
MBW:StageBased Population Model
From MathBio
Article review by Luke Pederson. An extended model from this review is presented in a later project by the same author.
D. T. Crouse, L. B. Crowder, and H. Caswell. A stagebased population model for loggerhead sea turtles and implications for conservation. Ecology, 68(5):14121423, October 1987. *All Information, Unless Otherwise Specified, Comes From This Source
Overview
 Mathematics used: Stagebased population model, Leslie and Lefkovitch matrices, dominant eigenvalues, eigenvectors, probabilities, sensitivity analysis
 Type of model: Population model, specifically on a loggerhead turtle population
 Biological system studied: Loggerhead turtle population dynamics, based on fecundity and survivability in each life stage
Summary
Introduction
 As people become more aware of the negative externalities of development without regulations to balance the marginal social const and marginal social benefits, there has been a shift in people’s mindset toward preservation or cost efficient use of rare and/or endangered animal species, natural resources, clean air, etc. Before the adoption of legislation to require cost benefit analysis of proposed preservation projects, efforts were typically been focused on simple cost analysis and in many cases have yielded poor results. With the adoption of costbenefit analysis, conservation efforts have been reconsidered to determine if the appropriate action was taken to optimally allocate resources to achieve the desired results. Crouse, Crowder, and Caswell's 1987 article on the stagebased population models for sea turtles affirms the necessity for examining the benefits of current and proposed conservation efforts and the implications that the findings pose on future efforts.
Methodology
 This article analyzed the impact of historic conservation methods and proposed efforts on the population of loggerhead sea turtles. To accomplish this, a sensitivity analysis on a stagebased population model was performed in order to determine where, in the turtles' life cycle, conservation efforts should be focused to prevent extinction. Stagebased population models are formed by a Lefkovitch Matrix and make the assumptions that all members within a group or stage are affected equally by the fecundity, mortality, and growth rates. The overall population growth/decline for these models is determined by examining the magnitude of the largest eigenvalue and the population distribution, in the long run, among stages, is determined by normalization of the eigenvector corresponding to the largest eigenvalue.
Sensitivity Analysis
 Sensitivity of the model to perturbations was analyzed by simulating changes to specific traits of a stage and examining the variation in the growth rate and population distribution. The findings of the article suggested that the population of loggerhead turtles is most sensitive to perturbations in the large juvenile and subadult stages and significantly less sensitive to changes in the fecundity rates and egg/hatchling survivability. These findings supported the observation at the time that, even with very careful egg protection, the populations were still falling. Today, after regulation requiring Trawler Efficiency Devices/Turtle Exclusion Devices (TEDs) in the US for shrimp nets, the population of loggerhead turtles has decreased in its rate of decline from ~3% in 1987 [1] to ~1.8 % today [2].
Conclusions
 Overall, the Crouse, Crowder, and Caswell's article demonstrated the importance of examining the benefits of conservation efforts on a population by examining its sensitivity to changes in a stage of development. Although current conservation methods may be insufficient to prevent the decline in sea turtle populations, the sensitivity analysis illustrates the significance of attempting to direct conservation efforts at the stages of development that are the most elastic, or sensitive to change.
Context & History
 Crouse, Crowder, and Caswell's article describes a model of the loggerhead turtle to compare the effects of various management techniques on the species survival. Conservation efforts for the loggerhead turtles started largely in the 1960s and continue today. Most efforts until the early 1990s were focused on protecting nest and beach management for nest survival. In the mid to late 1980s biologist began to consider that juvenile and adult mortality may be equally, if not more, important to the survival of the population. This hypothesis was one of the driving factors for the 1987 article. Many biologists have made observations to try to determine the rates of remigration, survival, mortality, etc. This article primarily utilized data from Richardson's 20 year Little Cucumber Island project with Frazer's analysis of the data [1].
Mathematical Model
StageBased Population Models
 Stagebased models are based on the concept that a large group can be broken into smaller relatively homogeneous groups that have similar parameters for each group and are somehow interconnected with the other groups. For a population model, it is common for the model to split up a group by age or characteristics. For example, a group of people could be broken into age groups such as infants (02), children (312), adolescents (1317), adults (1849), and elderly adults (50+). As time progresses, there must be an ability for each group to move to another group, remain in the same group, or produce offspring. If the probability of remaining in the same group is P, moving to the group is G, and producing offspring is F, the population model can be formed. Recalling the group of people example, if all groups after the children group produce one child per couple per ten year span, the probability F is 0.05 of a member of that group having a child. If the age cap is 100, people progress at a proportion of the age span, and no one dies in the non elderly adult stage, then the probability P would be [ 0.5, 0.111, 0.25, 0.032] and the probability G would be [ 0.5, 0.889, 0.75, .968, 0.98]. This would yield the matrix shown in equation 1.
Equation 1: Example StageBased Model
 Many assumptions must be made for this model to hold. The population must be completely homogeneous within each group. That is, there must be an equal amount of males and females in each state and each person’s capability to survive, reproduce, stay in the group, and leave the group must be equal for all group members. Also, for the experimental results to match with the mathematical model, the initial state should be close enough to the equilibrium distribution to prevent any physically impossible scenario such as a collection of infants growing up and progressing as the model predicts without anyone in any other group to take care of them. With the model generated, the dominant eigenvalue of the system can be extracted to determine the total population growth/decay rate. If the eigenvalue is less than one, the population will die out and conversely, if it is larger than one, the population will grow. In the example described by equation 1, the dominant eigenvalue is 1.013. This implies that as time progresses, the rate of population growth will approach ~1.3 %. Scaling the Eigenvector to one, the population distribution in the long run would be 3.95 % infants, 15.93 % children, 6.72 % adolescent, 37.30 % adult, and 36.09 % elderly adult.
 Sensitivity analysis can be performed on the model by examining the variation in the dominant eigenvalue caused by a change in one of the parameters. For example, the reproduction rate of one of the stages in equation 1 could be varied and the new eigenvalue determined. This could be repeated for all parameters to determine which group and which parameter of that group changes the dominant eigenvalue the most or least depending on what is desired. Suppose one wished to examine the effect of an all out overseas war on population growth where, effectively, all the male adult population is recruited (Ex. A worse version of WWII). Since many may die, the proportion of adults moving into the next category and the proportion staying in the adult category would fall (25% for example). Also, since stages are independent in regards to the parameters, the women in the adult stage would not be able to reproduce (by the model’s assumptions) and thus their contribution to the infant group would fall to zero. The updated model of the population can be seen in equation 2.
Equation 2: Example StageBased Model With War
 Analyzing this matrix (Equation 2), the dominant eigenvalue was found to be 0.9834 meaning that under this war time scenario, the population growth would cease and the population would actually begin falling. Assuming it was a long war, the population distribution would also have changed. The new steady state distribution would be 3.5 % infants, 18.64 % children, 8.86 % adolescents, 8.61 % adults, and 60.37 % elderly adults. This is a crude example; however, it does show how population dynamics change when affected by positive/negative externalities.
SetUp
 One of the principle goals of this article was to determine what stage was most likely causing the downfall of the turtles. The concept was that if the most sensitive factor could be isolated and conservation resourced focused on helping that stage, the decline of the turtle population could potentially be stopped. The model utilized in the article broke the turtle population into seven groups: 1st year (eggs and hatchlings), small juveniles, large juveniles, subadults, novice breeders, 1styr remigrants, and mature breeders. A stagebased model was used instead of an age based model as the data typically grouped the animals by size and very little data was available about the age of the turtles measured. Using the values derived primarily from Richardson's 20 year Little Cucumber Island project for reproductive output, survival probabilities, and probabilities of remaining in the same stage, a 7 by 7 Lefkovitch Matrix was generated to model the turtle population shown in equation 3.
Equation 3: Lefkovitch Model of Loggerhead Population [1]
Analysis of Base Model
 Examination of the model revealed that the dominant eigenvalue was 0.945. Since this is less than one, the population would decrease over time as suggested by the power method. Since the data showed that the population was diminishing towards extinction, it was not surprising that the dominant eigenvalue was less than one. To determine the population distribution over time, the eigenvector corresponding to the dominant eigenvalue was determined and scaled so that the sum of all elements was one. This yielded a long term population distribution as described in table 1.
Table 1
1st Year

Small Juveniles

Large Juveniles

SubAdults

Novice Breeders

1styr remigrants

Mature Breeders

20.651 % 
66.975 % 
11.460 % 
0.6623 % 
0.0363 % 
0.0005 % 
0.0029 %

 The response to a small initial perturbation from a steady state distribution was simulated using MATLAB to demonstrate the population response and distribution response. This can be seen in plots 1 and 2.
Plot 1: Population Response given Small Perturbation
Plot 2: Distribution Response given Small Perturbation
Sensitivity Analysis
 The benefit of a population matrix is that the growth and distribution variations can be determined for changes in any of the parameters.
Parameter Variations: Fecundity & Survivability
[1]
 The parameters chosen for the model were based on a range of estimates. Therefore, to ensure that the model was valid, Crouse, Crowder, and Caswell simulated the model with a 50% decrease in fecundity for each stage and propagated the changes throughout the model. This caused the rate of population to decay to increase (decay faster) for all stages. The resulting change in the population decay was greatest for the juvenile and subadult categories. This can be seen graphically in the top figure of figure 1 taken from the Crouse, Crowder, and Caswell article.
 The model was also simulated for a perfect survivability case. This is not realistic, but it serves to demonstrate the effect survivability plays on the populations. Similar to the 50% decrease case, the greatest change was noted in the juvenile and subadult stages. It is interesting to note that even with 100% survivability of all eggs, the population, according to the model, would still decrease, suggesting that the conservation efforts at the time the article was written (almost solely egg/beach management) could not stop the population decay regardless of effort exerted.
 The variations of all parameters, fecundity (F), survival with growth (G), and survival in the same class (P) for each stage was performed for the article and illustrated in figure 3. Elasticity or proportional sensitivity of the dominant eigenvalue is shown on the vertical axis. This figure demonstrates that fecundity has little effect on the population growth/decay, advancement to future states has a moderate effect, and that the most sensitive parameter was survival in the same stage. Once again, the juvenile, subadult, and mature adult populations demonstrated the most significant sensitivity to changing the growth/decay rate.
[1]
Population Management Efforts
 The primary cause of death to turtles in the southeastern US is believed to be caused by fishing nets. Crouse, Crowder, and Caswell's article was written before Trawl Efficiency Devices or Turtle Exclusion Devices (TEDs) were mandated. The article suggest that implementation of these devices may significantly reduce the deaths of juvenile and adult turtles. Simulations were performed to demonstrate the significant impact these devices could play in improving survivability and generating growth of the population.
 Today, the population of loggerhead turtles continues to fall even after the implementation of the TEDs. Recent articles by the Fish and Wildlife Service [2] suggest that this decline may be at least partially caused by the following: the first few generations of TEDs, especially those designed for small boats, were ineffective or were disabled by fisherman who believed that the device would reduce their catch: the TEDs were designed primarily for shrimp boats and not mandated for all other net utilizing boats so many turtles were still caught in those types of nets: and finally, turtles are intentionally caught in places like Mexico to use their shells as souvenirs or decorations. Hopefully as technology improves and enforcement/implementation of TED like devices stepped up, the conservation efforts can halt the drop in sea turtle populations and allow them to replenish. [2]
Conclusions
 Stagebased population models are a useful tool for evaluating conservation and management efforts. From the findings, it is clear that exact values are not always required to determine the general trends in the population and to which parameters it is most sensitive to. The data used to generate the turtle model varied significantly between estimates; however, even factoring in significant variations 50+%, for some values, the model still showed that the most sensitive stages to the turtles' long term population growth was the juvenile, subadult, and mature adult stages. The article found that there was very little effect on the long term growth/decay rate from even 100% effective egg and beach management. This finding was supported by the data at the time showing that, although efforts had been underway to protect nest for over 20 years at the time the article was written, the population was still falling. The article suggested that more emphasis on management be placed on increasing the survivability of the stages that demonstrated the highest sensitivity, possibly through the use of TEDs. Although TEDs are now in use and the populations are still declining, the concept of using a sensitivity analysis of a stagebased population model is still valid. Even if it does not perfectly represent the population, it still gives planners and policy makers an idea of how to most effectively allocate resources and conservation efforts to maximize desired results.
Further Reading
 The following is a summary of a paper by Enneson et al. (2008) (see below for paper), which cites the paper by Crouse et al. (1987). Both arrive at many of the same conclusions about turtle populations.
Like the paper by Crouse et al. (1987), "Using Longterm Data and a Stageclassified Matrix to Assess Conservation Strategies for an Endangered Turtle (Clemmys guttata)" uses stageclassified and ageclassified matrices to find the sensitivity and elasticity matrices to determine the most susceptible life stage of a certain population of turtle. Crouse et al. (1987) found that, for the Loggerhead turtle, changes to the survivability in juvenile stages made the most impact on the survivability of the entire species. Unlike these findings however, the paper by Enneson et al. (2008) found that increasing the survivability of the adult stages of spotted turtles would increase the entire survivability of the species the most. Small perturbations of the adult life stage of the spotted turtle produced large effects on the population of the turtles in Eastern Georgia Bay, Ontario, Canada. This was found to be the case by finding and interpreting the values in the elasticity matrix for the population of turtles. The elasticity matrix for the spotted turtle turned out to be:
One can see that the highest value in the elasticity matrix is for the survivability in the adult class. Since this number is the largest in the matrix, the authors reasonably concluded that small changes to the adult population of spotted turtles would cause large changes in the growth or decline of the total population of turtles. The second most important stage to protect is the survivability of the juvenile turtles. Though protecting this stage is not as critical as protecting the adult stage, any increase in this stage would prove beneficial to the population.
The authors compared their findings with other similar analysis done on other populations and breeds of turtles, and found that their findings were mostly consistent with those by others: an increase in the adult and juvenile stages would prove the most beneficial to the population of turtles. The authors then proposed many conservation techniques that would benefit this spotted turtle population the most. Many of these proposed solutions have problems, and the authors note this. The first conservation technique proposed by the authors was to simply protect the habitat of adult and juvenile spotted turtles. Since the habitat of juveniles is not fully understood, this would prove to be difficult. The second proposed conservation method describes “headstarting”. This involves capturing eggs and raising the turtles until they reach maturity. Since a spotted turtle does not reach maturity until age twelve, this method would be very costly, and may not even work. The last protection method proposed is to protect the turtle eggs. Unfortunately, the spotted turtle nests are very hard to find, so this method would also be hard to implement. Though the proposed protection methods seem unreasonable, the authors were able to recognize the life stages at which the spotted turtle is most vulnerable to increases or decreases in the population. Using stageclassified matrices, researchers can learn more about where to focus conservation efforts in order to save populations and species at risk of extinction.
External Links & References
 D. T. Crouse, L. B. Crowder, and H. Caswell. A stagebased population model for loggerhead sea turtles and implications for conservation. Ecology, 68(5):14121423, October 1987.
 Enneson, Jean J., and Jaqueline D. Litzgus. "Using Longterm Data and a Stageclassified Matrix to Assess Conservation Strategies for an Endangered Turtle (Clemmys guttata)." Biological Conservation 141 (2008): 1560568.
 National Marine Fisheries Service and U.S. Fish and Wildlife Service. 2008. Recovery Plan for the Northwest Atlantic Population of the Loggerhead Sea Turtle (Caretta caretta), Second Revision. National Marine Fisheries Service, Silver Spring, MD.
