Leslie matrices
A Leslie matrix is a discrete-time matrix model for an age-structured population.
Population vector
\[\mathbf n_t=\begin{pmatrix}n_1(t)\\n_2(t)\\n_3(t)\end{pmatrix}.\]Each component is the number of individuals in one age class.
Matrix update
\[\mathbf n_{t+1}=L\mathbf n_t,\]where a three-class Leslie matrix has the form
\[L=\begin{pmatrix}F_1&F_2&F_3\\S_1&0&0\\0&S_2&0\end{pmatrix}.\]The first row contains fertility terms \(F_i\), while the sub-diagonal entries \(S_i\) describe survival into the next age class.
Long-term behaviour
Repeated multiplication gives
\[\mathbf n_t=L^t\mathbf n_0.\]The dominant eigenvalue of \(L\) determines the asymptotic growth factor, while the corresponding eigenvector determines the stable age distribution when the usual conditions hold.
Key idea. Leslie matrices convert age-specific survival and fertility into a compact linear model of population change across discrete time steps.