Question 4
The age-specific population distribution of a particular species of animal is shown.
The Leslie matrix based on this data is
Not quite. The female populations make up the population vector that the matrix multiplies. They don’t go inside the Leslie matrix. In 2023, 9.3% of Queensland students chose (A).
Not quite. The class numbers 1 to 4 don’t belong in the matrix. The top row holds the breeding rates, and the survival rates sit below the main diagonal. In 2023, 1.0% of Queensland students chose (B).
Not quite. The two rows of rates have swapped. The breeding rates belong in the top row and the survival rates go below the main diagonal. In 2023, 6.6% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2023, 82.8% of Queensland students chose (D).
This is the Leslie matrix. Multiplying it by the population vector gives next year’s population in each age class, which is option (D).
This is option (C). The breeding and survival rates have swapped, so the newborn count uses survival rates.
This puts population counts in the matrix. The counts belong in the vector, so the products here have no meaning.
The same rates appear in both places. Each kind of rate has its own position in a Leslie matrix.
The top row of a Leslie matrix holds the breeding rates, one for each age class. Newborns come from every class that breeds.
The survival rates go just below the main diagonal. The survival rate of class 1 sits in row 2, column 1, the rate of class 2 in row 3, column 2, and so on.
The last survival rate, $0$, has nowhere to go, because there is no fifth class. Every other entry is $0$.
$$\begin{bmatrix}0&1.3&0.9&0.2\\0.6&0&0&0\\0&0.8&0&0\\0&0&0.4&0\end{bmatrix}$$
Breeding rates go across the top row and survival rates go below the main diagonal. The female populations form the population vector instead.
$$L=\begin{bmatrix}0&1.3&0.9&0.2\\0.6&0&0&0\\0&0.8&0&0\\0&0&0.4&0\end{bmatrix}$$
How Queensland students answered in 2023
From the QCAA Specialist Mathematics subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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