Math Insight

Loggerhead sea turtle project, part 2

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Total points: 30

Background

Detailed studies of loggerhead sea turtle populations indicate that there are really four different classes of juvenile loggerhead sea turtles: hatchlings, small juveniles, large juveniles, and subadults, each with slightly different survival probabilities. There are also three different classes of adults: novice (first-time) breeders, second-time breeders, and mature breeders, where each class has a different probability of survival and produces a different number of offspring on average.

Based on this, you decide to construct a second matrix model for loggerhead sea turtles with seven classes: hatchlings (stage 1), small juveniles (stage 2), large juveniles (stage 3), subadults (stage 4), novice breeders (stage 5), second-time breeders (stage 6), mature breeders (stage 7).

As before, only adults reproduce but now there are three adult classes each with a different fecundity (number of offspring produced): novice breeders have a fecundity of 127, second-time breeders have a fecundity of 4, and mature breeders have a fecundity of 80.

Hatchings either mature into small juveniles (with probability p=0.6747) or die (with probability p=0.3253), but do not stay in the hatchling class more than one year. Small juveniles mature into large juveniles (p=0.0486), stay as small juveniles (p=0.7370), or die (p=0.2144). Large juveniles mature into subadults (p=0.0147), stay as large juveniles (p=0.6610), or die (p=0.3243). Subadults either mature into novice breeders (p=0.0518), stay as subadults (p=0.6907), or die (p=0.257). Novice breeders either become second-time breeders (p=0.8091) or die (p=0.1909), but do not stay in the novice breeder class. Second-time breeders either become mature breeders (p=0.8091) or die (p=0.1909), but do not stay in the second-time breeder class. Mature breeders either remain mature breeders (p=0.8089 or die (p=0.1911).

For the steps below, turn in your R code as part of showing your work.

  1. Step 1: map from biology to math
    1. Using the data above, construct a matrix model with 7 classes for sea turtles.

  2. Step 2: analyze the model
    1. If you started with an initial population of loggerhead sea turtles today of 10 hatchlings, 10 small juveniles, 10 large juveniles, 10 subadults, 10 novice breeders, 10 second-time breeders, and 10 mature breeders, how many sea turtles of each class would you expect to see next year? Write a script in R to do this calculation.
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    2. How many total sea turtles would you expect to see next year?
    3. How many total sea turtles would you expect to see each year for the next 10 years? (calculate by iterating the model)
    4. After 10 years, how many sea turtles do you expect to see in each of the seven classes?
    5. Using the values from (c), calculate the ratio between the total population size in year 11 and the total population size in year 10.
    6. Calculate the eigenvalues and eigenvectors of the matrix using the R command eigen.
    7. What is the dominant eigenvalue? What is the eigenvector corresponding to this eigenvalue?
    8. How do the values in (g) compare to your values in (d) and (e)?

  3. Step 3: interpret the model analysis biologically
    1. What is the biological interpretation of the dominant eigenvalue?
    2. What is the biological interpretation of the eigenvector that corresponds to the dominant eigenvalue?
    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?
    4. What fraction of the population in 50 years time do you expect will be in each age class?
    5. Iterate your model for 50 years. What is the outcome? Does it match your expectations from (c) and (d)?

  4. Step 4: comparing possible conservation outcomes

    You have a limited number of sea turtle conservation funds that you can use to either protect breeding adults from trawlers or to protect hatchlings from predation. In order to evaluate which would be a more effective conservation strategy, you use your model.

    1. Try changing the probability that hatchings mature into small juveniles from 0.6747 to 1 (no hatchling mortality). What is the new dominant eigenvalue of the matrix? How does this compare to the value of the dominant eigenvalue that you calculated above?
    2. Try changing the survival of all breeding adults to 95%, i.e. change the probability that novice breeders become second-time breeders from 0.8091 to 0.95, change the probability that second-time breeders become mature breeders from 0.8091 to 0.95, and change the probability that mature breeders remain mature breeders from p=0.8089 to 0.95. What is the new dominant eigenvalue of the matrix? How does this compare to the value of the dominant eigenvalue that you calculated above?
    3. Based on your answers to (4a) and (4b), is protecting adults or juveniles a more effective conservation strategy for sea turtles. Why?

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Math 2241, Spring 2016