Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models

• Autores: Homero G. Díaz Marín, Osvaldo Osuna
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 23, Nº. 3, 2021, págs. 343-355
• Idioma: inglés
• DOI: 10.4067/S0719-06462021000300343
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• Resumen
  • español

    RESUMEN 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.

  • English

    ABSTRACT 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.

• Referencias bibliográficas
  • Barrios-O’Neill, D.,Dick, J. T. A.,Emmerson, M. C.,Ricciardi, A.,MacIsaac, H. J.. (2015). Predator-free space, functional responses and biological...
  • Cano, J.. (1993). An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms. Ann. Inst. Fourier (Grenoble)....
  • Demina, M. V.. (2018). Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems. Phys. Lett....
  • Demina, M. V.. (2018). Invariant algebraic curves for liénard dynamical systems revisited. Appl. Math. Lett.. 84. 42
  • Ferragut, A.,Gasull, A.. (2014). Non-algebraic oscillations for predator-prey models. Publ. Mat.. 58. 195-207
  • Giné, J.,Grau, M.. (2006). Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations. Nonlinearity....
  • Giné, J.,Llibre, J.. (2020). Strongly formal Weierstrass non-integrability for polynomial differential systems in C 2. Electron. J. Qual....
  • Giné, J.,Llibre, J.. (2020). Formal Weierstrass nonintegrability criterion for some classes of polynomial differential systems in C 2. Internat....
  • Hayashi, M.. (1996). On polynomial Li´enard systems which have invariant algebraic curves. Funkcial. Ekvac. 39. 403
  • Hille, E.. (1976). Ordinary Differential Equations in the Complex Domain. Dover Publications, Inc.. Mineola, NY.
  • Ince, E. L.. (1944). Ordinary Differential Equations. Dover Publications. New York.
  • Odani, K.. (1995). The limit cycle of the van der Pol equation is not algebraic. J. Differential Equations. 115. 146
  • Real, L. A.. (1977). The kinetics of functional response. The American Naturalist. 111. 289-300
  • Rosenbaum, B.,Rall, B. C.. (2018). Fitting functional responses: Direct parameter estimation by simulating differential equations. Methods...
  • Ryabchenko, V. A.,Fasham, M. J. R.,Kagan, B. A.,Popova, E. E.. (1997). What causes short-term oscillations in ecosystem models of the ocean...
  • Sugie, J.. (2000). Uniqueness of limit cycles in a predator-prey system with Holling-type functional response. Quart. Appl. Math.. 58. 577
  • Sugie, J.,Kohno, R.,Miyazaki, R.. (1997). On a predator-prey system of Holling type. Proc. Amer. Math. Soc.. 125. 2041
  • Upadhyay, R. K.,Iyengar, S. R. K.. (2013). Introduction to Mathematical Modeling and Chaotic Dynamics. CRC Press.