Extinción y sobrevivencia en sistemas competitivos de Lotka-Volterra con coeficientes constantes y retardos infinitos

• Montes de Oca, Francisco ^[2] ; Pérez, Liliana Rebeca ^[1]
  1. [1] Escuela Superior Politecnica del Litoral

     Escuela Superior Politecnica del Litoral

     Guayaquil, Ecuador

     
  2. [2] Universidad Centro Occidental Lisandro Alvarado
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 54, Nº. 1, 2020, págs. 75-91
• Idioma: español
• DOI: 10.15446/recolma.v54n1.89791
• Títulos paralelos:
  • Extinction and survival in competitive Lotka-Volterra systems with constant coefficients and infinite delays
• Enlaces
  • Texto completo
• Resumen
  • español

    Se estudian las propiedades cualitativas de un sistema competitivo no a utonomo de Lotka-Volterra con retardo infinito.Mediante el uso de un resultado de la teoría de matrices y del lema de fluctuaciones, se establecen una serie de condiciones algebraicas, fácilmente verificables, sobre los coeficientes y los núcleos, que son suficientes para garantizar la extinción y la sobrevivencia de un determinado número de especies. La parte sobreviviente se estabiliza alrededor de un punto de equilibrio globalmente estable de un subsistema del sistema en estudio. Estas condiciones también garantizan el comportamiento asintótico del sistema.

  • English

    The qualitative properties of a nonautonomous competitive Lotka-Volterra system with infinite delays are studied.By using a result of matrix theory and the fluctuation lemma, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernel, which are sufficient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.

• Referencias bibliográficas
  • S. Ahmad, On the nonautonomous Volterra{Lotka competition equations, Proc. Amer. Math. Soc. 117 (1993), 199-204.
  • S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems, Proc. Amer. Math.Soc 127 (1999), 2905-2910.
  • A. Battauz and F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems, J. Math. Anal. Appl 219 (1998), 179-199.
  • M. Brelot, Sur le probleme biologique hereditaire de deux especes devorante et devoree, Ann. Mat. Pura Appl.Ser, 1931.
  • R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with difusion, Houston J. Math 13 (1987),...
  • F. Chen, Z. Li, and Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Analysis:...
  • F. Chen, C. Shi, and Zhong Li, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,...
  • C. Cosner and A. C. Lazer, Stable Coexistence States in the Volterra-Lotka Competition Model with Difusion, SIAM J. Appl. Math 44 (1984),...
  • F. Montes de Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay, Nonlinear Analysis: Series...
  • F. Montes de Oca and L. Pérez, Balancing Survival and Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay,...
  • F. Montes de Oca and M. Vivas, Extinction in a two dimensional lotka-volterra systems with infinite delay, Nonlinear Analysis: Real world...
  • L. Dung and H. L. Smith, A Parabolic System Modeling Microbial Competition in an Unmixed Bio-reactor, Journal of Diferential Equations 130...
  • L. Dung and H. L. Smith, Steady states of models of microbial growth and competition with chemotaxis, J Math. Anal. Appl 229 (1999), 295-318.
  • K. Gopalsamy, Stability and Oscillations in Delay Diferential Equations of population dynamics, Mathematics and Its Applications, Kluwer Academic...
  • J. Hale and S. Verduyn Lunel, Introduction to functional diferential equations, Applied Mathematical Sciences, 1993.
  • X. He, Almost periodic solutions of a competition system with dominated infinite delays, Tohoku Math. J 50 (1998), 71-89.
  • W. Hirsch and J. Hanisch, Differential equation models of some parasitic infection-methods for the study of asymptotic behavior, Comm. Pure...
  • Z. Hou, Permanence, global attraction and stability, De Gruyter Ser.in Math. and Life Sc, 2013.
  • H. Hu, Z. Teng, and S. Gao, Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls, Nonlinear...
  • H. Hu, Z. Teng, and H. Jiang, On the permanence in non-autonomous lotka-volterra competitive system with pure-delays and feedback controls,...
  • A. C. Lazer and S. Ahmad, Average growth extinction in a competitive Lotka-Volterra system, Nonlinear Analysis 62 (2005), 545-557.
  • Z. Teng, On the nonautonomous Lotka-Volerra N-species competing systems, Appl. Math.Comp 114 (2000), 175-185.
  • A. Tineo, Asymptotic behavior of positive solutions of the nonautonomous Lotka-Volterra competetions equations, Differential and Integral...
  • A. Tineo, Necessary and sufficient conditions for extinction of one species, Advanced Nonlinear Studies 5 (2005), 57-71.
  • V. Volterra, Lecon sur la theorie mathematique de la lutte por la vie, Gauthier Villars, Paris, 1931.
  • F. Zanolin, Permanence and positive periodic solutions for kolmogorov competing species systems, Results Math 21 (1992), 224-250.
  • M. L. Zeeman, Extinction in competiive Lotka-Volterra Systems, Proc. Amer.Math.Soc 123 (1995), 87-96.
  • J. Zhao, L. Fu, and J. Ruan, Extinction in a nonautonomous competitive Lotka-Volterra system, Appl. Math. Letters 22 (2009), 766-770.
  • J. Zhao and J. Tiang, Average conditions for permanence and extinction in nonautonomous Lotka- Voltera systems, JMAA 299 (2004), 663-675.