Dinámicas de un modelo de depredación considerando respuesta funcional sigmoidea y alimento alternativo para los depredadores

• Tintinago-Ruiz, Paulo C. ^[1] ; González-Olivares , Eduardo ; Rojas-Palma, Alejandro ^[2]
  1. [1] Universidad del Quindío

     Universidad del Quindío

     Colombia

     
  2. [2] Universidad Católica del Maule

     Universidad Católica del Maule

     Provincia de Talca, Chile

     
• Localización: Selecciones Matemáticas, ISSN-e 2411-1783, Vol. 9, Nº. 2, 2022 (Ejemplar dedicado a: August - December), págs. 275-286
• Idioma: español
• DOI: 10.17268/sel.mat.2022.02.05
• Títulos paralelos:
  • Dynamics of a predation model considering sigmoid functional response and alternative food for predators
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Las interrelacciones entre dos especies son un tema básico en Dinámica de Poblaciones, particularmente la interacción entre los depredadores y sus presas. Esta importancia es debido a que ella permite una mejor comprensión del comportamiento de las cadenas alimentarias complejas.

    En este trabajo extendemos el análisis de un modelo depredador-presa del tipo Leslie-Gower asumiendo que la respuesta funcional es sigmoidea o de Holling tipo III y el depredador dispone de una comida alternativa.

    Mostramos que el sistema representando el modelo tiene hasta tres puntos de equilibrio positivos y establecemos condiciones para determinar la naturaleza de cada uno de los puntos de equilibrio.

    Además, mostramos la existencia de diferentes tipos de bifurcaciones, entre ellas las de Hopf y la homoclínica. Los resultados análiticos son discutidos desde una perspectiva ecológica.

  • English

    Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs.

    In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food.

    We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point.

    In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.

• Referencias bibliográficas
  • Aguirre P, González-Olivares E, Sáez E. Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal...
  • Bazykin AD. Nonlinear Dynamics of interacting populations, World Scientific, 1998.
  • Chicone C. Ordinary differential equations with applications. 2nd edition: Texts in Applied Mathematics 34, Springer 2006.
  • Dumortier F, Llibre J, Artés JC. Qualitative theory of planar differential systems, Springer, 2006.
  • Freedman HI. Deterministic Mathematical Model in Population Ecology, Marcel Dekker (1980).
  • Getz WM. A hypothesis regarding the abruptness of density dependence and the growth rate populations, Ecology. 1996; 77:2014-2026.
  • Goh B-S. Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, 1980.
  • Gaiko VA. Global Bifurcation theory and Hilbert sixteenth problem, Mathematics and its Applications 559, Kluwer Academic Publishers, 2003.
  • González-Olivares E, Rojas-Palma A. Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and...
  • González-Olivares E, Mena-Lorca J, Rojas-Palma A, Flores JD. Dynamical complexities in the Leslie-Gower predator-prey model as consequences...
  • González-Olivares E, Tintinago-Ruiz P, Rojas-Palma A. A Leslie-Gower type predator-prey model with sigmoid funcional response, International...
  • Holling CS. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist....
  • Kuznetsov YA. Elements of applied bifurcation theory. 3rd edition: Springer, 2004.
  • Leslie PH. Some further notes on the use of matrices in population mathematics, Biometrika. 1948; 35:213-245.
  • Leslie PH, Gower JC. The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika. 1960;...
  • Ludwig D, Jones DD, Holling CS. Qualitative analysis of insect outbreak systems: The spruce budworm and forest, Journal of Animal Ecology....
  • May RM, Beddington JR, Clark CW, Holt SJ, Laws RM. Management of Multispecies Fisheiries, Science. 1979; 20:267-277.
  • May RM. Stability and Complexity in Model Ecosystems. 2nd ed: Princeton University Press, 2001.
  • Middlemas SJ, Barton TR,D. Armstrong J, Thompson PM. Functional and aggregative responses of harbour seals to changes in salmonid abundance,...
  • Perko L, Differential equations and dynamical systems. 3rd edition: Springer-Verlag. 2001.
  • Shaikh AA, Das H, Ali N. Study of a predator-prey model with modified Leslie-Gower and Holling type III schemes, Modeling Earth Systems and...
  • Schenk D, Bacher S. Functional response of a generalist insect predator to one of its prey species in the field, Journal of Animal Ecology....
  • Spencer PD, Collie JS. A simple predator-prey model of exploited marine fish populations incorporating alternative prey, ICES J. Mar. Sci.,...
  • Taylor RJ. Predation, Chapman and Hall, 1984.
  • Turchin P. Complex population dynamics. A theoretical/empirical synthesis, Mongraphs in population biology 35 (2003).
  • Volterra V. Variazioni e fluttuazioni del numero de individui in specie animali conviventi. Memorie della R. Accademia dei Lincei, S.VI, IT....