
1
−
Q
/),
(4.73)
where
is the population's saturation level. Thus, the linear recursion relation (
4.67
)is
replaced with the nonlinear discrete logistic equation
Q
n
+
1
=
λ
Q
n
(
1
−
Q
n
/).
(4.74)
It is clear that, when the population is substantially below the saturation level, that is
Q
n
/
1, the growth is exponential since the nonlinear term is negligibly small com-
pared with the linear term
Q
n
/.
λ
Q
n
λ
However, at some point in time (iteration
number) the ratio
Q
n
/
will be of order unity and the rate of population growth will be
retarded. When
Q
n
/
=
1 there are no more births and the population stops growing.
Mathematically the regime
Q
n
/>
1 corresponds to a negative birth rate, but this does
not make biological sense and so we restrict the region of interpretation of this particular
model to
.
Finally, we reduce the number of parameters from two to one by introducing
the scaled variable
Y
n
=
(
1
−
Q
n
/) >
0
, the fraction of the saturation level achieved by the
population. In terms of this ratio the logistic equation becomes
Q
n
/
Y
n
+
1
=
λ
−
Y
n
)
,
Y
n
(
1
(4.75)
where the maximum population fraction is unity. In Figure
4.8
the nearly continuous
solution to the logistic map is given, showing a smooth growth to saturation. Note
that the saturation level is achieved when
Y
n
+
1
=
Y
n
, so that the saturation level is
Y
sat
=
1
−
1
/λ
rather than unity; or, in terms of the population variable, the saturation
level is not
.
The form of the saturation curve in Figure
4.8
ought to be familiar. It appears in
essentially all discussions of nonlinear phenomena that have a smooth switching behav-
ior from one value of the dependent variable to another. In this parameter regime the
mapping appears to be analogous to a discretization of the continuous logistic equation
dY
dt
=
kY
(
1
−
Y
).
(4.76)
