The sex of the turtles is determined by the incubation temperature of the eggs during the mid-trimester of development. In green sea turtles (Chelonia mydas), recent studies show that sex ratios are changing, producing a female-biased sex ratio within the population. We developed a novel continuous model to analyze the dynamics of the green sea turtle population long-term. We determine the safe operating space for the proportion of eggs that become male at which the population of green sea turtle can exist without going to extinction. When the proportion of male eggs leaves this range the overall turtles’ population collapses. Additionally, we examined how temperature changes affect the sex ratios of the green sea turtle population.

Keywords: Green sea turtle, Sex ratio, Temperature-dependent sex determination, Population dynamics

The life cycle of green sea turtles starts on land when they emerge from their nests as hatchlings and immediately travel to the sea. Upon entering the ocean, the hatchlings are relatively unseen until they reappear as juveniles in the open sea²⁰. This period between the hatchling and juvenile stage is composed of several years and is the life-stage that remains relatively unknown²⁰. Thus, scientists have coined the term “lost years” to describe this stage of life^20,21. Since these two first stages of the life cycle of the turtles do not carry relevant information for the development of the model, a simplified life cycle can be seen in Figure (1), which shows the “egg” stage as each individual that has not yet reached sexual maturity and the adult stage as only the sexually mature adults.

For green sea turtles, the mean age of sexual maturity is estimated to be within the range of 40-60 years^22–24. When sea turtles reach sexual maturity, females and males return to their natal beaches to mate and nest. Male sea turtles usually breed every year or every two years, whereas female sea turtles generally breed every 2 to 5 years^25,26. During mating season, sea turtles have multiple mates and lay several clutches of eggs at approximately 2-week intervals²⁷. This scheme of the life cycle focuses mainly on the relationship between adults during the mating period. This interaction eventually will result in the production of eggs that can later develop into either female or male mature sea turtles. Also, it shows how the death of the adults and the movement out of the reproductive stage decreases the population of adult sea turtles, while the end of eggs and juveniles reduces the people of the “eggs” class.

On the other hand, the successful mating of adults will increase the number of eggs deposited every season. Each clutch can be divided into female and male eggs depending on the incubation temperatures⁷. If eggs reach maturity, this will increase the number of adults in the population.

To describe the long-term behavior of the green sea turtle population, we study the dynamics of the community with a continuous model. The main objective is to figure out the sex ratio that would ultimately lead the overall population of green sea turtles to extinction. We chose the continuous model to be able to witness the change within the turtle’s entire life cycle (60-70 years)²⁴ and for the whole population over time. Modeling the distinctive aspects of the reproductive biology and nesting behavior of green sea turtles poses a significant challenge. The mating process is a complex interaction between male and female sea turtles that involves a variety of factors that could strongly affect the birth rate of the population. To clearly understand the principles involved in this interaction, we consider the mating process as a functional response, similar to continuous-time predator-prey models²⁸.

Such as in predator-prey models, we can conveniently classify most of our variables involved in the mating process: (1) density of female population, (2) density of male population, (3) behavioral response of females during mating process, (4) characteristics of the males, e.g., searching efficiency²⁹. To represent the successful number of mating interactions between males and females per unit time, which will eventually result in the production of eggs, we used the Holling type II functional response^29,30. In predator-prey models, researchers suggest that ratio-dependence is suitable when the predator must seek its prey and consequently compete for food ³¹. In modeling the process of mating interactions between green sea turtles, we have to consider that male sea turtles have to search for females, and that male turtles will compete with one another to mate with a female³². Moreover, we have to consider some aspects, such as coupling time, associated with each female being found by a male and the fact that not every male will interact with every female. Thus, it made more sense to use a ratio-dependent system to produce more accurate results³¹.

The following simulations were developed by using parameter estimations, as seen in Table (1). All parameter values in Table (1) are kept constant except for values of p .

We only analyzed our system for values of p that satisfy the equilibrium existence condition. In other words, since is a function of T, we analyzed our system for temperatures that allowed  p(T)to still be within the equilibrium condition (8). Since our system is now analyzed with the new p(T)function, the graphs in Figure (5) reflect how temperature changes lead to different proportions of males and how those proportions lead to different population numbers for the different stages.

                                             

According to the International Union for Conservation of Nature (UICN) Red List Threatened Species⁴¹, Green sea turtle is considered as an endangered species and its population is continuously decreasing. This population decline is due to many factors like illegal trade of eggs, turtle-shell trade, plastic, and other marine debris, ocean pollution, and global warming^42–45. Most studies focus on how these factors harm the adult population and have determined that we should maintain current efforts to reduce mortality of long juvenile and adult stages to keep a sustainable population⁴⁶. Our concern, in contrast, focuses on some aspects at the earliest stages of the life cycle that can harm the adult population long-term. Due to this, we studied how the temperature at the mid-trimester of the incubation period impacts the sex ratios of the population. Fluctuations in the incubation temperature can cause a female or male-biased sex ratios in the population. With this model, we analyzed how the sex ratio is decisive for the population growth and how this is affected by the temperature. We determined a safe operating space at which temperature can alter the sex ratios without taking the population to collapse. It has been shown that different populations of sea turtles have female-biased offspring production due to temperature increases during the incubation period^47,48. Regardless of that, our results show that the proportion of female eggs can go to extreme values without having a severe impact on the turtle population. However, our numerical results depend completely on the values that our parameters take since many of these are not found in the literature; most of them had to be estimated affecting the accuracy of the results. To avoid this, more fieldwork is necessary to strengthen the parameter estimations.

A system of ordinary differential equations was used to represent the life cycle of the turtle population. The emphasis of our study is the mechanics of sex structure on population stability. A closed-form analytic condition for the temperature to ensure species persistence was found. The previous literature relating to this subject has been relatively sparse, but our results seem to be in general in accordance with prior publications⁴⁹. At the moment, the lack of studies on the subject has left us without any current contrary findings.

Since we simplified our model to denote all pre-reproductive stages as the egg stage and the reproductive stage as the adult stage, we think that this might propose limitations on the details of each sub-stage in the life cycle, as well as including the post-reproductive stage. A dearth of information has been reported with regard to the searching efficiency and coupling time of green sea turtles. We believe that having inaccurate parameter estimations and a simplified model probably caused our condition on and our condition on to be so large, thus giving us a broad range where highly skewed proportions are viable. We also based our model in a very general sense, rather than focusing on a specific area such as a beach or ocean. Male sea turtles often never come back to land after they are born and females only come to lay their eggs, so much is unknown about green sea turtles, thus proposing further limitations for us to study population dynamics, this suggests uncertainties in our mathematical model. The well-mixed, homogeneous assumption has far more severe limitations based on our understanding of sea turtle biology than in other cases. Green sea turtles both male and female return, mate, and lay their eggs at their natal beaches. A discrete-time model of a single beach or a discrete-time model that takes into spatial dynamics would be far more appropriate. The limitations of our continuous-time approach cannot be stated until these other methods are explored.

Our model can be used as a basis for a more complex model involving details about each stage in the life cycle of males and females and determine which stage is the most critical for the survival of the population. It can also be used in future research to include the thermodynamics of microplastic in the sediment and see how it is directly affecting the sea turtle population sex-ratio. We believe that one can incorporate climate change into the model to show that many species and ecosystems are being affected by it. We also think it would be very useful to create a proportion of male eggs as a time-dependent function of temperature p(T(t)) to replace p(T). First, the random variation from mean daily temperatures in June and July would be the most appropriate complication added to the model. Secondly, it would be essential to include a general trend of climate change into the stochastic yearly variation as this is an observable trend. If we could find better information about the impact of microplastic proliferation on beaches on soil thermodynamics, then we could begin to include some heuristic to capture this effect. Lastly, a much more complicated model might be interested in studying how cyclical climate variations like El Niño on the long-term stability of turtle populations. It will be essential in the future to have broad trans-disciplinary collaborations to properly account for the edaphoclimatic factors that impact the turtles TSD. Lastly, it will be necessary for further models to include egg death induced by extreme temperatures⁴⁹.

We found that there is a limited safe operating space of temperature for green sea turtle sustainability, as seen from Equation (8). If this condition is not met, then the total population approaches extinction. However, it is essential to note that this threshold is substantial, so the probability that the temperature would be outside of this range is unlikely. Based on our simulations in Figure (5), we found that the proportion of eggs that become male can decrease dramatically without a severe impact on turtle populations. When the balance of males p approaches zero, the population of females can still survive in high numbers. Highly skewed sex ratios do not necessarily mean that the population is no longer viable because males will breed with many females, suggesting that just a few males are sufficient to father multiple clutches⁵⁰.

Moreover, male sea turtles breed twice as frequently as female sea turtles, meaning females can continue to find mates even when males are low in numbers⁴⁹. However, when the male proportion reaches zero, then the whole population of Green sea turtles will go extinct. Our research indicates that it is necessary to urge policymakers and environmental managers about the importance of monitoring beach temperatures to ensure the continued existence of Green sea turtles. If temperatures get too high and exceed our threshold, then the population will go extinct, thus hurting the two ecosystems that they are a part of.