Math Insight

Loggerhead sea turtle project

Group members:
Total points: 1
Grading rubric

To earn credit, a project must meet the following criteria.

CriterionMetNot met
Accurately construct a matrix model from the data
Accurately determine both the short-term and the long-term dynamics of the model and interpret those dynamics in terms of the sea turtle population
Form a recommendation for a sea turtle conservation strategy that is justified by the mathematical model
Project receives creditYESNO
Submitting project

Submit the following by the due date.

  1. Answers to the project questions (typed or handwritten)

Background

Loggerhead sea turtle
Loggerhead sea turtle, Caretta caretta
(Image: © ukanda, CC BY 2.0 wikimedia)

Loggerhead sea turtles (Caretta caretta) are an endangered species of marine turtle found in the tropics across the world. Sea turtles are relatively long-lived, but individuals live for several years before they are able to start reproducing. Predation, egg poaching, pollution, and trapping in trawling nets all contribute to sea turtle mortality. In order to determine how to best conserve and manage sea turtle populations, we must first understand their current population dynamics.

The overarching questions for this project are:

  • Are loggerhead sea turtle populations expected to increase, decrease or stay the same, over short- and long-term timescales?
  • How will different conservation strategies influence turtle populations?

The model

Detailed studies of loggerhead sea turtle populations indicate that there are 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 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).

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

Over the course of a year, each sea turtle either dies, matures into the subsequent class, or remains in the same class, and the probability of these events depends on the class. 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).

  1. Step 1: map from biology to math

    Using the above data, construct a matrix model with 7 classes for sea turtles.

  2. Step 2: analyze the model
    1. If you started with an initial population (in year 0) 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?
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    2. How many total sea turtles would you expect to see next year (in year 1)?
    3. How many total sea turtles would you expect to see each year for the next 10 years (in years 1-10)? (Calculate by iterating the model.)
    4. After 10 years (i.e,. in year 10), what fraction of the 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 10 and the total population size in year 9.
    6. What is the dominant eigenvalue? What is the eigenvector corresponding to this eigenvalue? To help it interpretation, the eigenvector should be scaled so that it sums up to 1.
    7. How do the eigenvalue and eigenvector in (f) compare to your values you calculated 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. Do you expect that in 50 years time there will be more or fewer sea turtles than in the population today? Explain how you arrived at this conclusion based on your answers to (a) and (b) and the results from step 2.
    4. What fraction of the population in 50 years time do you expect will be in each age class? Explain how you arrived at this conclusion based on your answers to (a) and (b) and the results from step 2.
    5. Iterate your model for 50 years and analyze your results at year 50. 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 2023