Section 6.3 – The Logistic Equation

Population GrowthWe have seen a model for the growth of a population (P)

when

This model represents exponential growth:

Now use separation of variables to find the general solution.

dPdt k P dPP k dt

ln P kt C kt CP e

Since C is arbitrary, ±eC represents an arbitrary nonzero number. We

can replace it with C:

C ktP e ektP Ce

Population Growth/DecayPopulations 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:

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 is very small:

The population is close to the constant L:

The population is very large:

0

lim 1 PLP

kP

1kP kPExponential Growth

lim 1 PLP L

kP

1 1kP 0Nearly no Growth

lim 1 PLP

kP

negative

kPnumber

The Growth Rate is Decreasing

Graphical Investigation of the Logistic Model

Consider the slope field for :

When is the population increasing?When the population (P) is

less than L.

L

P(t)

t

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.)

NOTE: Even though more are dying, they are still reproducing.

A Different Graphical ApproachIt is helpful to look at a graph of versus :

If P is the independent variable and dP/dt is the dependent variable, what is the shape

of the graph?

A concave down Parabola.

𝑑𝑃𝑑𝑡

=𝑘𝑃−𝑘𝑃2

𝐿What are the P intercepts of the graph?

0=𝑘𝑃 (1−𝑃𝐿 )

𝑃=0 𝑃=𝐿

𝑃𝐿

What does the population need to be for the growth to be increasing at the fastest rate?

𝑃=𝐿−02

¿𝐿2

At the vertex:

𝐿2

Only if the graph goes through this

point.

L/2 is also an

inflection point

Logistic ModelThe 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:

is a positive constant.

is the carrying capacity ()

reaches its maximum at .

ExampleThe population of a species satisfies the logistic differential equation , where the initial population and is the time in years.

a) What is ?

b) What population causes to have a maximum value? Does our population ever reach this rate of change? Explain.

c) When will the population reach 15,000? Explain.

d) Find the particular solution, , for the initial condition.

10,000

5,000. Yes. The initial population is below 5,000 and it is increasing. No. It can

not be greater than the carrying

capacity.We do not have the tools to complete

this yet.