Mathematical Model For Investigating The Viability Of Loggerhead Turtle Populations

Methods and model structure

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

Category	Years
Eggs and hatchlings	0 to 1 year old
Small juveniles	1 to 7 years old
Large Juveniles	8 to 15 years old
Sub adults	16 to 21 years old
First year breeders	22 years old
Returners after the first year of breeding	23 years old
Adults	24 years and older

 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

Category	Fecundity	Survival rate	Maturity rate
Eggs and hatchlings	0	0	0.6747
Small juveniles	0	0.737	0.0486
Large Juveniles	0	0.662	0.0147
Sub adults	0	0.6907	0.0518
First year breeders	127	0.0	0.8091
Returners after the first year of breeding	4	0.0	0.8091
Adults	80	0.8091	0

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

	Adults
Time	-0.4683081

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.