Question 14
The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year, but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year.
A Leslie matrix, $\mathbf{L}$, modelling the population distribution of the Tasmanian thornbill, has been partially completed.
$$\mathbf{L}=\begin{bmatrix}0&4&4\\x&0&0\\0&y&0\end{bmatrix}$$
State the values of $x$ and $y$.
[1 mark]At the start of 2021, a study began into the population of Tasmanian thornbills. The study:
- estimated that the initial female population was 510 in their first year, 480 in their second year and 420 in their third year
- found that the ratio of male to female was approximately 1:2.
Estimate the total population of Tasmanian thornbills at the start of 2025.
[4 marks]Move the females on four years, then add the males
Each year the survivors move up an age group and the chicks fill the youngest group. Multiplying by the Leslie matrix does both at once, for the females only.
The number of females jumps up and down from year to year, because first-year females have no chicks.
Project the thornbills to any year
Move the slider to choose a year. The readout is what your GDC gives for $\mathbf{L}^k\mathbf{N}_0$, where $k$ is the number of years after the start of 2021. Switch to the whole population to add the males on top.
This is $\mathbf{N}_0$, the initial female population from the study. There are no steps yet.
This is the start of 2025, four years after the study began, so the power is 4. These are the female numbers the marking guide uses.
The number of females rises and falls from year to year. First-year females have no chicks, so a big group of chicks only starts breeding a year later.
Male to female is $1:2$, so there is one male for every two females. The males are half the number of females, and the whole population is $\tfrac{3}{2}$ of the females.
Work through the solution one mark at a time
Try each step yourself before you reveal it. The steps follow the QCAA marking guide. This is a Technology-active question, so the matrix power comes from your GDC.
In a Leslie matrix the birth rates go across the top row, and you can see them already: 0 for first-years and 4 for each of the older groups. The survival rates sit on the diagonal just below the main diagonal. Row 2 is the first-years who survive into their second year, and row 3 is the second-years who survive into their third year.
The QCAA accept $0.25$ and $0.3$ written in that order, even if you don’t say which is $x$ and which is $y$. Percentages go in as decimals, so 25% becomes $0.25$.
Marker: correctly states the values of $x$ and $y$
The study’s female numbers at the start of 2021 are the initial population matrix. Write them as a column, youngest group at the top, in the same order as the rows of $\mathbf{L}$.
$$\mathbf{N}_0=\begin{bmatrix}510\\480\\420\end{bmatrix}$$
This mark can be implied by your later working.
Marker: correctly identifies the initial matrix
Each multiplication by $\mathbf{L}$ moves the population on one year. The start of 2025 is four years after the start of 2021, so the power is 4.
$$\mathbf{N}_4=\mathbf{L}^4\mathbf{N}_0=\begin{bmatrix}2166\\938.25\\81.45\end{bmatrix}$$
On your GDC, store $\mathbf{L}$ as a $3\times3$ matrix and $\mathbf{N}_0$ as a $3\times1$ matrix, then calculate $\mathbf{L}^4\times\mathbf{N}_0$. The QCAA accept rounded values such as $2166$, $938$ and $81$. This mark can be implied by your next line, and follow-through applies if your matrix was wrong.
If you label the start of 2021 as $\mathbf{N}_1$, as Game plan 5.4 does, the start of 2025 is $\mathbf{N}_5=\mathbf{L}^4\mathbf{N}_1$. The label changes, and the power is still 4.
Marker: determines female population in their first, second and third years at the start of 2025
Add the three age groups to get all the females.
$$\begin{aligned}&2166+938.25+81.45\\&=3185.7\ \text{females}\end{aligned}$$
The QCAA sample response rounds each group first, $2166+938+81=3185$. The decimal value $3185.7$ is accepted too, and this mark can be implied by your next line.
Marker: determines total female population at the start of 2025
A Leslie matrix only counts females, and the question asks for the total population. Male to female is $1:2$, so split the population into three equal parts: one part male and two parts female. The females are two parts, so one part is half of them and the whole population is three parts.
$$\begin{aligned}\text{total}&\approx\tfrac{3}{2}\times3185.7\\&\approx4778.55\end{aligned}$$
The mark needs a whole number. From the rounded total, $\tfrac{3}{2}\times3185=4777.5$, which the QCAA sample response gives as $4778$. Either answer is accepted, as are nearby values such as $4777$ and $4781$.
The marking guide prints this line as $\tfrac{3}{2}\times3186\approx4778$, but $\tfrac{3}{2}\times3186=4779$. The $4778$ comes from the total of $3185$ on the line before.
Marker: determines total population at the start of 2025, rounded to a whole number
Each phrase of the question gave you something
Reading a Leslie matrix from a description and projecting it forward on your GDC is the standard Leslie matrix question. Part a) is one look at where survival rates go, and part b) is one matrix power followed by two lines of arithmetic.
If I had to guess where you might lose marks, it would be the last line. The Leslie matrix only counts females, so stopping at 3185.7 leaves the final mark behind. Read the ratio carefully too. One male for every two females means multiplying by $\tfrac{3}{2}$. Multiplying by 3 treats the ratio as two males for every female. The other slip is the power: the study starts in 2021, so 2025 needs $\mathbf{L}^4$.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2021 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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