# The Rabbits and Foxes Problem

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.

#### Solving the Problem Numerically

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.

#### Solving the Problem Exactly

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.

# Markov Processes

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.

### The Car Rental Problem

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.

#### Numerical Approach to the Problem

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.

#### Exact Evaluation of this Limit

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

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