Weeks 4 and 5: The Model

We’ve captured, marked, and released 43 diamondback terrapins now; far more than we ever imagined we’d find.  Pretty soon I’ll be able to use the data we’ve gathered in a mathematical model to predict the future behavior of the population based on known survival rates of the various stage classes.

The model that I’m using to analyze the diamondback terrapin data is discrete time model, which means it predicts the population sizes at a certain time intervals (annually in my model).  The terrapin populations are divided into seven stage classes based on their life history traits: female hatchlings (Xh), female juveniles (Xj), female immature breeders (Xi), female mature breeders (Xm), male hatchlings (Yh), male juveniles (Yj), and male breeders (Ym).  Most stage classes last multiple years, and the members of a stage class all have similar likelihoods of survival, death, and production of offspring.  Unlike previously existing terrapin models, this model is nonlinear.  Because of this, the reproductive contributions of both male and female individuals are taken into account, instead of just that of the females.

In order to predict the number of individuals in each stage class in year n+1, we can use the number of individuals in year n and multiply them by certain parameters:

equations

The juvenile values are relatively easy to obtain; the number of hatchlings from the previous year is multiplied by the fraction that move on to the juvenile class, and the number of juveniles from the previous year is multiplied by the fraction that remain in that stage class in the next year.  The breeder stages follow a similar formula.  The parameter c represents the effects of crab potting, which only kills juveniles (of both sexes) and male breeders.

The hatchling stages are the most complicated.  We based our formulas off of similar ones in Caswell’s book Matrix Population Models (2001).  I won’t bore you with all the details, but I’ll go over the parameters really quickly.  The fraction of hatchlings that are female is a, the fraction of the eggs that survive is se, the clutch sizes of immature and mature breeding females are ki and km, the number of females that mate with one male is h, and l represents the degree of male preference for older breeding females.

As an example of what this model can do, I’ll run a couple of simulation under different levels of crab potting and show you the results.  First, let’s assume that we’re starting with a population of 20 terrapins in each stage class, and that there is no effect from crab potting.  Over the course of 100 years, a terrapin population would increase as shown here:

graph1

The x-axis is years, and the y-axis is population size

And at the end of that 100 years, the population distribution among the various stage classes would look like this:

graph

Now, if crab traps were killing off 35% of juveniles and male adults, the population would behave like this:

graph

The x-axis is years, and the y-axis is population size

graph

Similar simulations can be run varying other parameters, like male preference for mates, the number of females that a male will mate with, etc.  I’m really looking forward to using the data we collect this summer to experiment with the model!

– Sarah Gilliand

 

Sources:

Caswell, Hal. Matrix Population Models: Construction, Analysis, and Interpretation. Sunderland, MA: Sinauer Associates, 2001. Print.