# TBIL Activities for Understanding Linear Algebra

## Section4.4Dynamical Systems

### Activity4.4.0.1.

Suppose that we have a diagonalizable matrix $$A=PDP^{-1}$$ where
\begin{equation*} P = \left[\begin{array}{rr} 1 \amp -1 \\ 1 \amp 2 \\ \end{array}\right],\qquad D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -3 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

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
\begin{equation*} \begin{aligned} R_{k+1} \amp {}={} 0.9 R_k + 0.8 S_k \\ S_{k+1} \amp {}={} 0.2 R_k + 0.9 S_k\text{.} \\ \end{aligned} \end{equation*}
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 $$A = \left[\begin{array}{rr} 0.9 \amp 0.8 \\ 0.2 \amp 0.9 \\ \end{array}\right] \text{.}$$

#### (a)

Verify that
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:
\begin{equation*} \begin{aligned} R_{k+1} \amp {}={} 0.9 R_k + 0.8 S_k \\ S_{k+1} \amp {}={} 0.2 R_k + 0.9 S_k\text{.} \\ \end{aligned} \end{equation*}
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
\begin{equation*} \xvec_{k+1}=A\xvec_k\text{.} \end{equation*}
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.

### Activity4.4.0.4.

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

#### (a)

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
\begin{equation*} J_{k+1} = 0.4 A_k\text{.} \end{equation*}

#### (a)

Find similar expressions for $$Y_{k+1}$$ and $$A_{k+1}$$ in terms of $$J_k\text{,}$$ $$Y_k\text{,}$$ and $$A_k\text{.}$$

#### (b)

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

#### (c)

Find the eigenvalues of $$A\text{.}$$

#### (d)

What does the size of the complex eigenvalue tell you about its effect on the long-term behavior of the system?

### Activity4.4.0.9.

Use the same model for the herd as the previous activity.
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*}

#### (a)

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.