Leslie Matrices (VCE SSCE General Mathematics): Revision Notes

Leslie Matrices

Leslie matrices are mathematical tools used to model how populations change over time. They are particularly useful for tracking different age groups within a population and predicting future population sizes.

Background and purpose

These matrices help us understand:

• How population sizes change across different age groups
• The overall growth or decline of a population
• Long-term population trends

Understanding age groups

To use a Leslie matrix, we first divide the population into age groups. Each age group covers the same time period, and together they span the entire lifespan of the species.

Example with humans: If we use a time period of 10 years, we might have eleven age groups:

Age group $(i)$	1	2	3	4	$\cdots$	10	11
Age range (years)	0-10	10-20	20-30	30-40	$\cdots$	90-100	100-110

Key components of Leslie matrices

Birth rates

The birth rate, $b_i$, represents the average number of female offspring born to a mother in age group $i$ during one time period.

Survival rates

The survival rate, $s_i$, represents the proportion of females in age group $i$ who survive to move into age group $i + 1$.

Example: If the survival rate for age group 2 is $s_2 = 0.95$, this means 95% of females aged 10-20 years will survive to reach age group 3 (20-30 years).

Structure of a Leslie matrix

Simple example with three age groups

Let's examine a population with a lifespan of 9 years, divided into three age groups with a time period of 3 years:

Age group $(i)$	1	2	3
Age range (years)	0-3	3-6	6-9

For three age groups, the Leslie matrix is a $3 \times 3$ matrix:

$L = \begin{bmatrix} b_1 & b_2 & b_3 \\ s_1 & 0 & 0 \\ 0 & s_2 & 0 \end{bmatrix}$

This diagram shows how the population transitions between age groups, with curved arrows showing births and straight arrows showing survival.

Population state matrix

The population state matrix is a column vector that records the number of females in each age group at a specific time.

The initial population state matrix is denoted $S_0$.

Calculating future populations

Using the recursive formula

To find the population after one time period, we multiply the Leslie matrix by the initial state matrix:

$S_1 = LS_0 = \begin{bmatrix} 0 & 2.3 & 0.4 \\ 0.6 & 0 & 0 \\ 0 & 0.3 & 0 \end{bmatrix} \begin{bmatrix} 400 \\ 400 \\ 400 \end{bmatrix} = \begin{bmatrix} 1080 \\ 240 \\ 120 \end{bmatrix}$

For the second time period:

$S_2 = LS_1 = \begin{bmatrix} 0 & 2.3 & 0.4 \\ 0.6 & 0 & 0 \\ 0 & 0.3 & 0 \end{bmatrix} \begin{bmatrix} 1080 \\ 240 \\ 120 \end{bmatrix} = \begin{bmatrix} 600 \\ 648 \\ 72 \end{bmatrix}$

Total population after two time periods: $600 + 648 + 72 = 1320$

Pattern in calculations

Notice the pattern:

$S_1 = LS_0$

$S_2 = LS_1 = L^2S_0$

$S_3 = LS_2 = L^3S_0$

In general:

$S_{n+1} = LS_n = L^nS_0$

The explicit formula

To find the population state matrix after $n$ time periods, we use:

$S_n = L^nS_0$

General form of Leslie matrices

An $m \times m$ Leslie matrix (for $m$ age groups) has the form:

$L = \begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_{m-1} & b_m \\ s_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & s_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & s_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & s_{m-1} & 0 \end{bmatrix}$

where:

• $m$ is the number of age groups
• $s_i$ is the survival rate (proportion moving from age group $i$ to age group $i+1$)
• $b_i$ is the birth rate (average number of female offspring from age group $i$)

Formulas for population calculations

Worked example: Determining state matrices and life cycle diagrams

Worked example: Entering information into matrices

Long-term (limiting) behaviour

After many time periods, populations modelled by Leslie matrices can exhibit different behaviours:

• Increase without bound
• Decrease towards zero
• Stabilise at constant proportions
• Oscillate in cycles

Limiting behaviour with constant growth rate

Sometimes, after enough time periods, the proportion of the population in each age group stabilises. When this happens, we can find a constant growth rate $k$ such that:

$LS_n = kS_n$

This means each age group grows by the same factor $k$ each time period.

Different types of long-term behaviour

Consider $L = \begin{bmatrix} 0 & 0 & b_3 \\ 0.25 & 0 & 0 \\ 0 & 0.5 & 0 \end{bmatrix}$ with $S_0 = \begin{bmatrix} 1000 \\ 0 \\ 0 \end{bmatrix}$

Calculator techniques

Remember!