Section 6.3 – The Logistic
Equation
Population Growth
We have seen a model for the growth of a population (P)
𝑑𝑃
𝑑𝑡
= 𝑘𝑃
when 𝑘 > 0
This model represents exponential growth:
dP
dt

dP
P
 kP
Now use separation of variables to find the
general solution.
 k  dt
ln P  kt  C
kt  C
P e
C kt
P  e e
Since C is arbitrary, ±eC represents
an arbitrary nonzero number. We
can replace it with C:
P  Ce
kt
Population Growth/Decay
Populations may grow exponentially over short periods of
time, but it should be clear that no population can increase
without limit. (Imagine a bunny population or bacteria in a
petri dish.)
Population biologists use a variety of other differential
equations that take into account limitations to growth. One
widely used is based on the logistic differential equation:
𝑑𝑃
𝑃
= 𝑘𝑃 1 −
𝑑𝑡
𝐿
where k>0, P>0, and L>0
Let’s investigate this differential equation.
Analytical Investigation of the Logistic Model
Consider how the following limits affect the differential equation:
The population
lim kP 1 
is very small: P 0
P
L
The population
kP 1 
is close to the lim
PL
constant L:
P
L
The population lim kP 1 
P 
is very large:
P
L

 kP 1  kP
Exponential Growth
  kP  1  1
0
Nearly no Growth

 negative 
 kP  

 number 
The Growth Rate is Decreasing
Graphical Investigation of the Logistic Model
Consider the slope field for
P(t)
L
𝑑𝑃
𝑑𝑡
NOTE: Even though more are
dying, they are still reproducing.
= 𝑘𝑃 1 −
𝑃
𝐿
:
When is the population increasing?
When the population (P) is
less than L.
When is the population decreasing?
When the population (P) is
greater than L.
When is the population constant?
When the population (P) is
equal to L.
What happens as time increases?
The population (P) always
gets closer to L.
What does L represent graphically?
L is a vertical asymptote.
What does L represent contextually?
L is the carrying capacity.
(The equilibrium solution.)
A Different Graphical Approach
It is helpful to look at a graph of
𝑑𝑃
𝑑𝑡
𝑑𝑃
𝑑𝑡
L/2 is also
an
inflection
point
versus 𝑃:
𝑑𝑃
𝑃
= 𝑘𝑃 1 −
𝑑𝑡
𝐿
If P is the independent variable and dP/dt is
the dependent variable, what is the shape
of the graph?
𝑑𝑃
𝑘𝑃2 A concave down
= 𝑘𝑃 −
𝑑𝑡
𝐿
Parabola.
What are the P intercepts of the graph?
𝑃
0 = 𝑘𝑃 1 −
𝐿
𝑃=0
𝑃=𝐿
𝐿
2
What does the population need to be for the
growth to be increasing at the fastest rate?
Only if the graph
𝐿
𝐿−0
𝑃
At the
=
𝑃=
goes through this
𝐿
2
2
vertex:
point.
Logistic Model
The rate of change of the size of a population P is governed by
the logistic model when the population grows in the presence of
limited resources. If P represents the number of organisms in a
population at time t, the model is represented by the following
differential equation:
𝑑𝑃
𝑃
= 𝑘𝑃 1 −
𝑑𝑡
𝐿
𝑘 is a positive constant.
𝐿 is the carrying capacity ( lim 𝑃(𝑡) = 𝐿)
𝑑𝑃
𝑑𝑡
𝑡→∞
𝐿
2
reaches its maximum at 𝑃 = .
Example
The population 𝑃(𝑡) of a species satisfies the logistic
𝑑𝑃
𝑃
differential equation = 𝑃 2 −
, where the initial
𝑑𝑡
5000
population 𝑃 0 = 3,000 and 𝑡 is the time in years.
a) What is lim 𝑃(𝑡)?
𝑡→∞
10,000
𝑑𝑃
b) What population causes
to have a maximum value?
𝑑𝑡
Does our population ever reach this rate of change?
Explain.
5,000. Yes. The initial population is below 5,000
c) When will the population reach 15,000?
Explain.
and it is increasing.
No. It can
d) Find the particular solution, 𝑃(𝑡), for the initial condition.
not be
We do not have the tools to complete
this yet.
greater than
the carrying
capacity.