Activity 4.4.5
The following type of analysis has been used to study the population of a bison herd. We will divide the population of female bison into three groups: juveniles who are less than one year old; yearlings between one and two years old; and adults who are older than two years.
Each year,
80% of the juveniles survive to become yearlings.
90% of the yearlings survive to become adults.
80% of the adults survive.
40% of the adults give birth to a juvenile.
By \(J_k\text{,}\) \(Y_k\text{,}\) and \(A_k\text{,}\) we denote the number of juveniles, yearlings, and adults in year \(k\text{.}\) We have
Find similar expressions for \(Y_{k+1}\) and \(A_{k+1}\) in terms of \(J_k\text{,}\) \(Y_k\text{,}\) and \(A_k\text{.}\)
As is usual, we write the matrix \(\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{.}\) Write the matrix \(A\) such that \(\xvec_{k+1} = A\xvec_k\text{.}\)

We can write \(A = PEP^{1}\) where the matrices \(E\) and \(P\) are approximately:
\begin{equation*} \begin{aligned} E \amp {}={} \left[\begin{array}{rrr} 1.058 \amp 0 \amp 0 \\ 0 \amp 0.128 \amp 0.506 \\ 0 \amp 0.506 \amp 0.128 \\ \end{array}\right], \\ \\ P \amp {}={} \left[\begin{array}{rrr} 1 \amp 1 \amp 0 \\ 0.756 \amp 0.378 \amp 1.486 \\ 2.644 \amp 0.322 \amp 1.264 \\ \end{array}\right]\text{.} \end{aligned} \end{equation*}Make a prediction about the longterm behavior of \(\xvec_k\text{.}\) For instance, at what rate does it grow? For every 100 adults, how many juveniles, and yearlings are there?
Suppose that the birth rate decreases so that only 30% of adults give birth to a juvenile. How does this affect the longterm growth rate of the herd?
Suppose that the birth rate decreases further so that only 20% of adults give birth to a juvenile. How does this affect the longterm growth rate of the herd?
Find the smallest birth rate that supports a stable population.