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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.29 n.1 San José Jan./Jun. 2022 


Some applications of periodic orbits for competitive systems

Algunas aplicaciones de órbitas periódicas a sistemas competitivos

Homero G. Díaz-Marín1 

Osvaldo Osuna2 

1Universidad Michoacana, Facultad de Ciencias Físico-Matemáticas, Morelia, México;

2Universidad Michoacana, Instituto de Física y Matemáticas, Morelia, México;


We prove existence of periodic orbits for non-autonomous two dimensional competitive dynamical systems with periodic time dependence. The proof is an adaptation of a similar assertion stated for cooperative systems in [ 6 ]. We also give two main applications: one model for cancer cell populations under periodic chemotherapy as treated in [ 4 ] and [ 3 ] for the cooperative case, and another model for mosquito population replacement dynamics interacting with control sterile mosquitoes with periodic release [ 1 ] , for the competitive case.

Keywords: competitive systems; periodic orbit; angiogenesis; cancer treatment modeling; population replacement modeling; Aedes aegypti; Wolbachia.


Probamos la existencia de órbitas periódicas para sistemas dinámicos competitivos en dos dimensiones no autónomos con dependencia periódica respecto al tiempo. La prueba es una adaptación de un resultado similar para sistemas cooperativos en [ 6 ] . También damos dos aplicaciones: un modelo de población de celulas cancerosas sometidas a un tratamiento periódico de quimioterapia como se describe en [ 4 ] y [ 3 ] para el caso cooperativo, y otro modelo de poblaciones de mosquitos interactuando con mosquitos de control estériles liberados periódicamente [ 1 ] , para el caso competitivo.

Palabras clave: sistemas competitivos; órbita periódica; angiogénesis; modelado de tratamientos de cáncer; modelo de reemplazo de población; Aedes aegypti; Wolbachia.

Mathematics Subject Classification: 34C25, 37C60, 37C10, 92D25, 92C50.

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We thank the anonymous referees for some important suggestions.


L, Almeida; Y, Privat; M, Strugarek; N, Vauchelet. Optimal releases for population replacement strategies: Application to Wolbachia, SIAM Journal on Mathematical Analysis 51(2019), no. 4, 3170-3194. Doi: 10.1137/18M1189841 [ Links ]

H, Díaz-Marín; C,O, Osuna Castro. Periodic solutions for a model of cell population subjected to general periodic radiation, Revista Integración 38(2020), no. 2, 81-91. Doi: 10.18273/revint.v38n2-20200001 [ Links ]

A, d'Onofrio; A, Gandolfi. Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences 191(2004), no. 2, 159-184. Doi: 10.1016/j.mbs.2004.06.003 [ Links ]

P, Hahnfeldt; D, Panigrahy; J, Folkman; L, Hlatky. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer research 59(1999), no. 19, 4770-4775. [ Links ]

M,W, Hirsch. Systems of differential equations that are competitive or cooperative. V. Convergence in 3-dimensional systems, Journal of differential equations 80(1989), no. 1, 94-106. Doi: 10.1016/0022-0396(89)90097-1 [ Links ]

P, Korman. A periodic model for the dynamics of cell volume, arXiv, 2016. In: 1605.01324, 2016 math.DS [ Links ]

H,L, Smith. Dynamics of competition, in: V, Capasso (Eds.) Mathematics Inspired by Biology, Lecture Notes in Mathematics 1714, Springer, Berlin, 1999, pp. 191-240. Doi: 10.1007/BFb0092378 [ Links ]

Received: October 21, 2020; Revised: April 04, 2021; Accepted: September 16, 2021

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