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.

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

Setting $t=0$ in the above equation, \begin{align*} J_{1} &= (?) J_0 + (?) A_0\\ A_{1} &= (?) J_0 + (?) A_0 \end{align*} we can plug in the numbers for the population sizes \begin{align*} 25 &= (?) 100 + (?) 0\\ 1 &= (?) 100 + (?) 0 \end{align*}

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These last two numbers should determine the right two (?)'s in the model.

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.

Given the matrix \[{}\] and the vector `{}` (the blue arrow in the applet), their product is: \[{}\] The vector `{}` is represented by the dotted blue arrow in the applet.

The product of `{}` with the vector `{}` (the green arrow in the applet) is: \[{}\] The vector `{}` is represented by the dotted green arrow in the applet.

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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.)

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 = $

If the blue arrow of the applet is pointed in the direction of $\vc{u}$, then you can read off $A\vc{u}$ from the matrix-vector multiplication box. Divide the first component of $A\vc{u}$ by the first component of $\vc{u}$. The result is $\lambda_u$.

If the green arrow of the applet is pointed in the direction of $\vc{w}$, then you can read off $A\vc{w}$ from the matrix-vector multiplication box. Divide the first component of $A\vc{w}$ by the first component of $\vc{w}$. The result is $\lambda_w$.

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.