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|Age Class||Age interval|
Note that the intervals do not include the right-hand endpoint. For example, the age a of a female in class 3 must satisfy .
Next, we use the vector to represent the population at some particular point in time. That is, the component of is the number of females in the age class. As we have done before, we want to create a model that lets us relate the present population distribution to the population distribution in the past. To do so, we will use the following assumptions.
To make our model quantitative we define the following parameters. The birth coefficients ai, denote the average number of females born to a single female while she is a member of the age class. For example, if L=20 and n=5 as above, there would be five birth coefficients and a3, for example, would be the average number of female babies born to a female between the ages of 8 and 12. We also define the parameters bi, by saying that bi is the fraction of the females in class i that can be expected to survive and pass into the class. For example, suppose that at the last measurement period there were 100 females in age class 4 and that b4 = 3/4. Then we would predict that there would be 75 females in class 5 the next time the population is measured. Note that our assumption that no member of the population lives longer than L years means that bn = 0. Given these parameters, can be calculated from with the equation
where the matrix A looks like the following.
A matrix of this form is called a Leslie matrix and turns out to have some very special properties.
The case of a stable steady state is the easiest one to begin with. We already know that for there to be a steady state, there has to be a solution to the equation
So being able to find such a vector is necessary for there to be a stable population. However, even if we can find such a steady state, there is no guarantee that our simulations will approach the steady state.
It turns out that the key to stability, as well as the key to understanding why a population eventually grows or eventually decreases, is the notion of eigenvalue. We say that is an eigenvalue for an matrix A if we can find a vector satisfying the equation
The vector is called an eigenvector of the matrix A corresponding to the eigenvalue .
If we rewrite the equation above as the homogeneous system
then it should be clear that is an eigenvalue if and only if the following condition is satisfied.
It can be shown that is a polynomial of degree n in . This polynomial turns out to be so important in understanding the properties of the matrix A that it is given a special name. The polynomial is called the characteristic polynomial of the matrix A.
For example, suppose A is given by
then the characteristic polynomial is
The roots of the characteristic polynomial are the eigenvalues of the matrix A. In our simple example above, the eigenvalues are easily found to be -1 and -3. It isn't always so easy. In the exercises you will be working with matrices so the characteristic polynomials are fifth order, which usually can't be solved analytically. This means that you usually have to resort to numerical techniques, which can make the problem very difficult.
Now, what does this have to do with our population growth model? The answer is that a Leslie matrix, under certain conditions satisfied by all of the examples you will see in this project, has a single dominant eigenvalue . This eigenvalue is always a positive number and essentially determines the long-term behavior of the model as follows.
To get an idea of how this happens, suppose that is the dominant eigenvalue and is the corresponding eigenvector. If for some value of k we have , then we would have
On the next iteration, you would get , and so on, with the next population just being the previous population multiplied by . So, if , the population keeps getting larger and larger. If , the population stays the same and if , the population keeps decreasing.
This shows how the dominant eigenvalue controls the behavior if , but it doesn't tell the whole story. It is a little beyond the scope of this project, but it can be shown that no matter what the initial population vector is, it eventually approaches a multiple of the dominant eigenvector. When this happens, you get the behavior described above.
William W. Farr