Qualitative analysis of a mathematical model about population of green turtles on the Galapagos island
DOI:
https://doi.org/10.11145/j.biomath.2021.07.293Keywords:
Population dynamics, sex-structured continuous time model, Chelonia mydas, equi- librium point, local stability, global stabilityAbstract
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 and 2). The given condition on is necessary and sufficient for the unique nontrivial equilibrium point 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 is globally asymptotically stable.References
S.J. Morreale, G.J. Ruiz and E.A. Standora, Temperature dependent sex determination: current practices threaten conservation of sea turtles. Science, 216 (4551), 1245-1247, 1982.
V. Lewis-Winokur and R.M. Winokur, Incubation temperature affects sexual differentiation, incubation time, and posthatching survival in desert tortoises (Gopherus agassizi), Canadian Journal of Zoology, 73 (11), 2091-2097, 1995.
J.J. Bull, Temperaturesensitive periods of sex determination in a lizard: Similarities with turtles and crocodilians, Journal of Experimental Zoology, 241 (1), 143-148, 1987.
D.C. Deeming and M.W. Ferguson, The mechanism of temperature dependent sex determination in crocodilians: a hypothesis, American Zoologist, 29 (3), 973-985, 1989.
M.M Fuentes, M. Hamann, M. and C.J. Limpus, Past, current and future thermal profiles of green turtle nesting grounds: Implications from climate change, Journal of Experimental Marine Biology and Ecology, 383 (1), 56-64, 2010.
Y. Matsumoto, and D. Crews, Molecular mechanisms of temperature-dependent sex determination in the context of ecological developmental biology, Molecular and cellular endocrinology, 354 (1-2), 103-110, 2012.
S.Y. Ozdilek, B.E. S ??onmez, and Y. Kaska, Sex ratio estimations of Chelonia mydas hatchlings at Samanda Beach, Turkey, Turkish Journal of Zoology, 40 (4), 552-560, 2016.
L.I. Wright, K.L. Stokes, W.J. Fuller, B.J. Godley, A. McGowan, R. Snape and A.C. Broderick, Turtle mating patterns buffer against disruptive effects of climate change, Proceedings of the Royal Society B: Biological Sciences, 279 (1736), 2122-2127,
N. Mrosovsky and C.L. Yntema, Temperature dependence of sexual differentiation in sea turtles: implications for conservation practices, Biological Conservation, 18 (4), 271-280, 1980.
M.P. Jensen, C.D. Allen, T. Eguchi, I.P. Bell, E.L. LaCasella, W.A. Hilton and P.H. Dutton, Environmental warming and feminization of one of the largest sea turtle populations in the world, Current Biology, 28 (1), 154-159, 2018.
S. Yamaguchi and Y. Iwasa, Temperature-dependent sex determination, realized by hormonal dynamics with enzymatic reactions sensitive to ambient temperature, Journal of theoretical biology, 453, 146-155, 2018.
R. King, W.H. Cheng, C.T. Tseng, H. Chen and I.J. Cheng, Estimating the sex ratio of green sea turtles (Chelonia mydas) in Taiwan by the nest temperature and histological methods, Journal of experimental marine biology and ecology, 445, 140-147, 2013.
J.R. Spotila, E.A. Standora, S.J. Morreale and G.J. Ruiz, Temperature dependent sex determination in the green turtle (Chelonia mydas): effects on the sex ratio on a natural nesting beach, Herpetologica, 74-81, 1987.
A.B. Bolten, and G.H. Balazs, Biology of the early pelagic stagethe lost year.. Biology and Conservation of Sea Turtles, Revised edition, Smithsonian Institute Press, Washington, DC, 579, 1995.
A. Carr, Notes on the behavioral ecology of sea turtles. Biology and conservation of sea turtles, Smithsonian Institution press Washington, DC, 19-23, 1982.
F. Gantmacher, The Theory of Matrices, Vol 2, Chelsea Publishing, New York, 1974.
D. Green, The east Pacific green sea turtle in Galapagos, Noticias de Galpagos, Charles Darwin Research Station (28), 9-12, 1978
H.F. Hirth and D.A. Samson, Nesting behavior of green turtles (Chelonia mydas) at Tortuguero, Costa Rica. Comportamiento de anidamiento de las tortugas verde (Chelonia mydas) en Tortuguero, Costa Rica, Caribbean Journal of Science, 23 (3/4),
-379, 1987.
P.L. Lee, P. Luschi and G.C. Hays, Detecting female precise natal philopatry in green turtles using assignment methods, Molecular ecology, 16(1), 61-74, 2007.
D. Green, Growth rates of wild immature green turtles in the Galapagos Islands, Ecuador, Journal of Herpetology, 338-341, 1993.
J.J. Alava, P. Jimnez, M. Peafiel, W. Aguirre and P. Amador. Seaturtle strandings and mortality in Ecuador: 1994-1999, Marine Turtle Newsletter, 108, 4-7, 2005.
P. Zrate, P, K.A. Bjorndal, M. Parra, P.H. Dutton, J.A. Seminoff, and A.B. Bolten, Hatching and emergence success in green turtle Chelonia mydas nests in the Galpagos Islands, Aquatic Biology, 19 (3), 217-229, 2013
S. Zavala Montoya, J. Belmont, M. Hirschfeld and D. Alarcn, Estimacin de la proporcin de sexos de latortuga verde (Chelonia mydas) en reas de alimentacin en las islas Galpagos, In Simposio de Tortugas Marinas de Ecuador 2018.
L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics 7, Springer-Verlag, 1993.
J.K. Hale, Ordinary Differential Equations, Dover Publications, Mineola, New York, 2009.
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