This is a post from a series in my project.

Remember the sea turtles in Finding Nemo?

They are cute, aren’t they? It probably isn’t a surprise to most people that sea turtles (*Carretta carretta*) are endangered – they’ve been on the endangered species list for quite some time. What is a surprise is that you can use linear algebra (how to do stuff with matrices) to analyze the dynamics of a population of sea turtles to find out which life stage (hatchlings, juveniles, adults, or seniors) has the most impact on overall population viability. This post is a general overview of the findings of this research paper (1987) by Deborah T. Crouse, Larry B. Crowder and Hal Caswell.

Modeling populations with matrices is an important and useful technique in the field of mathematical biology. Matrices are used to determine which life stage(s) of a population are the most significant in population growth and decline. The authors describe how the matrix model is applied to the population of loggerhead sea turtles. In particular, they investigate how the population matrix aids in the prediction of population trends (growth or decline). Their findings lead them to make an argument that the most significant life stages for loggerhead sea turtle population survival are the juvenile and mature breeder stages. In other words, don’t focus on protecting beaches with brand new turtle eggs. Get the word out to fishermen, recreational boaters, commercial fishing operations, and anyone else who has a net: Don’t let juvenile or mature adult turtles get hurt. They are the ones who have the most impact on population survival.

# Life cycle

What are the stages in a turtle’s life? I admit, it’s an arbitrary line you’re drawing when you put certain turtles in some stages and others in different stages. All of these uncertainties and possible miscategorizations aside, the argument can be made that there are seven cycles.

The researchers divided up a turtle’s life into seven stages. These seven stages are supposed to characterize a turtle’s whole life, so they form the rows and columns of this population matrix. Each life stage has a certain probability of survival (P), fecundity (reproduction) (F), and growth (G). The arrows in the picture above show you the progression of a turtle as she passes through life. The self loops (labelled P) represent how probable it is that this turtle stays in its life stage for one time step. The arrows pointing from one stage to the next (labelled G) show you how probable it is that any given turtle in a particular life stage will advance to the next stage in the next time step. Finally, the arrows along the bottom (labelled F) represent how probable it is that a turtle will reproduce. It’s important to note that the turtles modeled in this simulation are all females because it turns out that they are the easiest to keep track of – given a turtle’s mobile and long lifestyle, it’s hard to track them.

The value of such a convoluted life cycle graph is that you can turn the relationships it shows into a matrix of numbers:

Note the columns of the matrix correspond to the current life stage of the turtle, and the rows represent the stages that follow. Each entry in the projection matrix can be thought of as corresponding to the transition from the stage-class in the given column to the stage-class in the given row. For example, the first row of the matrix lists the probability of reproducing for a turtle from a certain stage class. The red arrow shows the probability of a Stage 3 turtle surviving in its own stage, the purple arrow shows the probability of a Stage 3 turtle surviving and growing to the fourth stage, and the green arrow shows the probability of a Stage 6 turtle surviving in its own stage.

That’s it. That’s the model. Sort of anticlimactic, right? Well, the interesting bits are in the analysis that follows.

# Population projection

Basically what you do is you take this sparsely populated (pun intended) matrix and you multiply it by a vector of some initial population distribution of turtles in stage 1, stage 2, stage 3, … all the way to stage 7. Each time you do a multiplication step, it counts as one time step. Think of it as one year. Then, you add up all the elements in that vector to get an idea of how many turtles (across all life stages) are in that population at that time step. When you do this enough times, you get a plot that looks something like this:

If you take the present year to be considered Time = 0 years, then this plot is supposed to project into the next 100 years. Sadly, this model predicts that there will be basically zero loggerhead sea turtles in the year 2113. What can change this unfortunate trajectory? Which stage of turtles has the most impact on the survival of the species? How do we find out which life stage is the one we should target conservation efforts at?

# Sensitivity analysis

The researchers used matrix methods to analyze the stability of this model. In other words, they tried to find out which parameter in the model had the most effect on the long-term behavior of the model. This type of analysis is called a “sensitivity” analysis. The idea is to measure how much the population projection changes when you change certain parameters (reproductive rates, growth rates, or survival rates) of certain life stages (hatchlings, juveniles, adults, etc.), and then use this information to figure out which life stage has the most effect on the population’s survival. Sensitivity analysis paints a small picture of what’s behind the dynamics of this population model. Abstracting away the equations and numerical simulations, this is the result the researchers were looking at:

Basically, the plot shows that the survival (P) of the juvenile life stages (stages 2, 3, 4) and the survival of the mature breeders (stage 7) is the most “elastic.” Elasticity is another way of saying that these stages probably play the dominant roles in population growth. Let’s test the hypothesis! Ideally, if conservation efforts were successful, the loggerhead population would stabilize to a constant value as time goes on. Therefore, this was what the authors looked for when manipulating stage parameters.

Let’s examine the two highest values on the graph, survival (P) of stage 2 and stage 7, and the stage that conservation efforts before 1987 were focused on, growth (G) of stage 1. This is a projection of turtle populations over time if you increase the probability of survival of stage 7 (mature breeders) by about 17%:

This is a projection of turtle populations over time if you increase the probability of survival of stage 2 (juveniles) by about 22%:

To get the population projection to be as stable as the ones demonstrated above, it turns out that you have to increase the probability of growth of stage 1 (the hatchlings) by 175%:

What a difference that makes! Making a 175% effort is significantly more than making a 22% or a 17% effort for the same result: population conservation. Crouse et. al made a huge discovery when they published this paper in the 1980s. Their results were so influential that this paper is one of the most highly cited mathematical biology papers today. This team of researchers changed the way many people approached sea turtle conservation and impacted the way population biology research is carried out.