###### Activity4.4.5

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

\begin{equation*} J_{k+1} = 0.4 A_k\text{.} \end{equation*}
1. Find similar expressions for $$Y_{k+1}$$ and $$A_{k+1}$$ in terms of $$J_k\text{,}$$ $$Y_k\text{,}$$ and $$A_k\text{.}$$

2. As is usual, we write the matrix $$\xvec_k=\threevec{J_k}{Y_k}{A_k}\text{.}$$ Write the matrix $$A$$ such that $$\xvec_{k+1} = A\xvec_k\text{.}$$

3. 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 long-term behavior of $$\xvec_k\text{.}$$ For instance, at what rate does it grow? For every 100 adults, how many juveniles, and yearlings are there?

4. Suppose that the birth rate decreases so that only 30% of adults give birth to a juvenile. How does this affect the long-term growth rate of the herd?

5. Suppose that the birth rate decreases further so that only 20% of adults give birth to a juvenile. How does this affect the long-term growth rate of the herd?

6. Find the smallest birth rate that supports a stable population.

in-context