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We will now describe a mathematical analysis which will reveal two important results. First, it will provide the ratio of R to F, the composition or structure of the population. Second, it will give the finite rate of change of the biennial population, which will be seen to be equivalent to the finite rate of change (X) in equation 2.2. Therefore this analysis makes the important assumption about the square matrix, M, that it can be replaced by a single value (X) and therefore that Mvt = Xvt. If this is true then the matrix equation 4.11 can be written as the density-independent equation 2.1, except now that vt+i and vt are population vectors rather than single numbers:

You should note that in multiplying the vector, vt, by X, that all elements of the matrix are multiplied by X. (X is a scalar.) Equating the right-hand side of equations 4.11 and 4.12 - values at time t - we have:

It is helpful to have the right-hand side of equation 4.13 in a matrix form similar to the left-hand side. To do this we employ the identity matrix, I. Multiplying any matrix by the identity matrix leaves the matrix unchanged (therefore M • I = M on the left-hand side):

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