Mathematical Model For Investigating The Viability Of Loggerhead Turtle Populations

Methods and model structure

Introduction (6 marks) Describe the system you have chosen to model. Identify the scientific questions you will seek to answer with your model. This will likely require referring to some scientific literature, although referring to websites may also be acceptable. Methods and model structure (8 marks) Here, you should describe how you have decided to model the system. You should outline the simplifying assumptions you have chosen to make and describe why those simplifications have been made. As part of the methods, you should present your model is a flowchart or diagram. As an appendix to your report, include the R code you have used, with appropriate annotations (use the # commenting facility). 

  Results (8 marks) Describe, in words, the results of your modelling exercise. You should include some graphs here, but there should also be interpretation in words (for example, “figure 1 shows that…”). Each of your graphs should have an informative legend. 2 Discussion (8 marks). This is the essential step of relating your modelling findings back to the original problem. To what extent does your model answer the original question? How do the results depend on the values of the parameters you have used in your model? Which of the assumptions you have made do you think may be particularly problematical? How do your mathematical findings relate to the biology of the original problem? Total length: less than 1000 words, with a maximum of four graphs (not including the flowchart). Within each section, marks will be awarded for presentation, clarity and logical structure. Suggested Projects Some suggestions for projects follow. It may also be possible to do something different, but check with your tutor to make sure your idea is suitable.  

  3 Sea turtle population modelling This project is based on the following scientific paper, which you should download and read before you attempt this project. Crouse, D. T., L. B. Crowder, and H. Caswell. 1987. A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology 68:1412-1423. Note that the level of mathematical sophistication in the paper itself is far more than we need or will use for this course.   The scientific question The principal objective of this project is to use a discrete time population model to investigate rates of increase or decline in loggerhead turtles, and in particular to investigate how various conservation actions may influence the viability of these populations. 

Results

  “Head starting” is one option often suggested for increasing the growth rate of turtle populations. This is based on the idea that there are very high levels of mortality between the eggs being produced and the turtles reaching one year of age. This includes predation or harvesting of eggs on the beach and also the very high level of hatchling mortality as the baby turtles emerge from the nest, cross the beach and swim over the reef flat. Head starting is the idea that the eggs can be collected, reared in incubators and then released into the open sea, greatly reducing the mortality.   Another option to increase viability of turtle populations is to limit the mortality of adults from sources such as fishing bycatch.

This might be possible by employing “turtle exclusion devices” on fishing nets. Another challenge to the viability of turtle populations is the fact that they have temperature dependent sex determination – rather than sex been determined by the chromosomes, it is determined by the temperature at which the eggs are reared. You will need to do a little library research to see what the nature of this effect might be, but clearly if climate change causes most of the hatchlings to become males, this will have implications for population viability. Conversely, if temperature increase causes a higher percentage of females, this may make turtle populations more robust to other environmental challenges.

   The model Turtles and other long?lived vertebrates need to be modelled by including age structure in the models. Clearly, juveniles cannot reproduce and the substantial delay until adulthood has important implications for the population growth rate and also population structure.   Crouse et al. (1987) model turtles by considering a number of life history stages: 1. Eggs through to one year old 2. Small juveniles from the ages of 1 to 7 3. Large juveniles from the ages of 8 to 15   4. Sub adults from the age of 16 to 21   5. First year breeders (22 years old)   6. Returners after the first year of breeding (23 years old) 7. Adults (24 years and older) They assume that within each of these stages, survival is similar for all individuals, but survival differs between each of these life history stages. They also assume that only stages 5?7 are capable of reproducing. The model also follows females only. 4 To model this in R, we need to set up a matrix to hold the results of our population model.

Discussion

It will have as many columns as there are life history stages, and as many rows as there are years we want to simulate. Initially, let’s do it for 200 years. The question is to then use an iterative process to model how the numbers of turtles in year i+1 and stage j depend on the numbers of turtles in each of these stages in the previous year i. This is slightly more complex than the version the notes, but only a little. Here is the basic R code to run the model: Remember that the basic steps in running any iterative model are to: 1. Set up initial values and parameters 2. Set up a loop to run the iteration a number of times 3. Summarise and present the results First, set up a matrix with 200 rows (one for each year) and seven columns (one for each life history stage pop<-matrix(nrow=200,ncol=7) Second, specify the fecundity for each life history stage.

These are values I have taken from the original paper. Note that the four life history stages do not reproduce at all, the first time breeders have higher fecundity, very few breed the next year, and then the adults have lower fecundity than the first time breeders, city because not everyone returns to breed every year. fec<-c(0,0,0,0,127,4,80) Third, specify the survival rate for each of the life history stages. This requires a little explanation. It’s the chance of still being in that particular history stage next year, given that you were in the previous year. So, the “survival” for the first year is zero, not because everybody dies, but because if a turtle is between zero and one years old in year t, it can’t be between zero and one year old next year. Similarly, the first year breeders and the first year returners can’t stay two successive years in the same stages.

For all the other life history stages, it is possible for them to stay in the given stage. surv<-c(0,0.7370,0.6610,0.6907,0.0,0.0,0.8091) Fourth, we need to specify the maturation rates, which are the proportion of individuals in a given life history stage in year t that survive through to enter the next life history stage in year t+1. For the first stage, this is the survival rate through the life stage, as it is for the first time breeders and the returning breeders. The last life history stage (adults) don’t mature at all. mat<-c(0.6747,0.0486,0.0147,0.0518,0.8091,0.8091,0) Next, we need to set up the initial population. Here, I’ve used a little trick in R. pop[1,] specifies the whole first row of the pop matrix, so it has seven elements. I’ve chosen to start off with 100 adults and no individuals in the earlier life history stages. You should try changing this and see what it does to the predicted population.   pop[1,]<-c(0,0,0,0,0,0,100) Now, we can set up the main loop for the iteration. We need to go through the 200 years over which we simulate, and also iterate through the different life history stages each year.

for(i in 1:199){ 5 To get into the first life history stage, a turtle needs to be born, so you multiply the fecundity of each life history stage times the number of individuals in the life history stage in year I, and that becomes the number of juveniles year I +1 pop[i+1,1]<-fec[5]*pop[i,5]+fec[6]*pop[i,6]+fec[7]*pop[i,7] Now we need to loop through the life history stages other than the first. To be in life history stage J in year I +1, you need either to mature into that life history stage, having been in the previous one here before, or you need to stay in that life history stage, having been in it the previous year. for(j in 2:7){ pop[i+1,j]<-pop[i,j-1]*mat[j-1]+pop[i,j]*surv[j] And just close off both sets of curly brackets } } Next, you will need to present results. You can plot the number of adults versus time with   t?

The following mathematical model is primarily aimed at using a discrete model to investigate the rates of increase or decline in loggerhead turtles. Conversely, the mathematical model aims to find how various conservation actions may influence the viability of these populations.

To increase the growth rate of turtle population, one of the most common option has been “head starting” (Crouse et al., 1987). The assumption is that there is a very high level of mortality between the eggs being produced and the turtles reaching the age of one year. Another option of increasing viability of the population of loggerhead turtles is limiting the mortality of adults from sources such as fishing by-catch which is possible through turtle exclusion devices on fishing nets. Another challenge is that the viability of the turtle population is determined by temperature rather than by chromosomes (Ramsey & Crews, 2009). If the population becomes more male, then the viability of the population will be in question but if there are more females, then the population will be robust (Hawkes et al., 2009).

Methods and model structure

The turtles and other long-lived vertebrates were modeled through the inclusion of age structure in the models. Juvenile turtles cannot reproduce and yet there is a substantial delay into adulthood. As a result, this presents a crucial implication for the population growth rate and also population structure.  The structure of the life history of turtles is as seen in the table below:

Table 1: Turtle life history stages

 The mathematical model that was used to test the scientific model made use of looping in iterations. The initial matrix was set up with 200 rows as seen in appendix 1. The 200 rows were for each row while the seven columns were for each life history stage as seen in table 1. The fecundity survival rate and the maturation rates were also identified as set up as seen in table 2 below.

Table 2: Survival components

The looping for iteration was developed as seen in flowchart below:

 Figure 1: Looping flowchart

From the looping flowchart for iteration, the following codes were used in the R programming code in order to come up with the results.

># Setting up initial matrix with 200 rows>pop<-matrix(nrow=200,ncol=7)># Specifying fecundity>fec<-c(0,0,0,0,127,4,80)># Specifying survival rate>surv<-c(0,0.7370,0.6610,0.6907,0.0,0.0,0.8091)># Specifying maturity rate>mat<-c(0.6747,0.0486,0.0147,0.0518,0.8091,0.8091,0)># Setting up initial population>pop[1,]<-c(0,0,0,0,0,0,100)># Setting up looping for iteration>for(i in 1:199){+pop[i+1,1]<-fec[5]*pop[i,5]+fec[6]*pop[i,6]+fec[7]*pop[i,7]+for(j in 2:7){+pop[i+1,j]<-pop[i,j-1]*mat[j-1]+pop[i,j]*surv[j]+}+}># Presenting results in plot>t<-1:200>adults<-pop[,7]>plot(t,adults)

ResultsFigure 2: Adults against time plot

Results

Figure 2 shows that the population of the adult turtles has been on the decline for the past 20 years. The decline in population is rapid between the 1^st years to the 50^th year. From then, the population becomes relatively constant till the 200^th year.Figure 2: Regression line on plot

In figure 2, the regression line was obtained. It can be seen that the regression line is vertical showing a direct negative relationship between time and adults.

Discussion

A sensitivity test was carried out to see how the changes in time affects the number of adults. The test was carried out using a correlation analysis (Van Griensven et al., 2006). The results are as shown in the table below:

Table 1: Correlation

From the table above, it is evident that the relationship between time and the number of adults is negative. Thus, an increase in time decreases the number of adults over time.

From the regression line, it is seen that the survival rates of turtles is low. Turtles who make it past 24 years have a greater chance of living for long, a survival that could see them make to 200 years.  The regression analysis also give the conservatives in making efforts to save loggerhead turtle species. A 10% increase in juvenile survival has a greater chance effect on the population than a 10% increase in adult survival. Thus supporting the findings of Crouse et al. (1987) which indicate that an increase of 14% will increase the survival rates of large juvenile thereby allowing a simulated loggerhead population to increase.

The juvenile mortality is very high thus increasing their chances of survival will increase the chances of having more adults in the future. The plot shows that the rate of survival begins to slope as the number of years increase. Thus, an increase in juvenile survival will see an increase the number of adults. Consequently, other factors such as fecundity and maturity rate will see the number of turtles increase in the long term.

More data needs to be included on the other aspects of life cycle of loggerhead sea turtles in order to address the uncomfortable possibility that current conservation efforts of loggerhead sea turtles should focus on the part of the life history of the turtle so as to produce noticeable, long-tern results. 

References:

Crouse, D.T., Crowder, L.B. and Caswell, H., 1987. A stage?based population model for loggerhead sea turtles and implications for conservation. Ecology, 68(5), pp.1412-1423.

Hawkes, L.A., Broderick, A.C., Godfrey, M.H. and Godley, B.J., 2007. Investigating the potential impacts of climate change on a marine turtle population. Global Change Biology, 13(5), pp.923-932.

Ramsey, M. and Crews, D., 2009, May. Steroid signaling and temperature-dependent sex determination—Reviewing the evidence for early action of estrogen during ovarian determination in turtles. In Seminars in cell & developmental biology (Vol. 20, No. 3, pp. 283-292). Academic Press.

Van Griensven, A., Meixner, T., Grunwald, S., Bishop, T., Diluzio, M. and Srinivasan, R., 2006. A global sensitivity analysis tool for the parameters of multi-variable catchment models. Journal of hydrology, 324(1-4), pp.10-23.