### 4.6.2 Revised model

We can now use the above definitions to build the initial model. Firstly, we establish relationships giving us the numbers of females in each age group at time t+1. Take the lowest age group first, the number of members in this group is given by:

The numbers in the remaining classes all follow the same pattern which is given below:

These relationships are best expressed in matrix form as shown below.

A neater way of representing this matrix equation is

nt+1 = Mnt

from which it is clear that

n1 = Mn0

n2 = Mn1 = M2 n0

n3 = Mn2 = M3 n0

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so that in general, after p time periods, the population is described by the matrix equation

np = Mpn0

The matrix M is square and of order m+1 and has interesting mathematical properties which you will be asked to investigate later. In terms of the modelling process, we need to note that there are m+1 eigenvalues and corresponding eigenvectors x which satisfy

and that we can use certain properties of the matrix M and interpret them in terms of a population model.