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Una clase de modelo de depredación del tipo Leslie-Gower con respuesta funcional racional no monotónica y alimento alternativo para los depredadores

  • Tintinago-Ruiz, Paulo C. [1] ; Gallego-Berrío, Lina M. [1] ; González-Olivares, Eduardo [2]
    1. [1] Universidad del Quindío

      Universidad del Quindío

      Colombia

    2. [2] Pontificia Universidad Católica de Valparaíso

      Pontificia Universidad Católica de Valparaíso

      Valparaíso, Chile

  • Localización: Selecciones Matemáticas, ISSN-e 2411-1783, Vol. 6, Nº. 2, 2019 (Ejemplar dedicado a: Agosto-Diciembre), págs. 204-216
  • Idioma: español
  • DOI: 10.17268/sel.mat.2019.02.07
  • Títulos paralelos:
    • A class of Leslie-Gower type predator model with a non-monotonic rational functional response and alternative food for the predators
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  • Resumen
    • español

      Las interacciones entre dos especies son básicas en el estudio de cadenas alimentarias complejas, en particular la relación entre los depredadores y sus presas.El análisis de modelos simples, descritos por sistemas de tiempo continuo, en los cuales se incorporan algunos fenómenos ecológicos dando luces sobre esta interesante interrelación.En este trabajo, se analiza un modelo de depredador-presa del tipo Leslie-Gower, descrito por un sistema de ecuaciones diferenciales ordinarias (EDO) considerando dos aspectos: la presa se defiende de la depredación, formando grupo de defensa, y los depredadores disponen un alimento alternativo, cuando su alimento favorito escasea. Por lo tanto, se asume una respuesta funcional racional de Holling tipo IV y una modificación de la capacidad de carga de los depredadores para describir estos fenómenos.Determinamos las condiciones en el espacio de parámetros para la existencia de los equilibrios y la naturaleza de cada uno de ellos.Concluimos que el parámetro que indica la existencia de alimento alternativo para depredadores tiene una gran importancia en la dinámica del modelo, porque aparecen nuevos puntos de equilibrio y curvas de separatriz en el plano de fase.Por simulaciones numéricas comprobamos que existe un subconjunto de parámetros para los cuales hay un único punto de equilibrio positivo en el plano de fase, el cual es estable y está rodeado por dos ciclos límites originados por bifurcación de Hopf, el interior inestable y el exterior estable.

    • English

      The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.We determine conditions on the parameter space for the existence of the equilibria and their nature.Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilibrium point, which is stable and surrounded by two limit cycle, the innermost unstable and the outermost sable.We conclude that the parameter indicating the existence of alternative food for predator has a great importance on the dynamic of model, because appear new equilibrum points and separatrix curves in the phase plane.Some simulations are given to reinforce our findings the ecological interpretations of resultas are given.

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