# Math Insight

### Sea turtle introductory example

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1. A simple example of a stage-structured matrix population model is a two-stage model of a sea-turtle population. Imagine that we keep track of only the number of juvenile (non-reproductive) sea turtles and the number of adult (reproductive) sea turtles. We don't keep track of the ages of the turtles within each group, but just the total numbers.

Let $J_t$ be the number of juvenile sea turtles and $A_t$ be the number of adult sea turtles, where $t$ is the number of years that have passed. (We're using a discrete dynamical system where we let the number of turtles change only once per year.) The first step is to formulate the model from some observations.

The general form of the model will be \begin{align*} J_{t+1} &= (?) J_t + (?) A_t\\ A_{t+1} &= (?) J_t + (?) A_t \end{align*} for $t=0,1,2, \ldots$. We need to fill in four numbers for the (?)'s. Once you determine those four numbers from the following observations, fill them in here to complete the model.

$J_{t+1} =$
$J_t +$
$A_t$
$A_{t+1} =$
$J_t +$
$A_t$
1. Imagine that we started with $100$ juveniles and $0$ adults, i.e., that $J_0 = 100$ and $A_0=0$. After one year, we count that there are $J_1 = 25$ juveniles and $A_1=1$ adult. This means that $25$ juveniles survived a year and that $1$ of them grew into an adult. (The rest of the juveniles died.)

This information should be enough to determine the left two (?)'s in the model.

2. Since we didn't have any adults at first, we couldn't figure out the right two (?)'s. But now that $A_1 = 1$, we can determine those numbers. If we tell you that in year $t=2$ we count $J_2 = 56.25$ juveniles and $0.65$ adults, can you determine these remaining numbers?

2. We can rewrite the model in terms of a matrix as \begin{align*} \begin{bmatrix}J_{t+1}\\A_{t+1}\end{bmatrix} = \left[\begin{matrix}0.25 & 50\\0.01 & 0.4\end{matrix}\right] \begin{bmatrix} J_t \\ A_t \end{bmatrix} \end{align*}

If we let $\vc{v}_t = \begin{bmatrix}J_t\\A_t\end{bmatrix}$ be a vector describing both the number of juveniles and the number of adults in year $t$ and $M=\left[\begin{matrix}0.25 & 50\\0.01 & 0.4\end{matrix}\right]$, we can write the model simply as $$\vc{v}_{t+1} = M \vc{v}_t.$$

Each year, we multiply the vector $\vc{v}_t$ by the matrix $M$. We can visualize how $M$ transforms $\vc{v}_t$ into $\vc{v}_{t+1}$ graphically.

In the following applet, you can use either the blue solid vector or the green solid vector to represent $\vc{v}_t$. The applet then automatically multiplies the vector by $M$ to calculate $\vc{v}_{t+1}$, which is shown by the dotted arrow of the same color. You can see the calculation for $\vc{v}_{t+1}$ by opening the matrix-vector multiplication box below the applet.

Feedback from applet
eigenvectors:
Matrix-vector multiplication (Show)

It may seem strange we let values of $J_t$ and $A_t$ be negative, but it turns out negative values are useful for understanding what multiplication by $M$ is doing.

There are two special directions where multiplication by $M$ does not rotate the vector $\vc{v}$ but only stretches/shrinks the vector and/or flips it. Can you move the blue and green vectors until you find those directions? If the vectors lie along the same line, they don't count as pointing in the different directions.

These directions that are preserved under multiplication by $M$ are called the eigenvectors of $M$. The applet indicate if you found them when you click the below submit button.

Let $\vc{u}$ be the eigenvector where both components have the same sign (the one pointing upward and to the right if you make both component positive). What is this eigenvector?
$\vc{u} =$

Use the blue vector for $\vc{u}$ in the applet to make the text in the above matrix-vector multiplication box to match. (Since only the direction matters, stretching $\vc{u}$, i.e., multiplying both components by the same nonzero number, won't change the answer.)

Let $\vc{w}$ be the eigenvector where the components have the different sign (the one pointing upward and to the left, or, alternatively, downward and to the right). What is this eigenvector?
$\vc{w} =$

Use the green vector for $\vc{w}$ in the applet to make the text in the above matrix-vector multiplication box to match. (Since only the direction matters, stretching $\vc{w}$, i.e., multiplying both components by the same nonzero number, won't change the answer.)

3. Each eigenvector $\vc{u}$ and $\vc{w}$ is stretched and/or flipped when multiplied by $M$. The amount that it is stretched is called its eigenvalue. (If the eigenvalue is negative, the eigenvector is also flipped by $M$.)

By looking at the results from the matrix-vector multiplication, determine these eigenvalues.
The eigenvalue (or scaling factor) for $\vc{u}$ is $\lambda_u =$

The eigenvalue (or scaling factor) for $\vc{w}$ is $\lambda_w =$

4. Since $\lambda_u \gt 1$, the direction $\vc{u}$ gets stretched every time a vector is multiplied by $M$. On the other hand, since $|\lambda_w| \lt 1$, the direction $\vc{w}$ shrinks every time a vector is multiplied by $M$. Moreover, since $\lambda_w$ is negative, the direction is flipped with every multiplication by $M$.

You can explore the behavior of the vector $\vc{v}$ as it is repeatedly multiplied by $M$. Since the direction of $\vc{u}$ is stretched at each multiplication, eventually the vector points in the direction of the eigenvector $\vc{w}$.

We call the eigenvalue $\lambda_u$, the dominant eigenvalue, as it is the largest eigenvalue of $M$. It determines the growth rate of the population.

The eigenvector $\vc{u}$ of the dominant eigenvector determines stable stage distribution, i.e., the fraction of the population that will be juveniles and adults in the long term.