The Rabbits and Foxes Problem

In this example we are concerned with the populations of rabbits and foxes in a national park. Suppose that $1000$ rabbits and $1000$ foxes are introduced into the park which previously contained no rabbits or foxes and for every nonnegative integer $n,$ the numbers of rabbits and foxes in the path after $n$ months are $R\left( n\right) $ and $F\left( n\right) $, respectively. Suppose finally that for each $n$ we have
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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
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then, for each $n,$
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and so
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By pointing at the matrix $A$ and clicking on Eigenvalues we see that the two eigenvalues of this matrix are $1$ and $0.7.$ Since the eigenvalue $1$ has multiplicity only $1$ and the other eigenvalue has absolute value less than $1,$ the above theorem tells us that the sequence MATH is convergent.

Solving the Problem Numerically

By clicking on Evaluate Numerically we see that
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MATH

MATH

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Looking at these matrices we can conjecture that
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as MATH and that, in the long term, the numbers of rabbits and foxes will approach the coordinates of the vector
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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 $A$ we must rewrite it in a form that does not contain decimals. We therefore write the matrix $A$ as
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By pointing at the matrix $A$ and clicking on Eigenvectors we obtain $A$,
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We define
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and we supply this definition to Scientific Notebook by clicking on Define and New Definition. Since
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we deduce that if $n$ is any positive integer then
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Thus
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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 $i$ and $j$ in MATH we shall use the symbol $p_{ij}$ to describe the probability that a customer who has rented a car at location $j$ will return it to location $i.$ We observe that if MATH then a car rented at location $j$ must be returned to one of the three locations. Therefore
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Thus, if we define a matrix $P$ by the equation
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then the sum of the entries in each column of the matrix $P$ must be $1.$ 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 $P$ is called the transition matrix of the Markov process. The first column
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of the matrix $P$ 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
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of the matrix $P$ 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 $x_{1},$ $x_{2}$ and $x_{3}$, respectively. Note that $x_{1},$ $x_{2}$ and $x_{3}$ are nonnegative numbers and that MATH 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
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Repeating the process we see that if the vector
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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 $P^{2}\QTR{bf}{x.}$ In general, if $n$ is any nonnegative integer, the vector $P^{n}\QTR{bf}{x}$ lists the probabilities that the car will be at the locations 1, 2 and 3 respectively after it has been rented $n$ times.

We shall now consider the case in which the matrix $P$ is given by the equation
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Point at this equation and click on Define and New Definition in order to supply this definition of $P$ to Scientific Notebook.

Numerical Approach to the Problem

We observe that
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MATH

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Work out the matrix $P^{n}$ for some other positive integers $n.$ It seems clear that the sequence MATH approaches a limit matrix that is approximately
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and a particularly interesting feature of this limit matrix is that all of its columns are the same. In other words, if $n$ 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 $n$ times are
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After the company has been in business for a long, about $55.7\%$ of its cars will be at location 1, about $22.9\%$ of its cars will be at location 2 and about $21.3\%$ of its cars will be at location 2.

Exact Evaluation of this Limit

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


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and we supply its definition to Scientific Notebook. By pointing at the matrix $P$ and clicking on Eigenvectors we see that one eigenvector of $P$ is
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and that the other two are given in the form
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where $\rho $ is a root of the equation MATH Solving this equation and substituting its solutions in the preceding formula we see that the vectors
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are all eigenvectors of $P.$ We now define
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and, by pointing at the matrix $U^{-1}PU$ and clicking on Evaluate, we see that
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Thus, if $n$ is a positive integer we have
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and so
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