Math Insight

Loggerhead sea turtle warmup

Math 2241, Spring 2023
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Due date: Feb. 15, 2023, 11:59 p.m.
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Total points: 1

A warm-up model

The following is a simplified model of a loggerhead sea turtle population to help you get prepared for the loggerhead sea turtle project.

A simple loggerhead sea turtle population model is: \begin{align} \begin{bmatrix}J\\ A\end{bmatrix}_{t+1} = \begin{bmatrix}0.2&40\\0.01&0.4\end{bmatrix} \begin{bmatrix}J\\ A\end{bmatrix}_t \end{align} where $J$ is the number of juvenile (non-reproductive) sea turtles, $A$ is the number of adult (reproductive) sea turtles, and $t$ is the number of years that have passed.

  1. Step 1: map from math to biology
    1. What is the probability that a juvenile survives for one year and remains a juvenile (find the transition probability from $J$ to $J$)?
    2. What is the probability that a juvenile survives and matures into an adult the next year (find the transition probability from $J$ to $A$)?
    3. Based on your answers to (a) and (b), what is the probability that a juvenile dies in the following year (does not survive)?
      Need help?
    4. What is the probability that an adult survives and remains an adult for the following year (find the transition probability from $A$ to $A$)?
    5. What is the number of offspring that each adult produces in the next year (find the transition from $A$ to $J$)?
    6. What is the probability that an adult dies in the following year (does not survive)?

  2. Step 2: analyze the model
    1. If you started with an initial population of loggerhead sea turtles today of 10 juveniles and 10 adults, how many juveniles and adults would you expect to see next year? Number of juveniles =
      , Number of adults =
      . Is the total number of turtles increasing or decreasing?
      Need help?
    2. How many juveniles and adults would you expect to see after 2 years? Number of juveniles =
      , Number of adults =
      . Is the total number of turtles increasing or decreasing?
    3. Calculate the eigenvalues and eigenvectors of the matrix in equation 1 by hand.

      The eigenvalues are $\lambda_1=$
      and $\lambda_2=$

      (Enter the eigenvalues in increasing order. Include at least 5 significant digits in your response.)

      An eigenvector corresponding to $\lambda_1$ is

      An eigenvector corresponding to $\lambda_2$ is

  3. Step 3: interpret the model analysis biologically
    1. What is the biological interpretation of the dominant eigenvalue?

      It is the
      .

    2. What is the biological interpretation of the eigenvector that corresponds to the dominant eigenvalue?

      It is the
      .

    3. Based on your answers to (a) and (b), do you expect that in 50 years time there will be more or fewer sea turtles than in the population today?
      . At that point, by what percentage do you expect the population to be increasing or decreasing each year?
      %. (If the population is increasing, enter a positive number; if it is decreasing, enter a negative number. You answer must be correct to the nearest tenth.)

    4. What fraction of the population in 50 years time do you expect will be juveniles and what fraction will be adults? (Answer must be accurate to the nearest one-hundreth.)

      Expected fraction juveniles =

      Expected fraction adults =

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Math 2241, Spring 2023