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.

Reading fecundity and survival from a word problem

Exam questions give the rates in prose, and translating them correctly is where marks are won or lost. A fecundity rate is the average number of offspring a member of a given age class produces in one time period, and these go in the top row, one per age class. A survival rate is the proportion of an age class that lives to enter the next class, and these sit on the subdiagonal: the entry in row $2$, column $1$ is the fraction of class $1$ that survives to class $2$, and so on. Everything else is zero. A common trap is a fecundity rate quoted only for older classes (newborns rarely reproduce), so the first top-row entry is often $0$; read each rate to the class it belongs to before placing it.

The stable age distribution

When you project a Leslie model over many periods, two things settle down. First, the total population eventually multiplies by a constant growth factor each step, so successive totals form a geometric sequence. Second, the proportion of the population in each age class converges to a fixed ratio, the stable age distribution, no matter what the starting structure was. On a CAS you observe this by computing $L^n S_0$ for increasing $n$ and watching the ratios between the age classes stop changing. The growth factor is read as the long-run ratio of one total to the previous total, and a factor above $1$ signals sustained growth, exactly $1$ a steady state, and below $1$ decline.

Connection to transition matrices

The Leslie model shares its machinery with the transition matrix work: both project a column state matrix forward with $S_n = M^n S_0$ and both reveal long-term behaviour through high powers. The difference is interpretation. A transition matrix moves a fixed total between categories, so its columns sum to $1$, whereas a Leslie matrix can grow or shrink the total because births add new individuals and the oldest class dies out, so its columns do not sum to $1$. Recognising that the same matrix arithmetic underlies both models helps you set up calculations quickly and explains why the long-run analysis looks so similar.

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.