Servicios Personalizados
Revista
Articulo
nueva página del texto (beta)
Inglés (pdf)
Articulo en XML
Traducción automáticaIndicadores
-
Citado por SciELO -
Accesos
Links relacionados
-
Citado por Google -
Similares en
SciELO -
Similares en Google
Compartir
Permalink
Cubo (Temuco)
versión On-line ISSN 0719-0646
Cubo vol.23 no.3 Temuco dic. 2021
http://dx.doi.org/10.4067/S0719-06462021000300343
Articles
Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models
1 Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana, Edif. Alfa, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México. homero.diaz@umich.mx
2 Instituto de Física y Matemáticas, Universidad Michoacana, Edif. C-3, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México. osvaldo.osuna@umich.mx
We prove that for certain polynomial differential equations in the plane arising from predator-prey type III models with generalized rational functional response, any algebraic solution should be a rational function. As a consequence, limit cycles, which are unique for these dynamical systems, are necessarily trascendental ovals. We exemplify these findings by showing a numerical simulation within a system arising from zooplankton-phytoplankton dynamics.
Keywords and Phrases: Predator-prey models; functional-response: Puiseux series; Newton polygon, limit cycles; invariant algebraic curve
Probamos que para ciertas ecuaciones diferenciales polinomiales en el plano que aparecen a partir de modelos predador-presa de tipo III con respuesta funcional racional generalizada, toda solución algebraica debe ser una función racional. Como consecuencia, los ciclos límite, que son ´únicos para estos sistemas dinámicos, son necesariamente óvalos trascendentes. Ejemplificamos estos resultados mostrando una simulación numérica para un sistema que aparece en la dinámica de zooplancton-fitoplancton.
References
[1] D. Barrios-O’Neill, J. T. A. Dick, M. C. Emmerson, A. Ricciardi and H. J. MacIsaac, “Predator-free space, functional responses and biological invasions”, Functional Ecology, vol. 29, no. 3, pp. 377-384, 2015. [ Links ]
[2] J. Cano, “An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms”, Ann. Inst. Fourier (Grenoble), vol. 43, no. 1, pp. 125-142, 1993. [ Links ]
[3] M. V. Demina, “Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems”. Phys. Lett. A, vol. 382, no. 20, pp. 1353-1360, 2018. [ Links ]
[4] M. V. Demina, “Invariant algebraic curves for liénard dynamical systems revisited”, Appl. Math. Lett., vol. 84, pp. 42-48, 2018. [ Links ]
[5] A. Ferragut and A. Gasull. “Non-algebraic oscillations for predator-prey models”, Publ. Mat., vol. 58, suppl., pp. 195-207, 2014. [ Links ]
[6] J. Giné and M. Grau, “Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations”, Nonlinearity, vol. 19, no. 8, pp. 1939-1950, 2006. [ Links ]
[7] J. Giné and J. Llibre, “Strongly formal Weierstrass non-integrability for polynomial differential systems in C 2”, Electron. J. Qual. Theory Differ. Equ., no. 1, pp. 1-16, 2020. [ Links ]
[8] J. Giné and J. Llibre, “Formal Weierstrass nonintegrability criterion for some classes of polynomial differential systems in C 2”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30, no. 4, 7 pages, 2020. [ Links ]
[9] M. Hayashi, “On polynomial Li´enard systems which have invariant algebraic curves”, Funkcial. Ekvac., vol. 39, no. 3, pp. 403-408, 1996. [ Links ]
[10] E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications, Inc., Mineola, NY, 1976. [ Links ]
[11] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. [ Links ]
[12] K. Odani, “The limit cycle of the van der Pol equation is not algebraic”, J. Differential Equations, vol. 115, no. 1, pp. 146-152, 1995. [ Links ]
[13] L. A. Real, “The kinetics of functional response”, The American Naturalist, vol. 111, no. 978, pp. 289-300, 1977. [ Links ]
[14] B. Rosenbaum and B. C. Rall, “Fitting functional responses: Direct parameter estimation by simulating differential equations”, Methods in Ecology and Evolution, vol. 9, no. 10, pp. 2076-2090, 2018. [ Links ]
[15] V. A. Ryabchenko, M. J. R. Fasham, B. A. Kagan and E. E. Popova, “What causes short-term oscillations in ecosystem models of the ocean mixed layer?”, Journal of Marine Systems, vol. 13, no. 1, pp. 33-50, 1997. [ Links ]
[16] J. Sugie, “Uniqueness of limit cycles in a predator-prey system with Holling-type functional response”, Quart. Appl. Math., vol. 58, no. 3, pp. 577-590, 2000. [ Links ]
[17] J. Sugie, R. Kohno, and R. Miyazaki, “On a predator-prey system of Holling type”, Proc. Amer. Math. Soc., vol. 125, no. 7, pp. 2041-2050, 1997. [ Links ]
[18] R. K. Upadhyay and S. R. K. Iyengar, Introduction to Mathematical Modeling and Chaotic Dynamics, CRC Press, 2013 [ Links ]
Accepted: July 07, 2021; Received: January 08, 2021
This is an open-access article distributed under the terms of the Creative Commons Attribution License
