6.1.3.
Population Growth Models.
The simplest model for the population growth of an
organism is
N
0
=
rN
where
N
(
t
) is the population at time
t
and
r >
0 is the growth rate.
This model predicts exponential population growth
N
(
t
) =
N
0
e
rt
, where
N
0
=
N
(0). We
studied this model in
§
1.5
. Among other things, this model assumes that the organisms
have unlimited food supply. This assumption implies that the per capita growth
N
0
/N
=
r
is constant.
A more realistic model assumes that the per capita growth decreases linearly with
N
,
starting with a positive value,
r
, and going down to zero for a critical population
N
=
K >
0.
So when we consider the per capita growth
N
0
/N
as a function of
N
, it must be given by
the formula
N
0
/N
=

(
r/K
)
N
+
r
. This equation, when thought as a differential equation
for
N
is called the logistic equation model for population growth.
Definition 6.1.3.
The
logistic equation
describes the organisms population function
N
in time as the solution of the autonomous differential equation
N
0
=
rN
1

N
K
,
where the initial growth rate constant
r
and the carrying capacity constant
K
are positive.
We now use the graphical method to carry out a stability analysis of the logistic popu
lation growth model. Later on we find the explicit solution of the differential equation. We
can then compare the two approaches to study the solutions of the model.
Example
6.1.6
:
Sketch a qualitative graph of solutions for different initial data conditions
y
(0) =
y
0
to the
logistic equation
below, where
r
and
K
are given positive constants,
y
0
=
ry
1

y
K
.
230
G. NAGY – ODE
april 18, 2014
Solution:
The logistic differential equation for
population growth can be written
y
0
=
f
(
y
), where function
f
is the
polynomial
f
(
y
) =
ry
1

y
K
.
The first step in the graphical ap
proach is to graph the function
f
.
The result is in Fig.
39
.
The second step is to identify all crit
ical points of the equation. The crit
ical points are the zeros of the func
tion
f
. In this case,
f
(
y
) = 0 implies
y
0
= 0
,
y
1
=
K.
The third step is to find out whether
the critical points are stable or un
stable.
Where function
f
is posi
tive, a solution will be increasing,
and where function
f
is negative a
solution will be decreasing.
These
regions are bounded by the critical
points.
Now, in an interval where
f >
0 write a right arrow, and in
the intervals where
f <
0 write a left
arrow, as shown in Fig.
40
.
The fourth step is to find the re
gions where the curvature of a solu
tion is concave up or concave down.
That information is given by
y
00
. But
the differential equation relates
y
00
to
f
(
y
) and
f
0
(
y
).
We have shown
in Example
6.1.4
that the chain rule
and the differential equation imply,
y
00
=
f
0
(
y
)
f
(
y
)
So the regions where
f
(
y
)
f
0
(
y
)
>
0
a solution is concave up
(CU), and
the regions where
f
(
y
)
f
0
(
y
)
<
0 a
solution is concave down
(CD). The
result is in Fig.
41
.
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 Differential Equations, Derivative