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Qualitative analysis of a mathematical model about population of green turtles on the galapagos island

Abstract:

According to the IUCN, most sea turtles fall into one of the endangered categories. Since, sea turtles, like many other reptiles, present an unusual developmental process, marked by the determination of the sex of the offspring by environmental factors, more specifically by temperature. In the temperature sex determination (TSD) system the temperature of an embryo’s environment during incubation period will dictate the embryo’s sex development. This developmental process, together with the complex mating and nesting behavior and the vulnerability of sea turtles to threats of a natural or anthropogenic nature, naturally lead to the study of the population dynamics of the species. For this reason, in this paper, we have developed a continuous model given by a system of three ordinary differential equations to study the dynamics of the green sea turtle population long-term, focusing the mathematical simulations on the data obtained for the nesting species of Galapagos Islands. Through the qualitative analysis of the model, the following is demonstrated: 1) The flow induced by the system is positively invariant on the region of biological interest (Ω); and 2) The given condition onˆf is necessary and sufficient for the unique nontrivial equilibrium point (I∗) to be globally asymptotically stable in that region. When implementing the estimated values for our parameters in the numerical simulations, it was observed that indeed the population of Galapagos green sea turtles complies with the condition for which the nontrivial critical point (I∗) is globally asymptotically stable; that is, the asymptotic stabil-ity is maintained for any initial value within the set Ω. In contrast, when altering the estimated values of the parameters so that the established condition is not met, the trivial critical point (I0) becomes globally stable, and the population falls towards extinction regardless of the values taken within the positively invariant Ω set.