Question 2
The Leslie matrix for a certain endangered species is given.
$$\mathbf{L}=\begin{bmatrix}0.8&2.4&0.3\\0.4&0&0\\0&0.55&0\end{bmatrix}$$
A group of the species was moved into a secure property at the start of 2018. The initial female population is given.
$$\mathbf{N}_0=\begin{bmatrix}150\\80\\40\end{bmatrix}$$
The best estimate of the total female population at the start of 2025 is
Not quite. $3000$ is the estimate for the start of 2024. $\mathbf{L}^6\mathbf{N}_0$ has a total of about $2851$, which is six steps. From the start of 2018 to the start of 2025 is $2025-2018=7$ steps, so use $\mathbf{L}^7$.
That’s it. Step through the hints below to see why.
Not quite. $5000$ comes from multiplying in the wrong order. With the population as a row on the left, $\begin{bmatrix}150&80&40\end{bmatrix}\mathbf{L}^7$ has a total of about $5339$. Matrix multiplication isn’t commutative. The population is a column on the right: $\mathbf{N}_7=\mathbf{L}^7\mathbf{N}_0$.
Not quite. $6000$ is the estimate for the start of 2026. $\mathbf{L}^8\mathbf{N}_0$ has a total of about $6241$. Listing 2018, 2019, …, 2025 gives eight years, but there are only seven steps between the start of 2018 and the start of 2025.
Total number of females at the start of each year, from 2018 (’18) to 2026 (’26). The blue dashed lines are the four options.
This is the start of 2025, seven steps on from the start of 2018. The total is about $4221$, so the best estimate is $4000$. This is option (B).
This is the start of 2024, only six steps on. Its total of about $2851$ gives option (A). The start of 2025 is one more step.
This is the start of 2026, eight steps on. Its total of about $6241$ gives option (D). Listing the years 2018 to 2025 gives eight years, but there are only seven steps between their starts.
This is the right number of steps, but the population is a row multiplied on the left of $\mathbf{L}^7$. That mixes up the fecundity and survival rates. Its total of about $5339$ gives option (C).
With the population as a row on the left, the totals don’t follow the model. The population has to be a column on the right of $\mathbf{L}^n$.
Each step multiplies by $\mathbf{L}$ once and moves the population on one year. The question wants the start of 2025.
Multiplying by $\mathbf{L}$ moves the female population on one time step, here one year. After $n$ steps
$$\mathbf{N}_n=\mathbf{L}^n\mathbf{N}_0$$
The population stays a column matrix, on the right.
Count the steps between the two dates. $\mathbf{N}_0$ is the start of 2018.
$$n=2025-2018=7$$
Enter $\mathbf{L}$ and $\mathbf{N}_0$ in your GDC and calculate $\mathbf{L}^7\mathbf{N}_0$.
$$\mathbf{N}_7=\mathbf{L}^7\mathbf{N}_0\approx\begin{bmatrix}3079.6\\831.4\\310.4\end{bmatrix}$$
Add the three age groups for the total.
$$3079.6+831.4+310.4\approx4221$$
From the start of 2018 to the start of 2025 is $7$ steps.
$$\mathbf{N}_7=\mathbf{L}^7\mathbf{N}_0\approx\begin{bmatrix}3079.6\\831.4\\310.4\end{bmatrix}$$
$$\text{total}\approx3079.6+831.4+310.4\approx4221$$
The best estimate of the total female population at the start of 2025 is $4000$.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it