Set up a Leslie matrix from age-specific birth (fecundity) and survival rates, multiply it by an age-structure state matrix to project a population forward, and interpret the long-term growth and age distribution

What this dot point is asking

VCAA wants you to model an age-structured population with a Leslie matrix. The top row holds the birth (fecundity) rates for each age group, and the subdiagonal holds the survival rates that move individuals to the next age group. Multiplying the Leslie matrix by the current age-structure state matrix projects the population one time step forward, and repeated multiplication shows long-term growth and a stable age distribution. This extends the transition-matrix idea to populations that age.

Building the Leslie matrix

For three age classes with birth rates $b_1, b_2, b_3$ and survival rates $s_1$ (class 1 to 2) and $s_2$ (class 2 to 3), the Leslie matrix is

The top row generates newborns into class 1. Each survival rate sits on the subdiagonal, moving survivors down to the next age class. The oldest class has no survival entry, so it does not persist beyond its age band.

Projecting the population forward

Let $S_0$ be the starting numbers in each age class. The population after one step is $S_1 = L S_0$, and after $n$ steps $S_n = L^n S_0$.

Long-term behaviour

Raising $L$ to a large power and multiplying by $S_0$ shows the long-term picture: the total population eventually multiplies by a constant growth factor each step, and the proportions in each age class stabilise to a fixed stable age distribution, regardless of the starting numbers. On the calculator, project several steps and watch the ratios between age classes settle.

Why this matters for the exams

Leslie matrix questions test whether you can build the matrix correctly from birth and survival data and project a population forward, with the calculator doing the powers. Markers reward the right placement of fecundity and survival entries and a sensible interpretation of long-term growth. The model is a direct relative of the transition matrix, sharing the $S_n = M^n S_0$ machinery.