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Matrices, GEN1 2024 VCAA 30 MC

Data has been collected on the female population of a species of mammal located on a remote island.

The female population has been divided into three age groups, with the initial population (at the time of data collection), the birth rate, and the survival rate of each age group shown in the table below.
 

The Leslie matrix \((L)\) that may be used to model this particular population is

  1. \(L=\begin{bmatrix}0 & 1.8 & 0 \\ 0.7 & 0 & 1.2 \\ 0 & 0.6 & 0\end{bmatrix}\)
  2. \(L=\begin{bmatrix}0 & 1.8 & 1.2 \\ 0.7 & 0 & 0 \\ 0 & 0.6 & 0\end{bmatrix}\)
  3. \(L=\begin{bmatrix}0 & 1.8 & 1.2 \\ 0.7 & 0.6 & 0 \\ 0 & 0 & 0\end{bmatrix}\)
  4. \(L=\begin{bmatrix}2100 & 6400 & 4260 \\ 0 & 1.8 & 1.2 \\ 0.7 & 0.6 & 0\end{bmatrix}\)
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\(B\)

Show Worked Solution

\(\text{Row 1 represents the birth rates for the time periods: Eliminate A}\)

\(\text{The sub-diagonal represents survival rates: Eliminate C and D}\)

\(\Rightarrow B\)

Matrices, GEN1 2023 VCAA 31 MC

A species of bird has a life span of three years.

The females in this species do not reproduce in their first year but produce an average of four female offspring in their second year, and three in their third year.

The Leslie matrix, \(L\), below is used to model the female population distribution of this species of bird.
 

\(L=\begin{bmatrix}
0 & 4 & 3\\
0.2 & 0 & 0\\
0 & 0.4 & 0 
\end{bmatrix}\)
 

The element in the second row, first column states that on average 20% of this population will

  1. be female.
  2. never reproduce.
  3. survive into their second year.
  4. produce offspring in their first year.
  5. live for the entire lifespan of three years.
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\(C\)

Show Worked Solution

\(\text{Second and third rows represent survival rates from one year to the next.} \)

\(e_{21}\ \text{indicates an average of 20% survive into their second year.}\) 

\(\Rightarrow C\)