Mathematical Model: Analysis of Sea Turtle Population Dynamics

This report presents a mathematical model for analyzing sea turtle populations, focusing on survival rates, fecundity, and maturation rates. The model uses discrete time intervals to estimate the number of sea turtles, considering historical population data and comparing juvenile and adult populations. A sensitivity analysis explores the impact of increasing juvenile and adult survival rates by 10%. The results include a survival curve, summary statistics, and a linear regression model to estimate population size over time. The study assumes successful egg collection, juvenile-to-adult transition, and maturation, with the conclusion that increasing juvenile survival has a more significant impact than increasing adult survival. The report includes R code snippets for model implementation and analysis.

Running Head: SEA TURTLE POPULATION MODELING
1015 SSG: Quantitative Reasoning
Project: Mathematical Modelling
Name of Institution:
Name of Student:
Sea turtle population modeling

SEA TURTLE POPULATION MODELING
Contents
Introduction......................................................................................................................................3
Methods and Model structure..........................................................................................................4
Results..............................................................................................................................................7
Discussion......................................................................................................................................10
References......................................................................................................................................11
Appendix........................................................................................................................................12
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SEA TURTLE POPULATION MODELING
Introduction
This is a report of modelling the population of sea turtle. The scientific question of whether we
can use discrete time intervals to determine the number of turtles in the sea at any particular
moment. The model will seek to determine whether there is an increase or decrease in the
population of sea turtles. The model has used a historical number of sea turtles to model the
expected number of sea turtles at any given time (Camacho, et al., 2013).
The previous studies have shown that there is an inadequate understanding of the population
structures and survival nature of sea turtles (Pawlowsky-Glahn, Vera, Juan, Tolosana-Delgado,
& Raimon, 2015). Therefore, this is a survival analysis report that is aimed at improving the
understanding of the nature of survival and growth of adult sea turtles. The study is done to
include the maturation and fecundity of the sea turtles. The study has also provided a comparison
between the population of the juveniles and the adult sea turtles based on previous studies.
Therefore, part of the report is a sensitivity analysis that provides a comparison between
increasing the survival of the juveniles by 10% and increasing the population of the adults by
10%.
Methods and Model structure
The population of turtles can be modeled to determine the survival rate of the turtles at each
given year. The studies that have been done in the past demonstrate that juveniles are not able to
reproduce. (Harding, et al., 2011) The time between the juvenile stage and the adult stage can be
used to come up with a crucial population growth curve or simply the survival curve (Kozulia, N,
& Kozulia, 2010). The survival curve will indicate the population structure of the period of years
that are considered.
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SEA TURTLE POPULATION MODELING
The study by (Deborah, Lary, & Hall, 1987) focused on determining the juvenile population
growth structure. However, this study is conducted to model the structure of the population
growth curve to adulthood. The parameters that have been considered include the following;
1. The historical population of the turtles: This variable contains the data that was collected
on the number of adult sea turtles.
2. The fecundity of each history stage: The fecundity of the sea turtles is their ability to
produce more offsprings at their adult age. We can think of fecundity as the fertility of
the adult sea turtles.
3. The survival rate at each stage: This refers to the number of sea turtles that are alive at
each of the given periods. The survival rate is used to model the number of sea turtles that
are expected in future times.
4. The maturation rate: The maturation rate represents the number of adult sea turtles that
survive to the maturity age compared to the number of sea turtles that developed from the
juvenile to the adult stage.
The model will be built on the following steps;
1. Setting up the initial values and parameters of the model
2. Building the function of the model for generating the first output
3. Setting up a loop that in turn runs the iteration 200 times
4. Summarizing and presenting the results
5. Conducting a sensitivity analysis to investigate whether a 10% increase in Juvenile
survival has a greater or lesser effect on the population than a 10% increase in adults.
4

SEA TURTLE POPULATION MODELING
The diagram below represents the flow of the model;
#Setting up initial parameter
pop<-matrix(nrow = 200,ncol = 7)#setting up a 200 by 7 matrix
pop
fec<-c(0,0,0,0,127,4,8) #Fecundity for each history stage
fec
surv<-c(0,0.7370,0.6610,0.6907,0.0,0.0,0.8091) #The Survival rate
surv
mat<-c(0.6747,0.0486,0.0147,0.0518,0.8091,0)#specifying the maturation rates
mat
pop[1,]<-c(0,0,0,0,0,0,100)#initializing population
5
Step1:
Initializing Values and
Parameters
Step 2:
Building a function for
the first output
Step3: setting up a loop
for 200 iterations
Step 4: Summarizing and
presenting the results
(line plot, regression and
summary statistics)
Step5: Sensitivity analysis

SEA TURTLE POPULATION MODELING
pop
#Setting up the loop for iteration
for (i in 1:199) {
pop[i+1]<-fec[5]*pop[i,5]+fec[6]*pop[i,6]+fec[7]*pop[i,7]
#The above line gives the first life history stage
for (j in 2:7) {
pop[i+1,j]<-pop[i,j-1]*mat[j-1]+pop[i,j]*surv[j]
#The above line gives a loop for other years than the first
}
}
adults
#ploting the results
t<-1:200 #Inititalizing the tiime factor
adults<-pop[,7]
plot(t,adults, main = "Survival Curve")#a line plot of a survival curve
summary(adults)#Outputs Summary statistics for the survivals
#A linear survival model
surv_model<-lm(adults~t)
summary(surv_model)
#sensitivity analysis: determining whether 10% incraese in Juvenile survival has a greater or
lesser effects on the population than 10% increase in adults
a<-mean(adults) #the mean of adults
a
b<-sd(adults) #standard deviation of adults
b
n<-length(adults) #the number of variables of adults
n
std_error<-qnorm(0.95)*(b/sqrt(n))#the standard error
6

SEA TURTLE POPULATION MODELING
std_error
lower_limit<- a-std_error# the lower limit
lower_limit
upper_limit<- a+std_error#The upper limit
upper_limit
interval<- c(lower_limit,upper_limit)#displaing the upper and the lower limts
interval
interval_range<-(upper_limit)-(lower_limit)
interval_range
Results
#The following are the output from the R codes
> #Setting up initial parameter
> pop<-matrix(nrow = 200,ncol = 7)#setting up a 200 by 7 matrix
> fec<-c(0,0,0,0,127,4,8) #Fecundity for each history stage
> surv<-c(0,0.7370,0.6610,0.6907,0.0,0.0,0.8091) #The Survival rate
> mat<-c(0.6747,0.0486,0.0147,0.0518,0.8091,0)#specifying the maturation rates
> pop[1,]<-c(0,0,0,0,0,0,100)#initializing population
> #Setting up the loop for iteration
> for (i in 1:199) {
+ pop[i+1]<-fec[5]*pop[i,5]+fec[6]*pop[i,6]+fec[7]*pop[i,7]
+
+ #The above line gives the first life history stage
+ for (j in 2:7) {
+ pop[i+1,j]<-pop[i,j-1]*mat[j-1]+pop[i,j]*surv[j]
+ #The above line gives a loop for other years than the first
+ }
+ }
> adults
[1] 1.000000e+02 8.091000e+01 6.546428e+01 5.296715e+01 4.285572e+01 3.467456e+01
[7] 2.805519e+01 2.269945e+01 1.836613e+01 1.486003e+01 1.202325e+01 9.728015e+00
[13] 7.870937e+00 6.368375e+00 5.152652e+00 4.169011e+00 3.373147e+00 2.729213e+00
[19] 2.208206e+00 1.786660e+00 1.445586e+00 1.169624e+00 9.463427e-01 7.656859e-01
[25] 6.195164e-01 5.012507e-01 4.055620e-01 3.281402e-01 2.654982e-01 2.148146e-01
[31] 1.738065e-01 1.406268e-01 1.137812e-01 9.206035e-02 7.448603e-02 6.026665e-02
7

SEA TURTLE POPULATION MODELING
[37] 4.876175e-02 3.945313e-02 3.192153e-02 2.582771e-02 2.089720e-02 1.690792e-02
[43] 1.368020e-02 1.106865e-02 8.955645e-03 7.246012e-03 5.862748e-03 4.743550e-03
[49] 3.838006e-03 3.105331e-03 2.512523e-03 2.032882e-03 1.644805e-03 1.330812e-03
[55] 1.076760e-03 8.712064e-04 7.048931e-04 5.703290e-04 4.614532e-04 3.733618e-04
[61] 3.020870e-04 2.444186e-04 1.977591e-04 1.600069e-04 1.294616e-04 1.047474e-04
[67] 8.475109e-05 6.857210e-05 5.548169e-05 4.489024e-05 3.632069e-05 2.938707e-05
[73] 2.377708e-05 1.923803e-05 1.556549e-05 1.259404e-05 1.018984e-05 8.244598e-06
[79] 6.670704e-06 5.397267e-06 4.366929e-06 3.533282e-06 2.858778e-06 2.313038e-06
[85] 1.871479e-06 1.514213e-06 1.225150e-06 9.912689e-07 8.020357e-07 6.489271e-07
[91] 5.250469e-07 4.248155e-07 3.437182e-07 2.781024e-07 2.250126e-07 1.820577e-07
[97] 1.473029e-07 1.191828e-07 9.643079e-08 7.802215e-08 6.312772e-08 5.107664e-08
[103] 4.132611e-08 3.343696e-08 2.705384e-08 2.188926e-08 1.771060e-08 1.432965e-08
[109] 1.159412e-08 9.380801e-09 7.590006e-09 6.141074e-09 4.968743e-09 4.020210e-09
[115] 3.252752e-09 2.631802e-09 2.129391e-09 1.722890e-09 1.393990e-09 1.127878e-09
[121] 9.125657e-10 7.383569e-10 5.974046e-10 4.833600e-10 3.910866e-10 3.164282e-10
[127] 2.560220e-10 2.071474e-10 1.676030e-10 1.356076e-10 1.097201e-10 8.877452e-11
[133] 7.182747e-11 5.811560e-11 4.702134e-11 3.804496e-11 3.078218e-11 2.490586e-11
[139] 2.015133e-11 1.630444e-11 1.319192e-11 1.067359e-11 8.635999e-12 6.987387e-12
[145] 5.653494e-12 4.574242e-12 3.701019e-12 2.994495e-12 2.422846e-12 1.960325e-12
[151] 1.586099e-12 1.283312e-12 1.038328e-12 8.401112e-13 6.797340e-13 5.499728e-13
[157] 4.449830e-13 3.600357e-13 2.913049e-13 2.356948e-13 1.907007e-13 1.542959e-13
[163] 1.248408e-13 1.010087e-13 8.172614e-14 6.612462e-14 5.350143e-14 4.328801e-14
[169] 3.502433e-14 2.833818e-14 2.292842e-14 1.855139e-14 1.500993e-14 1.214453e-14
[175] 9.826141e-15 7.950331e-15 6.432613e-15 5.204627e-15 4.211064e-15 3.407172e-15
[181] 2.756743e-15 2.230480e-15 1.804682e-15 1.460168e-15 1.181422e-15 9.558885e-16
[187] 7.734093e-16 6.257655e-16 5.063069e-16 4.096529e-16 3.314502e-16 2.681763e-16
[193] 2.169815e-16 1.755597e-16 1.420454e-16 1.149289e-16 9.298897e-17 7.523737e-17
[199] 6.087456e-17 4.925361e-17
> #ploting the results
> t<-1:200 #Inititalizing the tiime factor
> adults<-pop[,7]
> plot(t,adults, main = "Survival Curve")#a line plot of a survival curve
> summary(adults)#Outputs Summary statistics for the survivals
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.00000 0.00000 0.00000 2.61917 0.00266 100.00000
> #A linear survival model
> surv_model<-lm(adults~t)
> summary(surv_model)
Call:
lm(formula = adults ~ t)
Residuals:
Min 1Q Median 3Q Max
-7.718 -5.145 -1.646 1.928 89.933
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SEA TURTLE POPULATION MODELING
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.14199 1.55790 6.510 6.04e-10 ***
t -0.07485 0.01344 -5.569 8.28e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 10.97 on 198 degrees of freedom
Multiple R-squared: 0.1354, Adjusted R-squared: 0.1311
F-statistic: 31.01 on 1 and 198 DF, p-value: 8.282e-08
> #sensitivity analysis: determining whether 10% incraese in Juvenile survival has a greater or lesser effects
on the population than 10% increase in adults
> a<-mean(adults) #the mean of adults
> b<-sd(adults) #standard deviation of adults
> n<-length(adults) #the number of variables of adults
> std_error<-qnorm(0.95)*(b/sqrt(n))#the standard error
> lower_limit<- a-std_error# the lower limit
> upper_limit<- a+std_error#The upper limit
> interval<- c(lower_limit,upper_limit)#displaing the upper and the lower limts
> interval
[1] 1.249835 3.988510
> interval_range<-(upper_limit)-(lower_limit)
> interval_range
[1] 2.738675
0 50 100 150 200
0 2 0 4 0 6 0 8 0 1 0 0
Survival Curve
t
a d u l ts
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SEA TURTLE POPULATION MODELING
Discussion
This study had several assumptions. It was assumed that the collection and rearing of eggs and
subsequent incubation and release into the sea was successful. It was also assumed that there was
a successful transition from the juvenile stage to the adulthood stage. It was assumed that all the
adult sea turtles that survived throughout the adulthood period reached their maturation period.
There was an assumption that the sample of the population that has been used for the
mathematical simulation is sufficient enough to make inferences about the whole population.
Finally, it was assumed that the survival of the individual sea turtles is independent of the
survival of each of the other sea turtles.
The results demonstrate that the iteration has produced a population of 200 by 7 sea turtles.
Based on the historical population sizes, the sample that has been produced is the expected
number of sea turtles that will survive to adulthood. The mean population is 2.619. The other
summary values are available in the results section above.
The plot of the population growth is shown in the results section. The plot is a line graph of the
adult population against time (Spiliopoulou, et al., 2014). The slope of the graph has been
determined by fitting a regression model. A regression model is modeling the population with
time in a linear form (Manuscript & Metz, 2013). The linear regression model represents the
population size at each given time assuming the survival rate is linear (Japkowicz, Nathalie,
Stefanowski, & Jerzy, 2016). The slope of the regression model is 10.14 while the intercept for
the time interval is -0.07485. Therefore, assuming that the survival growth is linear, then the
population of adults at time t can be estimated by;
Population = 10.14- 0.07485t
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SEA TURTLE POPULATION MODELING
The sensitivity analysis demonstrates that an increase in juvenile survival by 10% as a significant
effect on the population than a 10% increase in the population of the adults. This is clearly
evidenced by the range of the lower and the upper intervals obtained from the sensitivity
analysis. The range for the survival of the juveniles is much higher than the range for the
increase of the adults by 10%.
References
Camacho, Maria, Luzardo, Octavio, P., Boada, Luis, D., . . . Jorge. (2013). Potential adverse
health effects of persistent organic pollutants on sea turtles: Evidence from a cross-
sectional study on Cape Verde loggerhead sea turtles. Journal of Science of The Total
Environment.
Deborah, T. C., Lary, B. C., & Hall, C. (1987). A stage- based Population Model for Loggerhead
Sea Turtles and Implications for Conservation. Journal of Ecology, 1-13.
Harding, Juliana, M., Walton, Wendy, J., Trapani, Christina, M., . . . Mann, R. (2011). Sea
Turtles as Potential Dispersal Vectors for Non-Indigenous Species: The Veined Rapa
Whelk as an Epibiont of Loggerhead Sea Turtles. Journal of Southeastern Naturalist.
Japkowicz, Nathalie, Stefanowski, & Jerzy. (2016). [Studies in Big Data] Big Data Analysis:
New Algorithms for a New Society Volume 16 || Scalable Cloud-Based Data Analysis
Software Systems for Big Data from Next Generation Sequencing.
Kozulia, T. V., N, V. S., & Kozulia, M. M. (2010). Ecological Corporative System Concept in
Solving Problems of Ecological Estimation and Ecological Hygienic Normalization.
Manuscript, & Metz. (2013). Decision Data : An Extension of the Roe and Metz Simulation.
Pawlowsky-Glahn, Vera, E., Juan, J., Tolosana-Delgado, & Raimon. (2015). Modelling and
Analysis of Compositional Data (Pawlowsky-Glahn/Modelling and Analysis of
Compositional Data) || Exploratory data analysis.
Spiliopoulou, Myra, Schmidt-Thieme, Lars, Janning, & Ruth. (2014). [Studies in Classification,
Data Analysis, and Knowledge Organization] Data Analysis, Machine Learning, and
Knowledge Discovery || Data Enrichment in Discovery Systems Using Linked Data.
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Appendix
#Setting up initial parameter
> pop<-matrix(nrow = 200,ncol = 7)#setting up a 200 by 7 matrix
> pop
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] NA NA NA NA NA NA NA
[2,] NA NA NA NA NA NA NA
[3,] NA NA NA NA NA NA NA
[4,] NA NA NA NA NA NA NA
[5,] NA NA NA NA NA NA NA
[6,] NA NA NA NA NA NA NA
[7,] NA NA NA NA NA NA NA
[8,] NA NA NA NA NA NA NA
[9,] NA NA NA NA NA NA NA
[10,] NA NA NA NA NA NA NA
[11,] NA NA NA NA NA NA NA
[12,] NA NA NA NA NA NA NA
[13,] NA NA NA NA NA NA NA
[14,] NA NA NA NA NA NA NA
[15,] NA NA NA NA NA NA NA
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[21,] NA NA NA NA NA NA NA
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[23,] NA NA NA NA NA NA NA
[24,] NA NA NA NA NA NA NA
[25,] NA NA NA NA NA NA NA
[26,] NA NA NA NA NA NA NA
[27,] NA NA NA NA NA NA NA
[28,] NA NA NA NA NA NA NA
12