Year 12 Discrete Mathematics: Leslie matrices

Leslie matrices are a specialised type of transition matrix used for age-structured population models. They describe how a population changes from one time step to the next when individuals survive into the next age class and certain classes contribute newborns. What makes Leslie matrices distinctive is their structure: fertility rates appear in the top row, and survival rates usually appear on the subdiagonal. Once that structure is recognised, the interpretation becomes much more manageable.

In a Leslie matrix, the top row usually creates new individuals, while the subdiagonal usually shifts survivors into the next age class.

Subtopic 1: Reading the matrix structure

A Leslie matrix is not just any arrangement of numbers. The positions carry biological meaning. The top row tells you how many new individuals are produced by each age group. The entries directly below the main diagonal tell you what proportion survives into the next class. Zero entries often mean no direct contribution or no direct transition. If you remember that structure, many otherwise intimidating models become easier to read.

Worked example 1

Problem: Consider

L = [[0, 1.2, 1.8], [0.5, 0, 0], [0, 0.7, 0]]

. Explain the meaning of the entry

1.8

and the entry

0.7

The entry $1.8$ is in the top row, third column.
So it represents the number of new first-class individuals produced by each third-class individual per step.
The entry $0.7$ is in row 3, column 2.
So it represents the survival or transfer rate from class 2 to class 3 in one step.

Answer:

1.8

is a fertility contribution from class 3, and

0.7

is a survival transition from class 2 to class 3.

Subtopic 2: One-step population updates

Once the Leslie matrix and the current population vector are known, the next population vector comes from matrix multiplication. The arithmetic is similar to other transition matrices, but the interpretation is more specialised. The first component of the new vector counts births from the reproductive classes, while the lower components are produced by survival into older groups.

Worked example 2

Problem: Using the Leslie matrix above and current population vector

[[50], [20], [10]]

, find the next population vector.

First component: $0 (50) + 1.2 (20) + 1.8 (10) = 42$ .
Second component: $0.5 (50) + 0 (20) + 0 (10) = 25$ .
Third component: $0 (50) + 0.7 (20) + 0 (10) = 14$ .

Answer: the next population vector is

[[42], [25], [14]]

Why Leslie matrices matter

Leslie matrices show how discrete mathematics supports modelling in a very concrete way. Instead of tracking one undifferentiated population, they let you separate the population into meaningful stages. That can reveal whether growth is being driven by fertility in older groups, by survival in younger groups, or by both together. The point is not only to calculate the next vector. It is to understand which parts of the model are creating change and why.

Worked example 3

Problem: In the previous example, explain why the second component of the new vector does not depend on the second or third components of the old vector.

The second row of the matrix is $[0.5, 0, 0]$ .
This row multiplies the old vector to create the new second component.
Because only the first entry in the row is non-zero, only the first old population class contributes.

Answer: the model says only first-class individuals survive into the second class in one step.

Common traps

Forgetting the special structure of fertility in the top row and survival on the subdiagonal.
Treating the matrix as a generic transition matrix without interpreting the age classes.
Mixing up which old class contributes to which new class.
Giving a numerical answer without explaining its population meaning.

Revision focus

A good revision method is to label the age classes directly beside the state vector and matrix. This makes the biological meaning of every entry much clearer and reduces the chance of interpreting the multiplication incorrectly. The more explicit the labels are, the less abstract the model feels.

It also helps to explain each row in words before computing: "this row makes births," or "this row moves class 1 into class 2." That kind of verbal summary often reveals whether your matrix setup matches the context before you spend time on arithmetic.

Practice links

Level 1: Leslie matrices