54
Economic Dynamics
express the diffusion process as
dN
(
t
)
dt
=
bN
(
t
)(ln
m
−
ln
N
(
t
))
(2.26)
or
dF
(
t
)
dt
=
bF
(
t
)(
−
ln
F
(
t
))
Suppressing the time variable for convenience, then the two models are
˙
F
=
(
a
+
bF
)(1
−
F
)
and
˙
F
=
bF
(
−
ln
F
)
Pursuing the logistic equation, we can graph
˙
F
against
F
. When
F
=
0 then
˙
F
=
a
and when
˙
F
=
0 then (
a
+
bF
)(1
−
F
)
=
0 with solutions
F
1
=−
b
/
a
and
F
2
=
1
Since
˙
F
denotes the rate of diffusion, then the diffusion rate is at a maximum
(penetration is at its maximum rate) when
¨
F
=
0, i.e., when
d
2
F
/
dt
2
=
0. Differ
entiating and solving for
F
, which we denote
Fp
(for maximum penetration rate),
we obtain
Fp
=
1
2
−
a
2
b
implying
Np
=
m
·
Fp
=
m
±
1
2
−
a
2
b
²
=
m
2
−
am
2
b
In order to Fnd the time
tp
when
F
(
tp
) is at a maximum penetration rate, we must
Frst solve for
F
(
t
). This we indicated above. Since we need to Fnd the value of
t
satisfying
F
(
t
)
=
Fp
, then we need to solve
1
−
e
−
(
a
+
b
)
t
1
+
(
b
/
a
)
e
−
(
a
+
b
)
t
=
1
2
−
a
2
b
for
t
, which we can do using a software package. This gives the time for the
maximum penetration of
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 Fall '11
 Dr.Gwartney
 Economics, maximum penetration

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