A Digression into Matrix Algebra

One way to project the behavior of populations with these matrices is simply to write a little program, throw in the numbers and see what happens. It is possible, however, to be a bit more sophisticated. For any $n \times n$ matrix $\mathbf A$ there are $n$ pairs of scalars $\lambda_k$ and vectors $\mathbf x_k$ satisfying the following equation

$${\mathbf Ax}_k = \lambda_k{\mathbf x}_k$$

The $\lambda_k$ are conventionally numbered from the largest in absolute value to the smallest, and $\lambda_k$ is referred to as the $k$th eigenvalue, while ${\mathbf x}_k$ is the corresponding eigenvector.

Let $\mathbf X$ be the $n \times n$ matrix in which column $k$ is composed of ${\mathbf x}_k$, and let $\mathbf \Lambda$ be a diagonal matrix in which the $k$th diagonal element is $\lambda_k$.¹¹ Then

$$\begin{eqnarray*} {\mathbf AX} &=& {\mathbf X\Lambda} \\ {\mathbf A} &=& {\mathbf X\Lambda \mathbf X}^{-1} \\ \end{eqnarray*}$$

Thus,

$$\begin{eqnarray*} {\mathbf A}^t &=& ({\mathbf X\Lambda \mathbf X}^{-1})^t \\ &... ...=& \sum_{k=1}^n\lambda_k^t{\mathbf x}_k{\mathbf x}_k^{-1} \quad, \end{eqnarray*}$$

where ${\mathbf x}_k^{-1}$ refers to the vector forming the $k$th row of ${\mathbf X}^{-1}$.

We can apply these results to the projection of populations from one generation to the next. Recall that with both Leslie and Lefkovitch matrices the population dynamics can be summarized as

$${\mathbf n}(t) = {\mathbf A}{\mathbf n}(t-1)$$ (1)

$$= {\mathbf A}^t{\mathbf n}(0)$$ (2)

Thus,

$${\mathbf n}(t) =~\sum_{k=1}^n\lambda_k^t{\mathbf x}_k{\mathbf x}_k^{-1}{\mathbf n}(0)$$

All of the age categories grow asymptotically at the rate $\lambda_1$, because if $\lambda_1 > \lambda_k$, then $\lambda_1^t \gg \lambda_k^t$, which implies that ${\mathbf n}(t) \approx \lambda_1{\mathbf n}(t-1)$. The largest eigenvalue gives the asymptotic rate of population increase. Moreover, ${\mathbf n}(t) \approx \lambda_1{\mathbf n}(t-1)$ implies that ${\mathbf n}(t-1) \propto {\mathbf x}_1$. When the population has reached its asymptotic growth rate, the age-structure of the population is proportional to ${\mathbf x}_1$. The eigenvector corresponding to the largest eigenvalue gives the stable age structure.