Dynamical systems

Suppose we have two species $R$ and $S$ 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 $R_0$ and $S_0$; after $k$ years, the populations are $R_k$ and $S_k$.

If we know the populations in one year, suppose that the populations in the following year are determined by the expressions This is an exam…

In this activity, we will consider several ways in which two species might interact with one another. Throughout, 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 .

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

   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…

Consider the matrices

1. Find the eigenvalues of $A$. To which of the six types does the system $\xvec_{k+1}=A\xvec_{k}$ belong?

2. Using the eigenvalues of $A$, we can write $A=PEP^{-1}$ for some matrices $E$ and $P$. What is the matrix $E$ and what geometric effect does multiplication by $E$ have on vectors in the plane?

3. If we remember that $A^k = PE^kP^{-1}$, determine the smallest positive value of $k$ for which $A^k=I$.

4. Find the eigenvalues of $B$.

5. Find a matrix $E$ such that…

Suppose we have the female population of a species is divided into juveniles, yearlings, and adults and that each year

• 90% of the juveniles live to be yearlings.

• 80% of the yearlings live to be adults.

• 60% of the adults survive to the next year.

• 50% of the adults give birth to a juvenile.

1. Set up a system of the form $\xvec_{k+1}=A\xvec_k$ that describes this situation.

2. Find the eigenvalues of the matrix $A$.

3. What prediction ca…

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 $\xvec_{k+1} = A\xvec_k$.

1. If the $2\times2$ matrix $A$ has a complex eigenvalue, we cannot make a prediction about the behavior of the trajectories.

2. If $A$ has eigenvalues whose absolute value is smaller than 1, then all the trajectories are pulled in toward the ori…

The Fibonacci numbers form the sequence of numbers that begins $0, 1, 1, 2, 3, 5, 8, 13, \ldots$. If we let $F_n$ denote the $n^{th}$ Fibonacci number, then . In general, a Fibonacci number is the sum of the previous two Fibonacci numbers; that is, $F_{n+2} = F_{n}+F_{n+1}$ so that we have

1. If we write $\xvec_n = \twovec{F_{n+1}}{F_n}$, find the matrix $A$ such that $\xvec_{n+1} = A\xvec_n$.

2. Show that $A$ has eigenvalues with associated eigenvectors $\vvec_1=\twovec{\lambda_1}{1}$ and $\vvec_2=\twovec{\lambda_2}{1}$.

3. Classify this dynamical system as one of the six types that we ha…

This exercise is a continuation of the previous one.

The Lucas numbers $L_n$ are defined by the same relationship as the Fibonacci numbers: $L_{n+2}=L_{n+1}+L_n$. However, we begin with $L_0=2$ and $L_1=1$, which leads to the sequence $2,1,3,4,7,11,\ldots$.

1. As before, form the vector $\xvec_n=\twovec{L_{n+1}}{L_n}$ so that $\xvec_{n+1}=A\xvec_n$. Express $\xvec_0$ as a linear combination of $\vvec_1$ and $\vvec_2$, eigenvectors of $A$.

2. Explain why

3. Explain why $L_n$ is the closest integer to $\phi^n$ when $n$ is large, where $\phi = \lambda_1$ is the golden rat…

Gil Strang defines the Gibonacci numbers $G_n$ as follows. We begin with $G_0 = 0$ and $G_1=1$. A subsequent Gibonacci number is the average of the two previous; that is, $G_{n+2} = \frac12(G_{n}+G_{n+1})$. We then have

1. If $\xvec_n=\twovec{G_{n+1}}{G_n}$, find the matrix $A$ such that $\xvec_{n+1} = A\xvec_n$.

2. Find the eigenvalues and associated eigenvectors of $A$.

3. Explain why this dynamical system does not neatly fit into one of the six types that we saw in this section.

4. Write $\xvec_{0}$ as a linear combination o…

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,

• Half of the juveniles live to be yearlings.

• One quarter of the yearlings live to be adults.

• Adult females produce eight female offspring.

• None of the adults survive to the next year.

1. Writing the populations of juveniles, yearlings, and a…