Leslie matrices and population models: VCE General Mathematics Unit 4 Matrices

What this dot point is asking

The Study Design includes the "use of the matrix recurrence relation: $S_0 =$ initial state matrix, $S_{n+1} = TS_n$ or $S_{n+1} = LS_n$ where $T$ is a transition matrix, $L$ is a Leslie matrix, and $S$ is a column state matrix, to generate a sequence of state matrices (assuming the next state only relies on the current state)". For Leslie matrices that means you should be able to:

1. build a Leslie matrix from birth and survival rates given in words, a table or a life-cycle diagram;
2. interpret any element of the matrix in context;
3. project the age structure forward with $S_{n+1} = LS_n$ (or $S_n = L^nS_0$), including totals, percentages and rounding;
4. describe the long term: the growth rate of the total population and the proportions in each age group.

The Leslie questions on recent papers have been one-mark multiple-choice items: build the matrix from a life-cycle diagram (2025 Examination 1 Question 29, $84\%$) and interpret one element (2023 Examination 1 Question 31, $74\%$). Both are quick if you know where each kind of rate goes. Projection questions use the same working as transition matrices.

The answer

What the model assumes

A Leslie model divides the female population into age groups of equal width, and uses a time step equal to that width (for example, age groups of one year and a time step of one year). In each time step:

• every female in age group $j$ produces, on average, $b_j$ female offspring, who join age group 1;
• a proportion $s_j$ of the females in age group $j$ survive and move up to age group $j + 1$;
• females in the oldest age group do not move on (in the basic model they die at the end of the step).

Only females are counted because the number of births depends on the number of females.

The structure of $L$

For four age groups:

• Row 1 holds the birth rates. $b_1$ is often 0 (the youngest group does not breed). Birth rates can be greater than 1.
• The subdiagonal holds the survival rates. The element in row $j + 1$, column $j$ is the proportion of group $j$ that survives into group $j + 1$. Survival rates are between 0 and 1.
• Everything else is 0, including the leading diagonal (no one stays in the same age group) and the last column below row 1.

Unlike a transition matrix, the columns of a Leslie matrix do not add to 1: births add individuals and deaths remove them, so the total population can grow or shrink.

Reading a life-cycle diagram

A life-cycle diagram draws each age group as a vertex. Straight arrows from group $j$ to group $j + 1$ carry the survival rates; curved arrows back to group 1 carry the birth rates. The diagram below is the one from 2025 Examination 1 Question 29.

Reading it: the top row is $\begin{bmatrix} 0 & 2.1 & 4.6 & 1.8 \end{bmatrix}$ (group 1 has no birth arrow) and the survival rates $0.9$, $0.7$ and $0.2$ go in positions $(2, 1)$, $(3, 2)$ and $(4, 3)$.

Projecting forward

With $S_0$ the initial column matrix of the number of females in each age group,

Multiplying out one step shows what each row does:

• row 1 gives the newborns: $b_1 \times (\text{group 1}) + b_2 \times (\text{group 2}) + \dots$;
• row $j + 1$ gives the survivors moving up: $s_j \times (\text{group } j)$.

The total population after $n$ periods is the sum of the elements of $S_n$. Keep full accuracy in the matrix and round only the final answer (numbers of animals are usually rounded to the nearest whole number).

To go backwards, from $S_1$ to $S_0$, use $S_0 = L^{-1}S_1$. This needs $\det L \neq 0$, which happens when the oldest group has a non-zero birth rate and all survival rates are non-zero.

The long term: growth rate and stable age distribution

If you calculate $S_n$ for larger and larger $n$ (CAS: $L^{50}S_0$, $L^{51}S_0$, and so on), two things usually settle down:

1. the ratio of successive totals approaches a constant $r$. The population then grows (or declines) by the same percentage every period: $(r - 1) \times 100\%$;
2. the proportion of the population in each age group stops changing. These proportions are the stable age distribution.

If $r > 1$ the population grows, if $r = 1$ it is stable, and if $r < 1$ it declines. Both $r$ and the stable distribution come from the matrix $L$, not from the starting population, although the starting population affects how quickly they appear.

The Study Design asks for this informally: "no noticeable change from one state matrix to the next" once you allow for the growth factor. One caution: when only one age group breeds, the numbers cycle (each individual takes exactly the same number of steps from birth to breeding), so check several consecutive ratios rather than just one.

Leslie matrices and transition matrices

Both models use a recurrence relation of the form $S_{n+1} = MS_n$ and powers of a matrix to look ahead, and in both the long-term state appears when the state matrix stops changing (for Leslie models, when the proportions stop changing). The differences are in what the matrix means:

• a transition matrix moves a fixed total between states, so each column adds to 1 and the total stays the same;
• a Leslie matrix adds newborns and loses animals that die, so its columns do not add to 1 and the total grows or shrinks.

Culling and restocking are modelled by adding a column matrix to each step, $S_{n+1} = MS_n + B$ (see the transition matrices page for that form).