In a previous page, we studied the movement between the city and suburbs. Indeed, if *I* are *S* are the initial population of the inner city and the suburban area, and if we assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city, then after one year the populations are given by

The matrix

is very special. Indeed, the entries of each column vectors are positive and their sum is 1. Such vectors are called

A Markov chain is a process that consists of a finite number of

Of particular interest is a probability vector

This system reduces to the equation -0.4

So the vector is a steady state vector of the matrix above. So if the populations of the city and the suburbs are given by the vector , after one year the proportions remain the same (though the people may move between the city and the suburbs).

Let us discuss another example on population dynamics.

**Example: Age Distribution of Trees in a Forest**

Trees in a forest are assumed in this simple model to fall into four age
groups: *b*(*k*) denotes the number of baby trees in the forest (age group
0-15 years) at a given time period *k*; similarly *y*(*k*),*m*(*k*) and *o*(*k*)
denote the number of young trees (16-30 years of age), middle-aged trees
(age 31-45), and old trees (older than 45 years of age), respectively.
The length of one time period is 15 years.

How does the age distribution change from one time period to the next?
The model makes the following three assumptions:

- A certain percentage of trees in each age group dies.
- Surviving trees enter into the next age group; old trees remain old.
- Lost trees are replaced by baby trees.

Note that the total tree population does not change over time.

We obtain the following difference equations:

b(k+1) |
= | (1) | |

y(k+1) |
= | (1-d_{b}) b(k) |
(2) |

m(k+1) |
= | (1-d_{y}) y(k) |
(3) |

o(k+1) |
= | (1-d_{m}) m(k) + (1-d_{o}) o(k) |
(4) |

Here 0 <

Let

be the ``age distribution vector". Consider the matrix

Then we have

Note that the matrix

If,

After easy calculations, we find the steady state vector for the age distribution in the forest:

Assume a total tree population of 50,000 trees. Suppose the forest is newly planted, i.e.

After 15 years, the age distribution in the forest is given by

After 30 years, we have

and after 45 years

After 15n years, where , the age distribution in the forest is given by

So the problem is to find the

The calculations on the example above becomes tedious. Let us illustrate the problem on a small matrix.

**Example.** Consider the stochastic matrix

Note this is a symmetric matrix. The characteristic polynomial of

An eigenvector associated to 1 is

and an eigenvector associated to 0.6 is

If we set

then we have

So, we have

When

So the sequence of vectors defined by

will get closer to

when

Note that the vector is proportional to the unique steady state vector of

This is not surprising. In fact there is a general result similar to the one above for any stochastic matrix.

**
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

**Author**: M.A. Khamsi

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