In this example we are concerned with the populations of rabbits and foxes in
a national park. Suppose that
rabbits and
foxes are introduced into the park which previously contained no rabbits or
foxes and for every nonnegative integer
the numbers of rabbits and foxes in the path after
months are
and
,
respectively. Suppose finally that for each
we have

We want to determine what will happen to the populations of rabbits and foxes
in the long term.

We begin our study of this problem with the observation that if

then, for each

and so

By pointing at the matrix
and clicking on Eigenvalues we see that the two
eigenvalues of this matrix are
and
Since the eigenvalue
has multiplicity only
and the other eigenvalue has absolute value less than
the above theorem
tells us that the sequence
is convergent.

By clicking on Evaluate Numerically we see that

Looking at these matrices we can conjecture that

as
and that, in the long term, the numbers of rabbits and foxes will approach the
coordinates of the vector

In other words, in the long term, there will be twice as many rabbits in the
park as there are foxes.

In order to work exactly with the matrix
we must rewrite it in a form that does not contain decimals. We therefore
write the matrix
as

By pointing at the matrix
and clicking on Eigenvectors we obtain
,

We define

and we supply this definition to *Scientific Notebook* by clicking on
Define and New Definition.
Since

we deduce that if
is any positive integer then

Thus

showing that the conjecture we made previously was correct.

In this section we give a brief and elementary introduction to the concept of
Markov processes in finite probability spaces and we suggest some ways in
which the computing features of *Scientific Notebook* can be used to
draw conclusions about these Markov processes.

We begin with a simple application of Markov processes that will help to motivate the theory.

Let us suppose that a car rental agency has three locations: 1, 2 and 3 and
that a customer can rent a car at any of the three locations and return it to
any of the locations. For all
and
in
we shall use the symbol
to describe the probability that a customer who has rented a car at location
will return it to location
We observe that if
then a car rented at location
must be returned to one of the three locations. Therefore

Thus, if we define a matrix
by the equation

then the sum of the entries in each column of the matrix
must be
This process of observing a car as it moves from location to location as it is
repeatedly rented is known as a **Markov process **and the matrix
is called the **transition matrix **of the Markov process. The
first column

of the matrix
lists the probabilities that a car that was originally at location 1 will be
at the locations 1, 2 and 3 after it has been rented once. Similarly, the
second and third columns

of the matrix
list the probabilities that a car that was originally at location 2 or 3 will
be at the locations 1, 2 and 3 after it has been rented once.

Now suppose that a car could have originated at any of the three locations and
that the probabilities that the car originated in the locations 1, 2 and 3 are
written as
and
,
respectively. Note that
and
are nonnegative numbers and that
The probabilities that this car will be at the locations 1, 2 and 3 after it
has been rented once are the coordinates of the vector

Repeating the process we see that if the vector

lists the probabilities that a car originated at the locations 1, 2 and 3,
then the probabilities that this car will be at the locations 1, 2 and 3 after
it has been rented twice are the coordinates of the vector
In general, if
is any nonnegative integer, the vector
lists the probabilities that the car will be at the locations 1, 2 and 3
respectively after it has been rented
times.

We shall now consider the case in which the matrix
is given by the equation

Point at this equation and click on Define and
New Definition in order to supply this definition of
to *Scientific Notebook.*

We observe that

Work out the matrix
for some other positive integers
It seems clear that the sequence
approaches a limit matrix that is approximately

and a particularly interesting feature of this limit matrix is that all of its
columns are the same. In other words, if
is a sufficiently large positive integer then the probabilities that a car
that originated at any of the three locations will be at the locations 1, 2
and 3, respectively after it has been rented
times are

After the company has been in business for a long, about
of its cars will be at location 1, about
of its cars will be at location 2 and about
of its cars will be at location 2.

In order to work exactly with the matrix we must write it in a form that does not involve any decimals. We write in the form

and
we supply its definition to *Scientific Notebook*. By pointing at the
matrix
and clicking on Eigenvectors we see that one
eigenvector of
is

and
that the other two are given in the form

where
is a root of the equation
Solving this equation and substituting its solutions in the preceding formula
we see that the vectors

are
all eigenvectors of
We now define

and,
by pointing at the matrix
and clicking on Evaluate, we see that

Thus,
if
is a positive integer we have

and
so