Find the eigenvalues of \(A\) and find a basis for the associated eigenspaces.

(b)

Form a basis \(\bcal\) of \(\real^2\) consisting of eigenvectors of \(A\) and write the vector \(\xvec = \twovec{1}{4}\) as a linear combination of basis vectors.

(c)

Write \(A\xvec\) as a linear combination of basis vectors.

(d)

What is \(\coords{\xvec}{\bcal}\text{,}\) the representation of \(\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

(e)

What is \(\coords{A\xvec}{\bcal}\text{,}\) the representation of \(A\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

(f)

What is \(\coords{A^4\xvec}{\bcal}\text{,}\) the representation of \(A^4\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

Activity4.4.0.2.

Suppose we have two species \(R\) and \(S\) that interact with one another and that we record the change in their populations from year to year. When we begin our study, the populations, measured in thousands, are \(R_0\) and \(S_0\text{;}\) after \(k\) years, the populations are \(R_k\) and \(S_k\text{.}\)

If we know the populations in one year, they are determined in the following year by the expressions

We will combine the populations in a vector \(\xvec_k = \twovec{R_k}{S_k}\) and note that \(\xvec_{k+1} = A\xvec_k\) where \(A = \left[\begin{array}{rr}
0.9 \amp 0.8 \\
0.2 \amp 0.9 \\
\end{array}\right]
\text{.}\)

are eigenvectors of \(A\) and find their respective eigenvalues.

(b)

Suppose that initially \(\xvec_0 = \twovec{2}{3}\text{.}\) Write \(\xvec_0\) as a linear combination of the eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)

(c)

Write the vectors \(\xvec_1\text{,}\)\(\xvec_2\text{,}\) and \(\xvec_3\) as a linear combination of eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)

(d)

When \(k\) becomes very large, what happens to the ratio of the populations \(R_k/S_k\text{?}\)

Activity4.4.0.3.

We will contnue to use the population model from the previous problem:

Where we have the corresponding matrix equation with \(\xvec_k = \twovec{R_k}{S_k}\) and \(\xvec_{k+1} = A\xvec_k\) where \(A = \left[\begin{array}{rr}
0.9 \amp 0.8 \\
0.2 \amp 0.9 \\
\end{array}\right]
\text{.}\)

(a)

Begin instead with \(\xvec_0 =
\twovec{4}{4}\text{.}\) What eventually happens to the ratio \(R_k/S_k\) as \(k\) becomes very large?

(b)

Explain what happens to the ratio \(R_k/S_k\) as \(k\) becomes very large no matter what the initial populations are.

(c)

After a very long time, by approximately what factor does the population of \(R\) grow every year? By approximately what factor does the population of \(S\) grow every year?

In the next few activities, we will consider several ways in which two species might interact with one another. We will consider two species \(R\) and \(S\) whose populations in year \(k\) form a vector \(\xvec_k=\twovec{R_k}{S_k}\) and which evolve according to the rule

Based on the eigenvalues of the \(2\times2\) matrix defining a dynamical system of the form \(\xvec_{k+1}=A\xvec_k\text{,}\) we can classify the system as follows:

\(|\lambda_1|, |\lambda_2| \lt 1\) produces an attractor so that trajectories are pulled in toward the origin.

\(|\lambda_1| \gt 1\) and \(|\lambda_2| \lt 1\) produces a saddle in which most trajectories are pushed away from the origin and in the direction of \(E_{\lambda_1}\text{.}\)

\(|\lambda_1|, |\lambda_2| \gt 1\) produces a repellor in which trajectories are pushed away from the origin.

Explain why the species do not interact with one another.

(b)

Which of the six types of dynamical systems do we have?

(c)

What happens to both species after a long time?

Activity4.4.0.5.

Suppose now that \(A = \left[\begin{array}{rr}
0.7 \amp 0.3 \\
0 \amp 1.6 \\
\end{array}\right]
\text{.}\)

(a)

Explain why \(S\) is a beneficial species for \(R\text{.}\)

(b)

Which of the six types of dynamical systems do we have?

(c)

What happens to both species after a long time?

Activity4.4.0.6.

Suppose now that \(A = \left[\begin{array}{rr}
0.7 \amp 0.5 \\
-0.4 \amp 1.6 \\
\end{array}\right]
\text{.}\)

(a)

Explain why this describes a predator-prey system. Which of the species is the predator and which is the prey?

(b)

Which of the six types of dynamical systems do we have?

(c)

What happens to both species after a long time?

Activity4.4.0.7.

Suppose now that \(A = \left[\begin{array}{rr}
0.5 \amp 0.2 \\
-0.4 \amp 1.1 \\
\end{array}\right]
\text{.}\)

(a)

Compare this predator-prey system to the one in the previous activity.

(b)

Which of the six types of dynamical systems do we have?

(c)

What happens to both species after a long time?

Activity4.4.0.8.

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

Use this information to give an eigenvector associated with the real eigenvalue.

(b)

Make a prediction about the long-term behavior of \(\xvec_k\text{.}\) For instance, at what rate does it grow?

(c)

For every 100 adults, how many juveniles, and yearlings are there?

Activity4.4.0.10.

Use the same model for the bison herd as the previous activities, but suppose that the birth rate decreases so that only 30% of adults give birth to a juvenile.

(a)

How does this affect the long-term growth rate of the herd?

(b)

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?

(c)

Find the smallest birth rate that supports a stable population.