Let's start by looking at the first week of butterfly populations. We'll use R to compute these values. First, create a matrix for the system and a matrix to represent the initial condition vector. The matrix corresponding to the butterfly model is
$$A=\left[\begin{matrix}0.2 & 0 & 0 & 100\\0.08 & 0.47 & 0 & 0\\0 & 0.06 & 0.58 & 0\\0 & 0 & 0.1 & 0.73\end{matrix}\right]$$
Enter A=matrix(c(0.2, 0, 0, 100, 0.08, 0.47, 0, 0, 0, 0.06, 0.58,0, 0, 0, 0.1, 0.73),4,4,byrow=TRUE) . For the initial condition vector, enter b=c(250000, 800, 300, 400). You may wish to verify that the matrices are correct by entering just the name on the next line (A or b, respectively).
Now, to find the next day's population, b=A%*%b . The %*% is the R command for matrix multiplication. Written this way, we are overwriting the value of the vector b with the next day's populations. To see the actual value, enter b on the next line. Repeat these two steps for each day, and enter the values you get below (include at least one decimal place).
Write the state vectors below, with $\vc{x}_n=\begin{bmatrix} e_n \\ c_n \\ p_n \\ b_n \end{bmatrix}$.
$\vc{x}_1=$
$\vc{x}_2=$
$\vc{x}_3=$
$\vc{x}_4=$
$\vc{x}_5=$
$\vc{x}_6=$
$\vc{x}_7=$
Based on this first week, do you think the population will be larger or smaller in several years?
Are you confident about your answer?
It's good a thing we aren't relying on visualizing the effects of $A$ on the system. Since we have for state variables, we are dealing with four-dimensional vectors. Our visualizations don't work so well in four dimensions! Even so, we can easily work with higher dimensional vectors by just thinking of them as lists of numbers, in this case representing population sizes.
Now let's consider the fraction of the population which are adults in several years. The fraction on day $n$ is given by $\frac{b_{n}}{b_{n} + c_{n} + e_{n} + p_{n}}$. Suppose the state variable is an eigenvector. What is the fraction on day $n+1$? Write the fraction in terms of e_n, c_n, p_n, b_n and lambda, where lambda denotes the corresponding eigenvalue.
Estimate the fraction of adults in several years. Include at least five significant digits in your answer.
Hint
This might seem a little trickier, but the question just synthesizes what you know about linear systems and eigenvectors. If you start on an eigenvector, what happens to the relative sizes of each component? If you start off an eigenvector, what do the relative sizes of the components approach?
Write the ratio as "number of x" / "number of y".
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