Modelos de poblaciones con crecimiento logístico y memoria.
ilustraciones, diagramas
- Autores:
-
Ramírez Granada, Jonnathan
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/80309
- Palabra clave:
- 510 - Matemáticas
Modelos biológicos
Ecuaciones diferenciales
Differential equations
Biological models
Sistemas dinámicos
Derivada de Caputo
Memoria
Teorema de Matignon
linealización
Crecimiento lógístico
Dynamical systems
Caputo derivative
Memory
linearization
Matignon Theorem
logistic growth
- Rights
- openAccess
- License
- Atribución-NoComercial-CompartirIgual 4.0 Internacional
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| dc.title.spa.fl_str_mv |
Modelos de poblaciones con crecimiento logístico y memoria. |
| dc.title.translated.eng.fl_str_mv |
Populations models with logistic growth and memory. |
| title |
Modelos de poblaciones con crecimiento logístico y memoria. |
| spellingShingle |
Modelos de poblaciones con crecimiento logístico y memoria. 510 - Matemáticas Modelos biológicos Ecuaciones diferenciales Differential equations Biological models Sistemas dinámicos Derivada de Caputo Memoria Teorema de Matignon linealización Crecimiento lógístico Dynamical systems Caputo derivative Memory linearization Matignon Theorem logistic growth |
| title_short |
Modelos de poblaciones con crecimiento logístico y memoria. |
| title_full |
Modelos de poblaciones con crecimiento logístico y memoria. |
| title_fullStr |
Modelos de poblaciones con crecimiento logístico y memoria. |
| title_full_unstemmed |
Modelos de poblaciones con crecimiento logístico y memoria. |
| title_sort |
Modelos de poblaciones con crecimiento logístico y memoria. |
| dc.creator.fl_str_mv |
Ramírez Granada, Jonnathan |
| dc.contributor.advisor.none.fl_str_mv |
Mejía-Salazar, Carlos Enrique |
| dc.contributor.author.none.fl_str_mv |
Ramírez Granada, Jonnathan |
| dc.contributor.researchgroup.spa.fl_str_mv |
Computación Científica |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas |
| topic |
510 - Matemáticas Modelos biológicos Ecuaciones diferenciales Differential equations Biological models Sistemas dinámicos Derivada de Caputo Memoria Teorema de Matignon linealización Crecimiento lógístico Dynamical systems Caputo derivative Memory linearization Matignon Theorem logistic growth |
| dc.subject.lem.none.fl_str_mv |
Modelos biológicos |
| dc.subject.lemb.none.fl_str_mv |
Ecuaciones diferenciales Differential equations Biological models |
| dc.subject.proposal.spa.fl_str_mv |
Sistemas dinámicos Derivada de Caputo Memoria Teorema de Matignon linealización Crecimiento lógístico |
| dc.subject.proposal.eng.fl_str_mv |
Dynamical systems Caputo derivative Memory linearization Matignon Theorem logistic growth |
| description |
ilustraciones, diagramas |
| publishDate |
2021 |
| dc.date.accessioned.none.fl_str_mv |
2021-09-27T13:58:36Z |
| dc.date.available.none.fl_str_mv |
2021-09-27T13:58:36Z |
| dc.date.issued.none.fl_str_mv |
2021-06-24 |
| dc.type.spa.fl_str_mv |
Trabajo de grado - Maestría |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/masterThesis |
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info:eu-repo/semantics/acceptedVersion |
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Text |
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http://purl.org/redcol/resource_type/TM |
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acceptedVersion |
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https://repositorio.unal.edu.co/handle/unal/80309 |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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https://repositorio.unal.edu.co/handle/unal/80309 https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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spa |
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spa |
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Abdelouahab, M.-S., Hamri, N.-E., y Wang, J. (2012). Hopf bifurcation and chaos in fractionalorder modified hybrid optical system. Nonlinear Dynamics, 69(1):275-284. Aguila-Camacho, N., Duarte-Mermoud, M. A., y Gallegos, J. A. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9):2951-2957. Ahmed, E., El-Sayed, A., y El-Saka, H. A. (2006). On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Physics Letters A, 358(1):1-4. Akrami, M. H. y Atabaigi, A. (2020). Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. Journal of Applied Mathematics and Computing, 64:615-633. Bacäer, N. (2011). A short history of mathematical population dynamics. Springer Science & Business Media. Baleanu, D., Diethelm, K., Scalas, E., y Trujillo, J. J. (2012). Fractional calculus: models and numerical methods, volume 3. World Scientific. Belmahi, N. y Shawagfeh, N. (2020). A new mathematical model for the glycolysis phenomenon involving Caputo fractional derivative: Well posedness, stability and bifurcation. Chaos, Solitons & Fractals, page 110520. Boukhouima, A., Hattaf, K., Lotfi, E. M., Mahrouf, M., Torres, D. F., y Yousfi, N. (2020). Lyapunov functions for fractional-order systems in biology: Methods and applications. Chaos, Solitons & Fractals, 140:110224. De Vries, G., Hillen, T., Lewis, M., Müller, J., y Schönfisch, B. (2006). A course in mathematical biology: quantitative modeling with mathematical and computational methods. SIAM. Deshpande, A. y Daftardar-Gejji, V. (2016). Local stable manifold theorem for fractional systems. Nonlinear Dynamics, 83(4):2435-2452. Deshpande, A. S., Daftardar-Gejji, V., y Sukale, Y. V. (2017). On hopf bifurcation in fractional dynamical systems. Chaos, Solitons & Fractals, 98:189-198. Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media. Diethelm, K., Ford, N. J., y Freed, A. D. (2004). Detailed error analysis for a fractional Adams method. Numerical algorithms, 36(1):31-52. Doetsch, G. (2012). Introduction to the Theory and Application of the Laplace Transformation. Springer Science & Business Media. Elsadany, A. y Matouk, A. (2015). Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization. Journal of Applied Mathematics and Computing, 49(1):269-283. Gallegos, J. A. y Duarte-Mermoud, M. A. (2016). On the Lyapunov theory for fractional order systems. Applied Mathematics and Computation, 287:161-170. Garrappa, R. y Popolizio, M. (2018). Computing the matrix Mittag-Leffler function with applications to fractional calculus. Journal of Scientific Computing, 77(1):129-153. Ghosh, U., Pal, S., y Banerjee, M. (2020). Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis. Chaos, Solitons & Fractals, 143. Gorenflo, R. y Mainardi, F. (2008). Fractional calculus: integral and differential equations of fractional order. arXiv:0885.3823v1. Hartman, P. (1982). Ordinary Differential Equations: Second Edition. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). Hassouna, M., Ouhadan, A., y El Kinani, E. H. (2018). On the solution of fractional order SIS epidemic model. Chaos, Solitons & Fractals, 117:168-174. Kaczorek, T. (2011). Selected problems of fractional systems theory. Springer Science & Business Media. Krabs, W. y Pickl, S. (2010). Dynamical systems: stability, controllability and chaotic behavior. Springer Science & Business Media. Kumar, R. y Kumar, S. (2014). A new fractional modelling on susceptible-infected-recovered equations with constant vaccination rate. Nonlinear Engineering, 3(1):11-19. Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory. Springer Science & Business Media. Layek, G. (2015). An introduction to dynamical systems and chaos. Springer. Li, C. y Ma, Y. (2013). Fractional dynamical system and its linearization theorem. Nonlinear Dynamics, 71(4):621-633. Li, H.-L., Zhang, L., Hu, C., Jiang, Y.-L., y Teng, Z. (2016). Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing, 54(1-2):435-449. Li, X. y Wu, R. (2014). Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dynamics, 78(1):279-288. Lynch, S. (2004). Dynamical systems with applications using MATLAB. Springer. Ma, L. y Li, C. (2016). Center manifold of fractional dynamical system. Journal of Computational and Nonlinear Dynamics, 11(2). Machado, J. T., Kiryakova, V., y Mainardi, F. (2011). Recent history of fractional calculus. Communications in nonlinear science and numerical simulation, 16(3):1140-1153. Machado, T., Kiryakova, V., y Mainardi, F. (2010). A poster about the recent history of fractional calculus. Fractional Calculus and Applied Analysis, 13(3):329p-334p. Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications, volume 2, pages 963-968. Lille, France. Milici, C., Drăgănescu, G., y Machado, J. T. (2019). Introduction to fractional differential equations. Springer. Müller, J. y Kuttler, C. (2015). Methods and models in mathematical biology. Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Heidelberg, Germany. Perko, L. (2013). Differential equations and dynamical systems. Springer Science & Business Media. Popolizio, M. (2019). On the Matrix Mittag-Leffler Function: Theoretical Properties and Numerical Computation. Mathematics, 7(12):1140. Rivero, M., Trujillo, J. J., Vázquez, L., y Velasco, M. P. (2011). Fractional dynamics of populations. Applied Mathematics and Computation, 218(3):1089-1095. Rosenzweig, M. L. y MacArthur, R. H. (1963). Graphical representation and stability conditions of predator-prey interactions. The American Naturalist, 97(895):209-223. Sadeghi, A. y Cardoso, J. R. (2018). Some notes on properties of the matrix Mittag-Leffler function. Applied Mathematics and Computation, 338:733-738. Saeedian, M., Khalighi, M., Azimi-Tafreshi, N., Jafari, G., y Ausloos, M. (2017). Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model. Physical Review E, 95(2):022409. Sayevand, K. (2016). Fractional dynamical systems: A fresh view on the local qualitative theorems. International Journal of Nonlinear Analysis and Applications, 7(2):303-318. Stanislavsky, A. (2000). Memory effects and macroscopic manifestation of randomness. Physical Review E, 61(5):4752. Tian, J., Yu, Y., y Wang, H. (2014). Stability and bifurcation of two kinds of three-dimensional fractional Lotka-Volterra systems. Mathematical Problems in Engineering, 2014. Vargas-De-León, C. (2015). Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1-3):75-85. Wiggins, S. (2003). Introduction to applied nonlinear dynamical systems and chaos. Springer Science & Business Media. Witelski, T. y Bowen, M. (2015). Methods of mathematical modelling. Springer. Yazdani, M. y Salarieh, H. (2011). On the existence of periodic solutions in time-invariant fractional order systems. Automatica, 47(8):1834-1837. |
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vii, 89 páginas |
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application/pdf |
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Colombia |
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Universidad Nacional de Colombia |
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Medellín - Ciencias - Maestría en Ciencias - Matemática Aplicada |
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Escuela de matemáticas |
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Facultad de Ciencias |
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Medellín |
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Universidad Nacional de Colombia - Sede Medellín |
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Universidad Nacional de Colombia |
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Atribución-NoComercial-CompartirIgual 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Mejía-Salazar, Carlos Enrique857d01016c4a3b5adaea406d23f10665600Ramírez Granada, Jonnathancf5ccf708bdef859b968ffeb5f10e7a2Computación Científica2021-09-27T13:58:36Z2021-09-27T13:58:36Z2021-06-24https://repositorio.unal.edu.co/handle/unal/80309Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramasLa modelación matemática de sistemas biológicos está basada en el uso de diferentes herramientas, en particular en ecuaciones diferenciales y, por tanto, en sistemas dinámicos. Recientemente, se ha buscado que los modelos biológicos consideren el concepto de memoria, lo cual ha llevado a formular sistemas de ecuaciones en derivadas de orden fraccionario como el operador diferencial de Caputo. Con base en esto, se plantean varios objetivos dentro de esta investigación, incluyendo reconocer los resultados respecto a la estabilidad de las soluciones de equilibrio existentes, comparar el comportamiento cualitativo de los sistemas ordinario y fraccionario y aplicar estos resultados en modelos de poblaciones y transmisión de enfermedades. En aras de ello, se realiza el estudio de diferentes sistemas por medio de linealización y la construcción de los diagramas de fase usando el método predictor-corrector de Adams-Bashforth-Moulton. Los modelos estudiados corresponden a modelos predador-presa, de competición y transmisión de epidemias, incluyendo algunos casos con crecimiento logístico. Respecto al comportamiento de los diferentes sistemas, se puede ver que el orden de la derivada es determinante en la estabilidad de los mismos, obteniendo casos en los que este valor corresponde a un parámetro de bifurcación, llegando inclusive a obtener comportamientos consistentes con bifurcaciones de Hopf. (Texto tomado de la fuente)Mathematical modeling of biological systems is based on the use of different tools, particularly differential equations and, therefore, dynamical systems. Recently, biological models have sought to consider the concept of memory, which has led to the formulation of systems of fractional order derivative equations such as the Caputo differential operator. Based on this, several objectives are raised within this research, including recognizing the results regarding the stability of existing equilibrium solutions, comparing the qualitative behavior of ordinary and fractional systems, and applying these results in models of populations and disease transmission. For this purpose, the study of different systems is carried out by means of linearization and the construction of phase diagrams using the Adams-Bashforth-Moulton predictor-corrector method. The studied models correspond to predator-prey, competition and epidemics transmission models, including some cases with logistic growth. Regarding the behavior of the different systems, it can be seen that the order of the derivative is determinant in their stability, obtaining cases in which this value corresponds to a bifurcation parameter, even obtaining behaviors consistent with Hopf bifurcations.MaestríaMagíster en Ciencias: Matemática AplicadaLínea de Investigación: Sistemas dinámicosvii, 89 páginasapplication/pdfspaUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - Matemática AplicadaEscuela de matemáticasFacultad de CienciasMedellínUniversidad Nacional de Colombia - Sede Medellín510 - MatemáticasModelos biológicosEcuaciones diferencialesDifferential equationsBiological modelsSistemas dinámicosDerivada de CaputoMemoriaTeorema de MatignonlinealizaciónCrecimiento lógísticoDynamical systemsCaputo derivativeMemorylinearizationMatignon Theoremlogistic growthModelos de poblaciones con crecimiento logístico y memoria.Populations models with logistic growth and memory.Trabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMColombiaAbdelouahab, M.-S., Hamri, N.-E., y Wang, J. (2012). Hopf bifurcation and chaos in fractionalorder modified hybrid optical system. Nonlinear Dynamics, 69(1):275-284. Aguila-Camacho, N., Duarte-Mermoud, M. A., y Gallegos, J. A. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9):2951-2957. Ahmed, E., El-Sayed, A., y El-Saka, H. A. (2006). On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Physics Letters A, 358(1):1-4. Akrami, M. H. y Atabaigi, A. (2020). Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. Journal of Applied Mathematics and Computing, 64:615-633. Bacäer, N. (2011). A short history of mathematical population dynamics. Springer Science & Business Media. Baleanu, D., Diethelm, K., Scalas, E., y Trujillo, J. J. (2012). Fractional calculus: models and numerical methods, volume 3. World Scientific. Belmahi, N. y Shawagfeh, N. (2020). A new mathematical model for the glycolysis phenomenon involving Caputo fractional derivative: Well posedness, stability and bifurcation. Chaos, Solitons & Fractals, page 110520. Boukhouima, A., Hattaf, K., Lotfi, E. M., Mahrouf, M., Torres, D. F., y Yousfi, N. (2020). Lyapunov functions for fractional-order systems in biology: Methods and applications. Chaos, Solitons & Fractals, 140:110224. De Vries, G., Hillen, T., Lewis, M., Müller, J., y Schönfisch, B. (2006). A course in mathematical biology: quantitative modeling with mathematical and computational methods. SIAM. Deshpande, A. y Daftardar-Gejji, V. (2016). Local stable manifold theorem for fractional systems. Nonlinear Dynamics, 83(4):2435-2452. Deshpande, A. S., Daftardar-Gejji, V., y Sukale, Y. V. (2017). On hopf bifurcation in fractional dynamical systems. Chaos, Solitons & Fractals, 98:189-198. Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media. Diethelm, K., Ford, N. J., y Freed, A. D. (2004). Detailed error analysis for a fractional Adams method. Numerical algorithms, 36(1):31-52. Doetsch, G. (2012). Introduction to the Theory and Application of the Laplace Transformation. Springer Science & Business Media. Elsadany, A. y Matouk, A. (2015). Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization. Journal of Applied Mathematics and Computing, 49(1):269-283. Gallegos, J. A. y Duarte-Mermoud, M. A. (2016). On the Lyapunov theory for fractional order systems. Applied Mathematics and Computation, 287:161-170. Garrappa, R. y Popolizio, M. (2018). Computing the matrix Mittag-Leffler function with applications to fractional calculus. Journal of Scientific Computing, 77(1):129-153. Ghosh, U., Pal, S., y Banerjee, M. (2020). Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis. Chaos, Solitons & Fractals, 143. Gorenflo, R. y Mainardi, F. (2008). Fractional calculus: integral and differential equations of fractional order. arXiv:0885.3823v1. Hartman, P. (1982). Ordinary Differential Equations: Second Edition. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). Hassouna, M., Ouhadan, A., y El Kinani, E. H. (2018). On the solution of fractional order SIS epidemic model. Chaos, Solitons & Fractals, 117:168-174. Kaczorek, T. (2011). Selected problems of fractional systems theory. Springer Science & Business Media. Krabs, W. y Pickl, S. (2010). Dynamical systems: stability, controllability and chaotic behavior. Springer Science & Business Media. Kumar, R. y Kumar, S. (2014). A new fractional modelling on susceptible-infected-recovered equations with constant vaccination rate. Nonlinear Engineering, 3(1):11-19. Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory. Springer Science & Business Media. Layek, G. (2015). An introduction to dynamical systems and chaos. Springer. Li, C. y Ma, Y. (2013). Fractional dynamical system and its linearization theorem. Nonlinear Dynamics, 71(4):621-633. Li, H.-L., Zhang, L., Hu, C., Jiang, Y.-L., y Teng, Z. (2016). Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing, 54(1-2):435-449. Li, X. y Wu, R. (2014). Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dynamics, 78(1):279-288. Lynch, S. (2004). Dynamical systems with applications using MATLAB. Springer. Ma, L. y Li, C. (2016). Center manifold of fractional dynamical system. Journal of Computational and Nonlinear Dynamics, 11(2). Machado, J. T., Kiryakova, V., y Mainardi, F. (2011). Recent history of fractional calculus. Communications in nonlinear science and numerical simulation, 16(3):1140-1153. Machado, T., Kiryakova, V., y Mainardi, F. (2010). A poster about the recent history of fractional calculus. Fractional Calculus and Applied Analysis, 13(3):329p-334p. Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications, volume 2, pages 963-968. Lille, France. Milici, C., Drăgănescu, G., y Machado, J. T. (2019). Introduction to fractional differential equations. Springer. Müller, J. y Kuttler, C. (2015). Methods and models in mathematical biology. Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Heidelberg, Germany. Perko, L. (2013). Differential equations and dynamical systems. Springer Science & Business Media. Popolizio, M. (2019). On the Matrix Mittag-Leffler Function: Theoretical Properties and Numerical Computation. Mathematics, 7(12):1140. Rivero, M., Trujillo, J. J., Vázquez, L., y Velasco, M. P. (2011). Fractional dynamics of populations. Applied Mathematics and Computation, 218(3):1089-1095. Rosenzweig, M. L. y MacArthur, R. H. (1963). Graphical representation and stability conditions of predator-prey interactions. The American Naturalist, 97(895):209-223. Sadeghi, A. y Cardoso, J. R. (2018). Some notes on properties of the matrix Mittag-Leffler function. Applied Mathematics and Computation, 338:733-738. Saeedian, M., Khalighi, M., Azimi-Tafreshi, N., Jafari, G., y Ausloos, M. (2017). Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model. Physical Review E, 95(2):022409. Sayevand, K. (2016). Fractional dynamical systems: A fresh view on the local qualitative theorems. International Journal of Nonlinear Analysis and Applications, 7(2):303-318. Stanislavsky, A. (2000). Memory effects and macroscopic manifestation of randomness. Physical Review E, 61(5):4752. Tian, J., Yu, Y., y Wang, H. (2014). Stability and bifurcation of two kinds of three-dimensional fractional Lotka-Volterra systems. Mathematical Problems in Engineering, 2014. Vargas-De-León, C. (2015). Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1-3):75-85. Wiggins, S. (2003). Introduction to applied nonlinear dynamical systems and chaos. Springer Science & Business Media. Witelski, T. y Bowen, M. (2015). Methods of mathematical modelling. Springer. Yazdani, M. y Salarieh, H. (2011). On the existence of periodic solutions in time-invariant fractional order systems. Automatica, 47(8):1834-1837.InvestigadoresLICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80309/1/license.txtcccfe52f796b7c63423298c2d3365fc6MD51ORIGINAL1045048487.2021.pdf1045048487.2021.pdf"Tesis de Maestría en Ciencias: Matemática Aplicada"application/pdf3846279https://repositorio.unal.edu.co/bitstream/unal/80309/5/1045048487.2021.pdfdc0460bbe10e553e64c69e96fde0db51MD55THUMBNAIL1045048487.2021.pdf.jpg1045048487.2021.pdf.jpgGenerated Thumbnailimage/jpeg4006https://repositorio.unal.edu.co/bitstream/unal/80309/6/1045048487.2021.pdf.jpg83633c94a3c4a6669dafdfdb6bf3b734MD56unal/80309oai:repositorio.unal.edu.co:unal/803092023-07-28 23:03:28.793Repositorio Institucional Universidad Nacional de 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