The last section demonstrated ways in which we may relate a matrix, and the effect that multiplication has on vectors, to a simpler form. For instance, if there is a basis of consisting of eigenvectors of , we saw that is similar to a diagonal matrix . As a result, the effect of multiplying vectors by , when expressed using the basis of eigenvectors, is the same as multiplying by .
In this section, we will put these ideas to use as we explore discrete dynamical systems, first encountered in Subsection 2.5.3. Recall that we used a state vector to characterize the state of some system at a particular time, such as the distribution of delivery trucks between two locations. A matrix described the transition of the state vector with characterizing the state of the system at a later time. Since we would like to understand how the state vector evolves over time, we are interested in studying the sequence of vectors .
We will begin with a dynamical system that illustrates how the ideas we’ve been developing can help us understand the populations of two interacting species. There are several possible ways in which two species may interact. For example, wolves on Isle Royale in northern Michigan prey on moose so this interaction is often called a predator-prey relationship. Other interactions between species, such as bees and flowering plants, are mutually beneficial for both species.
Suppose we have two species and that interact with each 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 and ; after years, the populations are and .
This is an example of a mutually beneficial relationship between two species. If species is not present, then , which means that the population of species decreases every year. However, species benefits from the presence of species , which helps to grow by 80% of the population of species . In the same way, benefits from the presence of .
are eigenvectors of and find their respective eigenvalues.
Suppose that initially . Write as a linear combination of the eigenvectors and .
Write the vectors ,, and as linear combinations of the eigenvectors and .
What happens to after a very long time?
When becomes very large, what happens to the ratio of the populations ?
After a very long time, by approximately what factor does the population of grow every year? By approximately what factor does the population of grow every year?
If we begin instead with , what eventually happens to the ratio as becomes very large?
This activity demonstrates the type of systems we will be considering. In particular, we will have vectors that describe the state of the system at time and a matrix that describes how the state evolves from one time to the next: . The eigenvalues and eigenvectors of provide the key that helps us understand how the vectors evolve and that enables us to make long-range predictions.
The coordinate system defined by the basis can be used to express the state vectors. For instance, we can write the initial state vector , which means that . Moreover, so that
Thinking about this geometrically, we begin with the vector . Subsequent vectors are obtained by scaling horizontally by a factor of and scaling vertically by a factor . Notice how the points move along a curve away from the origin becoming ever closer to the horizontal axis. After a very long time, .
To recover the behavior of the sequence , we change coordinate systems using the basis defined by and . Here, the points move along a curve away from the origin becoming ever closer to the line defined by .
In addition, so that and , which tells us that both populations are multiplied by 1.3 every year meaning the annual growth rate for both populations is about 30%.
In the same way, we can consider other possible initial populations as shown in Figure 4.4.1. Regardless of , the population vectors, in the coordinates defined by , are scaled horizontally by a factor of and vertically by a factor of . The sequence of points , called trajectories, move along the curves, as shown on the left. In the standard coordinate system, we see that the trajectories converge to the eigenspace .
Figure4.4.1.The trajectories of the dynamical system formed by the matrix in the coordinate system defined by , on the left, and in the standard coordinate system, on the right.
We conclude that, regardless of the initial populations, the ratio of the populations will approach 2 to 1 and that the growth rate for both populations approaches 30%. This example demonstrates the power of using eigenvalues and eigenvectors to rewrite the problem in terms of a new coordinate system. By doing so, we are able to predict the long-term behavior of the populations independently of the initial populations.
Diagrams like those shown in Figure 4.4.1 are called phase portraits. On the left of Figure 4.4.1 is the phase portrait of the diagonal matrix while the right of that figure shows the phase portrait of . The phase portrait of is relatively easy to understand because it is determined only by the two eigenvalues. Once we have the phase portrait of , however, the phase portrait of has a similar appearance with the eigenvectors replacing the standard basis vectors .
In the previous example, we were able to make predictions about the behavior of trajectories by considering the eigenvalues and eigenvectors of the matrix . The next activity looks at a collection of matrices that demonstrate the types of behavior a dynamical system can exhibit.
We will now look at several more examples of dynamical systems. If , we note that the columns of form a basis of . Given below are several matrices written in the form for some matrix . For each matrix, state the eigenvalues of and sketch a phase portrait for the matrix on the left and a phase portrait for on the right. Describe the behavior of as becomes very large for a typical initial vector .
Suppose that has two real eigenvalues and and that both . In this case, any nonzero vector forms a trajectory that moves away from the origin so we say that the origin is a repellor. This is illustrated in Figure 4.4.2.
Figure4.4.2.The origin is a repellor when .
Suppose that has two real eigenvalues and and that . In this case, most nonzero vectors form trajectories that converge to the eigenspace . In this case, we say that the origin is a saddle as illustrated in Figure 4.4.3.
Figure4.4.3.The origin is a saddle when .
Suppose that has two real eigenvalues and and that both . In this case, any nonzero vector forms a trajectory that moves into the origin so we say that the origin is an attractor. This is illustrated in Figure 4.4.4.
Figure4.4.4.The origin is an attractor when .
Suppose that has a complex eigenvalue where . In this case, a nonzero vector forms a trajectory that spirals away from the origin. We say that the origin is a spiral repellor, as illustrated in Figure 4.4.5.
Figure4.4.5.The origin is a spiral repellor when has an eigenvalue with .
Suppose that has a complex eigenvalue where . In this case, a nonzero vector forms a trajectory that moves on a closed curve around the origin. We say that the origin is a center, as illustrated in Figure 4.4.6.
Figure4.4.6.The origin is a center when has an eigenvalue with .
Suppose that has a complex eigenvalue where . In this case, a nonzero vector forms a trajectory that spirals into the origin. We say that the origin is a spiral attractor, as illustrated in Figure 4.4.7.
Figure4.4.7.The origin is a spiral attractor when has an eigenvalue with .
This list includes many types of expected behavior, but there are other possibilities if, for instance, one of the eigenvalues is 0. The next section explores the situation when one of the eigenvalues is 1.
In this activity, we will consider several ways in which two species might interact with one another. Throughout, we will consider two species and whose populations in year form a vector and which evolve according to the rule
Explain why the species do not interact with one another. Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Suppose now that .
Explain why is a beneficial species for . Which of the six types of dynamical systems do we have? What happens to both species after a long time?
If , explain why this describes a predator-prey system. Which of the species is the predator and which is the prey? Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Suppose that . Compare this predator-prey system to the one in the previous part. Which of the six types of dynamical systems do we have? What happens to both species after a long time?
Up to this point, we have focused on systems. In fact, the general case is quite similar. As an example, consider a system where the matrix has eigenvalues ,, and . Since the eigenvalues are real and distinct, there is a basis consisting of eigenvectors of so we can look at the trajectories in the coordinate system defined by . The phase portraits in Figure 4.4.8 show how some representative trajectories will evolve. We see that all the trajectories will converge into the eigenspace .
In the same way, suppose we have a system with complex eigenvalues and . Since the complex eigenvalues satisfy , there is a two-dimensional subspace in which the trajectories spiral in toward the origin. The phase portraits in Figure 4.4.9 show some of the trajectories. Once again, we see that all the trajectories converge into the eigenspace .
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.
Find similar expressions for and in terms of ,, and .
As is usual, we write the matrix . Write the matrix such that and find its eigenvalues.
We can write where the matrices and are approximately:
.
Make a prediction about the long-term behavior of . 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 long-term growth rate of the herd?
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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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Find the smallest birth rate that supports a stable population.
We have been exploring discrete dynamical systems in which an initial state vector evolves over time according to the rule . The eigenvalues and eigenvectors of help us understand the behavior of the state vectors. In the case, we saw that
For each of the matrices below, find the eigenvalues and, when appropriate, the eigenvectors to classify the dynamical system . Use this information to sketch the phase portraits.
For the different values of , determine which types of dynamical system results. For what range of values do we have an attractor? For what range of values do we have a saddle? For what value does the transition between the two types occur?
where is a parameter. As we saw in the text, this dynamical system represents a typical predator-prey relationship, and the parameter represents the rate at which species preys on . We will denote the matrix .
If , determine the eigenvectors and eigenvalues of the system and classify it as one of the six types. Sketch the phase portraits for the diagonal matrix to which is similar as well as the phase portrait for .
If , determine the eigenvectors and eigenvalues of the system. Sketch the phase portraits for the diagonal matrix to which is similar as well as the phase portrait for .
For what values of is the origin a saddle? What can you say about the populations when this happens?
Describe the evolution of the dynamical system as begins at and increases to .
Find the eigenvalues of . To which of the six types does the system belong?
Using the eigenvalues of , we can write for some matrices and . What is the matrix and what geometric effect does multiplication by have on vectors in the plane?
If we remember that , determine the smallest positive value of for which .
Find the eigenvalues of .
Find a matrix such that for some matrix . What geometric effect does multiplication by have on vectors in the plane?
Determine the smallest positive value of for which .
Set up a system of the form that describes this situation.
Find the eigenvalues of the matrix .
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What prediction can you make about these populations after a very long time?
If the birth rate goes up to 80%, what prediction can you make about these populations after a very long time? For every 100 adults, how many juveniles, and yearlings are there?
Determine whether the following statements are true or false and provide a justification for your response. In each case, we are considering a dynamical system of the form .
Gil Strang defines the Gibonacci numbers as follows. We begin with and . A subsequent Gibonacci number is the average of the two previous; that is, . We then have
Consider a small rodent that lives for three years. Once again, we can separate a population of females into juveniles, yearlings, and adults. Suppose that, each year,