Dynamical Systems

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};

after

k

years, the populations are

R_{k}

and

S_{k} .

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

\begin{aligned} R_{k + 1} & = 0.9 R_{k} + 0.8 S_{k} \\ S_{k + 1} & = 0.2 R_{k} + 0.9 S_{k} . \end{aligned}

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We will combine the populations in a vector

x_{k} = [\begin{array}{r} R_{k} \\ S_{k} \end{array}]

and note that

x_{k + 1} = A x_{k}

where

A = [\begin{array}{rr} 0.9 & 0.8 \\ 0.2 & 0.9 \end{array}] .

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(a)

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Verify that

v_{1} = [\begin{array}{r} 2 \\ 1 \end{array}], v_{2} = [\begin{array}{r} - 2 \\ 1 \end{array}]

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are eigenvectors of

A

and find their respective eigenvalues.

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(b)

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Suppose that initially

x_{0} = [\begin{array}{r} 2 \\ 3 \end{array}] .

Write

x_{0}

as a linear combination of the eigenvectors

v_{1}

and

v_{2} .

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(c)

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Write the vectors

x_{1},

x_{2},

and

x_{3}

as a linear combination of eigenvectors

v_{1}

and

v_{2} .

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(d)

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When

k

becomes very large, what happens to the ratio of the populations

R_{k} / S_{k} ?

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Activity 4.4.0.3.

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

\begin{aligned} R_{k + 1} & = 0.9 R_{k} + 0.8 S_{k} \\ S_{k + 1} & = 0.2 R_{k} + 0.9 S_{k} . \end{aligned}

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Where we have the corresponding matrix equation with

x_{k} = [\begin{array}{r} R_{k} \\ S_{k} \end{array}]

and

x_{k + 1} = A x_{k}

where

A = [\begin{array}{rr} 0.9 & 0.8 \\ 0.2 & 0.9 \end{array}] .

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(a)

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Begin instead with

x_{0} = [\begin{array}{r} 4 \\ 4 \end{array}] .

What eventually happens to the ratio

R_{k} / S_{k}

k

becomes very large?

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(b)

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Explain what happens to the ratio

R_{k} / S_{k}

k

becomes very large no matter what the initial populations are.

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(c)

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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?

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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

x_{k} = [\begin{array}{r} R_{k} \\ S_{k} \end{array}]

and which evolve according to the rule

x_{k + 1} = A x_{k} .

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Based on the eigenvalues of the

2 \times 2

matrix defining a dynamical system of the form

x_{k + 1} = A x_{k},

we can classify the system as follows:

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$| λ_{1} |, | λ_{2} | < 1$ produces an attractor so that trajectories are pulled in toward the origin.
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$| λ_{1} | > 1$ and $| λ_{2} | < 1$ produces a saddle in which most trajectories are pushed away from the origin and in the direction of $E_{λ_{1}} .$
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$| λ_{1} |, | λ_{2} | > 1$ produces a repellor in which trajectories are pushed away from the origin.

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Activity 4.4.0.4.

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Suppose that

A = [\begin{array}{rr} 0.7 & 0 \\ 0 & 1.6 \end{array}] .

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(a)

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Explain why the species do not interact with one another.

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(b)

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

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(c)

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What happens to both species after a long time?

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Activity 4.4.0.5.

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Suppose now that

A = [\begin{array}{rr} 0.7 & 0.3 \\ 0 & 1.6 \end{array}] .

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(a)

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Explain why

S

is a beneficial species for

R .

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(b)

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

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(c)

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What happens to both species after a long time?

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Activity 4.4.0.6.

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Suppose now that

A = [\begin{array}{rr} 0.7 & 0.5 \\ - 0.4 & 1.6 \end{array}] .

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(a)

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

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(b)

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

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(c)

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What happens to both species after a long time?

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Activity 4.4.0.7.

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Suppose now that

A = [\begin{array}{rr} 0.5 & 0.2 \\ - 0.4 & 1.1 \end{array}] .

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(a)

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

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(b)

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

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(c)

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What happens to both species after a long time?

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Activity 4.4.0.8.

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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.

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Each year,

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80% of the juveniles survive to become yearlings.
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90% of the yearlings survive to become adults.
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80% of the adults survive.
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40% of the adults give birth to a juvenile.

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J_{k},

Y_{k},

and

A_{k},

we denote the number of juveniles, yearlings, and adults in year

k .

We have

J_{k + 1} = 0.4 A_{k} .

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(a)

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Find similar expressions for

Y_{k + 1}

and

A_{k + 1}

in terms of

J_{k},

Y_{k},

and

A_{k} .

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(b)

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As is usual, we write the matrix

x_{k} = [\begin{array}{r} J_{k} \\ Y_{k} \\ A_{k} \end{array}] .

Write the matrix

A

such that

x_{k + 1} = A x_{k} .

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(c)

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Find the eigenvalues of

A .

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(d)

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

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Activity 4.4.0.9.

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

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We can write

A = P E P^{- 1}

where the matrices

E

and

P

are approximately:

\begin{aligned} E & = [\begin{array}{rrr} 1.058 & 0 & 0 \\ 0 & - 0.128 & - 0.506 \\ 0 & 0.506 & - 0.128 \end{array}], \\ P & = [\begin{array}{rrr} 1 & 1 & 0 \\ 0.756 & - 0.378 & 1.486 \\ 2.644 & - 0.322 & - 1.264 \end{array}] . \end{aligned}

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(a)

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

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(b)

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Make a prediction about the long-term behavior of

x_{k} .

For instance, at what rate does it grow?

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(c)

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

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Activity 4.4.0.10.

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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.

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(a)

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

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(b)

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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?

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(c)

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Find the smallest birth rate that supports a stable population.