Model change between generations

A Leslie matrix acts on a column vector of age-class populations. Fertility rates occupy the first row; survival proportions move populations into the next age class. Repeated multiplication gives later populations.

A dominance matrix can encode one-step wins; squaring it counts two-step paths. Model outputs depend on assumptions: fixed survival, fixed fertility and a consistent time interval. Other matrix applications can be investigated in class; external assessment focuses on dominance and Leslie models.

\displaystyle \mathbf n_{t+1}=L\mathbf n_t,\quad\mathbf n_t=L^t\mathbf n_0

WORKED EXAMPLE

Use $L=\begin{pmatrix}0&2\\0.5&0\end{pmatrix}$ and $\mathbf n_0=(100,40)^T$ to find the next generation.

Young population: 0(100) + 2(40) = 80.
Older population: 0.5(100) + 0(40) = 50.
The next population vector is (80, 50)ᵀ.

Assumed knowledge

Multiply matrices and column vectors, calculate matrix powers, and interpret proportions and rates. Keep the order of categories and the time interval consistent in a numerical model.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Model change between generations

Use L=(020.50)L=\begin{pmatrix}0&2\\0.5&0\end{pmatrix}L=(00.5​20​) and n0=(100,40)T\mathbf n_0=(100,40)^Tn0​=(100,40)T to find the next generation.

Put the idea to work.

Use $L=\begin{pmatrix}0&2\\0.5&0\end{pmatrix}$ and $\mathbf n_0=(100,40)^T$ to find the next generation.