MA 36600 LECTURE NOTES: FRIDAY, JANUARY 30
Population Dynamics
Logistic Equations.
We discuss an application of autonomous equations by considering population growth.
Say that we have a population which has a size
P
=
P
(
t
) at time
t
. Assume that the rate of change of
the size is proportional to both the size of the population and the difference of the size from some maximal
sustainable size
K
. That is
dP
dt
∝
P
(
K

P
) =
K
·
P
1

P
K
.
Hence there exists a positive constant
r
such that
dP
dt
=
r P
1

P
K
.
This is known as the
logistic equation
. The equilibrium solutions
P
=
P
L
can be found by computing the
critical points of the function
f
(
P
) =
r P
1

P
K
.
Clearly these are
P
L
= 0 and
P
L
=
K
. Sometimes we refer to
K
as the
saturation level
or the
environmental
carrying capacity
. Today we will study the solution to this problem.
Solution to Initial Value Problem.
We solve this problem explicitly. To this end, it will be useful to
consider the “normalized” function and the “normalized” initial condition
y
(
t
) =
P
(
t
)
K
and
y
0
=
P
0
K
.
We substitute
P
=
K y
into the differential equation above:
dP
dt
=
r P
1

P
K
d
dt
[
K y
] =
r K y
1

K y
K
K
dy
dt
=
r K y
(1

y
)
=
⇒
dy
dt
=
r y
(1

y
)
.
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 Spring '09
 MA
 Equations, Constant of integration, dt, yl

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