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Lecture 11
1.1
Logistic equation and population models
Let
B
(
t
)
and
D
(
t
)
denote the number of births and deaths that have oc
curred since
t
= 0
by the time
t
.
The birth rate
(
t
)
=
lim
t
!
0
B
(
t
t
)
B
(
t
)
P
(
t
t
=
1
P
(
t
)
B
0
(
t
)
The death rate
±
(
t
)
=
lim
t
!
0
D
(
t
t
)
D
(
t
)
P
(
t
t
=
1
P
(
t
)
D
0
(
t
)
;
where
P
(
t
)
is population at time
t
. So,
(
t
)
and
±
(
t
)
represent the number
of births and death per unit of population and per unit of time. Population
P
[
(
t
)
±
±
(
t
tP
(
t
)
. So,
P
0
= (
±
±
)
P:
Notice that
and
±
may not be constants.
Assumption
(birth rate decreases as population increases):
(
t
) =
&
0
±
1
P;
where
&
0
1
=
const
,
&
0
>
0
1
>
0
:
±
=
±
0
=
const:
Then
P
0
= (
&
0
±
1
P
±
±
0
)
P
)
P
0
=
kP
(
M
±
P
)
;
where
k
=
1
;M
=
&
0
±
±
0
1
:
The ODE
P
0
=
kP
(
M
±
P
)
is called the
logistic equation.
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 Fall '08
 Staff
 Differential Equations, Equations

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