Question 19
An experiment researching the population changes of a certain species of insect was conducted over a four-week period. The insect has two distinct stages in its two-week lifespan. Each stage is approximately one week in length.
A constant proportion of females survive from stage 1 into stage 2.
The ratio of the reproduction rate for females in stage 2 to females in stage 1 is 2:1. All offspring are born into stage 1.
The number of females in each stage was measured initially and then again after two weeks as shown.
| Female population | Stage 1 | Stage 2 |
|---|---|---|
| Initially | 48 | 32 |
| After two weeks | 25 | 21 |
Use a matrix approach to estimate the total number of females after four weeks.
Two weeks of data let you work out both rates
Watch the life cycle turn into a matrix. You will see how the week 2 numbers pin down $x$ and $y$, which then give you week 4.
Each arrow in the life cycle becomes one entry of the matrix, which is why you need two unknowns and not four.
Choose the rates that match the week 2 count
Move both sliders. You need the predicted week 2 populations to land on the measured 25 and 21, shown as dashed outlines.
Work through the solution one mark at a time
Try each step yourself before you reveal it. The last mark is for how you set out your working, so you should define your variables as you go.
You need a birth rate and a survival rate. Let $x$ be the birth rate for stage 1 females. The ratio 2:1 means stage 2 females have a birth rate of $2x$. Let $y$ be the proportion of stage 1 females that survive into stage 2.
$$L=\begin{bmatrix}x&2x\\y&0\end{bmatrix}$$
Births go across the top row, because all offspring are born into stage 1. The bottom-right entry is 0, because stage 2 is the end of the lifespan. You can use any letters you like, and this mark can be implied by your later working.
Marker: correctly determines an appropriate Leslie matrix
Each stage lasts a week, so one multiplication by $L$ is one week. Let $P_{n}$ be the population after $n$ weeks. Two weeks is two steps, so you need $L^{2}$.
$$P_{0}=\begin{bmatrix}48\\32\end{bmatrix},\qquad P_{2}=L^{2}P_{0}$$
$$L^{2}=\begin{bmatrix}x^{2}+2xy&2x^{2}\\xy&2xy\end{bmatrix}$$
$$\begin{aligned}L^{2}P_{0}&=\begin{bmatrix}48x^{2}+96xy+64x^{2}\\48xy+64xy\end{bmatrix}\\&=\begin{bmatrix}112x^{2}+96xy\\112xy\end{bmatrix}=\begin{bmatrix}25\\21\end{bmatrix}\end{aligned}$$
Follow-through marks are allowed here, and this mark can be implied by your later working.
Marker: determines a matrix equation linking the initial population with the population after two weeks
Two matrices are equal when each entry matches, so you get one equation from each row.
$$112x^{2}+96xy=25\quad\ldots(1)$$
$$112xy=21\quad\ldots(2)$$
This mark can be implied by your later working.
Marker: determines two simultaneous equations in terms of the relevant birth and survival rates
Equation (2) gives you $xy$ straight away. You can put it into (1) as a single number, without finding $y$ first.
$$xy=\frac{21}{112}=0.1875\quad\ldots(3)$$
$$112x^{2}+96(0.1875)=25\ \Rightarrow\ 112x^{2}=7\ \Rightarrow\ x^{2}=0.0625$$
$$x=0.25\qquad y=\frac{0.1875}{0.25}=0.75$$
You reject $x=-0.25$, because a birth rate cannot be negative. Giving $x=\frac{1}{4}$ and $y=\frac{3}{4}$ is also accepted.
Marker: determines appropriate values of $x$ and $y$
Now you put the rates into $L$ and run it for four weeks on your calculator.
$$P_{4}=L^{4}P_{0}=\begin{bmatrix}0.25&0.5\\0.75&0\end{bmatrix}^{4}\begin{bmatrix}48\\32\end{bmatrix}\approx\begin{bmatrix}13.6\\12.6\end{bmatrix}$$
Add the two stages to get about 26 females. You can also get there with $L^{2}P_{2}$ from the week 2 numbers. Answers of 25 or 27 are accepted depending on your rounding, and so is 26.2.
Marker: determines the total number of females at the conclusion of the experiment
The last mark is for how you set out your working. You earn it by defining every letter you use and setting out the matrices so that each step is easy to follow.
About 26 females after four weeks
Finish with a sentence in context. A bare number at the bottom of the page does not show the marker that you have answered the question.
Marker: shows logical organisation of a fully attempted solution, communicating key steps
Each phrase of the question gave you something
You are not given the Leslie matrix. You have to build it from a description and then use two sets of data to work out the rates. Only after that can you run it forwards to answer the question.
The thing I would check first in your working is the power of $L$. Two weeks is two steps, so you need $L^{2}$ here. The QCAA love to space the measurements two steps apart so that you have to square the matrix before you can equate anything.
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 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.