Un modelo de depredación del tipo Leslie-Gower considerando depredadores generalistas y efecto Allee en las presas

• Martínez-Jeraldo, Nicole ^[1] ; Rozas-Torres, Elizabeth ^[2] ; González-Olivares, Eduardo ^[3]
  1. [1] Universidad Católica del Maule

     Universidad Católica del Maule

     Provincia de Talca, Chile

     
  2. [2] Universidad Austral de Chile

     Universidad Austral de Chile

     Valdivia, Chile

     
  3. [3] Pontificia Universidad Católica de Valparaíso

     Pontificia Universidad Católica de Valparaíso

     Valparaíso, Chile

     

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• Localización: Selecciones Matemáticas, ISSN-e 2411-1783, Vol. 8, Nº. 1, 2021 (Ejemplar dedicado a: Enero-Julio), págs. 147-160
• Idioma: español
• DOI: 10.17268/sel.mat.2021.01.14
• Títulos paralelos:
  • A Leslie-Gower-type predation model considering generalist predators and the Allee effect on prey
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• Resumen
  • español

    La característica principal de los modelos del tipo Leslie-Gower, es que la ecuación de crecimiento de los depredadores es descrita por la función de logística. Por lo tanto, es un modelo que supone implícitamente la competencia entre los depredadores.

    En este trabajo se analiza la dinámica de un modelo derivado del modelo de Leslie-Gower,considerando dos aspectos importantes: (i) los depredadores capturan un alimento alternativo cuando la cantidad de presas es escasa y (ii) la población de presas se ve afectada por un efecto Allee.

    Considerando un sistema equivalente topológico, se establecen las principales propiedades del modelo. Se determinan las condiciones necesarias y suficientes para la existencia y la estabilidad local de los equilibrios. Además, se prueba la existencia de una órbita homoclínica y de al menos un ciclo límite.

    Cuando los depredadores son generalistas, la dinámica del modelo difiere bastante con respecto al modelo donde los depredadores son especialistas. Dinámicamente aparecen más puntos de equilibrio y una órbita homoclínica.

  • English

    The main feature of the Leslie-Gower-type predation model is that the predator’s growth function is one of logistic-type. Thus, it is a model assuming implicitly the competition among predators.

    In this work the dynamics of a modified Leslie-Gower type predator-prey model is analyzed, considering two important aspects: (i) the predators capture an alternative food when the quantity of prey is scarce and (ii) the prey population is affected by an Allee effect.

    Considering a topological equivalent system, the main properties of the system are established. Necessary and sufficient conditions for the existence and local stability of equilibria are determined, also showing the existence of a homoclinic orbit and of at least a limit cycle.

    When the predators are generalists the dynamics of the model differ enough respecting the model considering predators specialist since appearing more equilibrium points and the mentioned homoclinic orbit.

• Referencias bibliográficas
  • Aguirre P, Gonzalez-Olivares E, Sáez E. Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis:...
  • Aguirre P, González-Olivares E, Sáez E. Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal...
  • Andronov AA, Leontovich EA, Gordon I, Maier AG. Qualitative theory of second-order dynamics systems, A Halsted Press Book, John Wiley &...
  • Arancibia-Ibarra C, González-Olivares E. A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect...
  • Arancibia-Ibarra C. The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee effect, Nonlinear Analysis. 2019;...
  • Arrowsmith DK, Place CM. Dynamical System. Differential equations, maps and chaotic behaviour. Chapman and Hall; 1992.
  • Bazykin AD. Nonlinear Dynamics of interacting populations, World Scientific Publishing Co. Pte. Ltd., 1998.
  • Berec L, Angulo E, Courchamp F. Multiple Allee effects and population management. Trends in Ecology and Evolution. 2007; 22:185-191.
  • Berryman AA, Gutierrez AP, Arditi R. Credible, parsimonious and useful predator-prey models - A reply to Abrams, Gleeson and
  • Sarnelle. Ecology. 1995; 76:1980-1985.
  • Boukal DS, Berec L. Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters. J. of Theoret. Biology....
  • Boukal DS, Sabelis MW, Berec L. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population...
  • Cheng K-S. Uniqueness of a limit cycle for a predator-prey system. SIAM J. on Math. Anal. 1981; 12:541-548.
  • Chicone C. Ordinary differential equations with applications. (2nd edition), Texts in Applied Mathematics 34. New York: Springer;
  • Clark CW. Mathematical Bioeconomic: The optimal management of renewable resources, (second edition). New York: JohnWiley and Sons; 1990.
  • Coleman CS. Hilbert’s 16th. Problem: How Many Cycles? In: M. Braun, CS. Coleman and D. Drew (Ed). Differential Equations Model, Springer Verlag;...
  • Courchamp F, Clutton-Brock T, Grenfell B. Inverse density dependence and the Allee effect, Trends in Ecology and Evolution. 1999; 14(10):405-410.
  • Courchamp F, Berec L, Gascoigne J. Allee effects in Ecology and Conservation, Oxford University Press 2007.
  • Dumortier F, Llibre J, Art´es JC. Qualitative theory of planar differential systems, Springer, 2006.
  • Freedman HI. Deterministic Mathematical Model in Population Ecology, Marcel Dekker New York 1980.
  • Gaiko VA. Global Bifurcation theory and Hilbert’s sixteenth problem, Mathematics and its Applications 559, Kluwer Academic, Publishers New...
  • Gaiko VA, Vuik V. Global dynamics in the Leslie-Gower model with the Allee effect, Int. J. of Bifurcation and Chaos. 2018, Vol. (12:1850151)...
  • Goh B-S. Management and Analysis of Biological Populations. Elsevier Scientific Publishing Company; 1980.
  • González-Olivares E, González-Yañez B, Mena-Lorca J, Ramos-Jiliberto R. Modelling the Allee effect: are the different mathematical forms proposed...
  • 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, Mena-Lorca J, Rojas-Palma A, Flores JD. Erratum to Dynamical complexities in the Leslie-Gower predator-prey model as...
  • González-Olivares E, Rojas-Palma A, González-Yañez B. Multiple limit cycles in a Leslie-Gower type predator-prey model considering weak Allee...
  • González-Olivares E, Rojas-Palma A. Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist...
  • González-Yañez B, González-Olivares E, Mena-Lorca J. Multistability on a Leslie-Gower type predator-prey model with nonmonotonic functional...
  • Korobeinikov A. A Lyapunov function for Leslie-Gower predator-prey models. Applied Mathematical Letters. 2001; 14:697-699.
  • Kuznetsov YA. Elements of applied bifurcation theory. (3rd edition), Springer; 2004.
  • Leslie PH. Some further notes on the use of matrices in population mathematics, Biometrica. 1948; 35:213-245.
  • Leslie PH, Gower JC. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrica; 1960;...
  • Liermann M, Hilborn R. Depensation: evidence, models and implications. Fish and Fisheries; 2001; 2:33-58.
  • Martínez-Jeraldo N, Aguirre P. Allee effect acting on the prey species in a Leslie-Gower predation model. Nonlinear Analysis:
  • Real World Applications; 2019; 45:895-917.
  • May RM. Stability and complexity in model ecosystems (2nd edition), Princeton: Princeton University Press; 2001.
  • Mena-Lorca J, González-Olivares E, González-Yañez B. The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a...
  • Monzón P. Almost global attraction in planar systems, Systems & Control Letters, 2005; 54:753-758.
  • Perko L. Differential equations and dynamical systems (3rd edition), Springer-Verlag, 2001.
  • Rantzer A. A dual to Lyapunov’s stability theorem, Systems and Control Letters; 2001; 42(3):161-168.
  • Sáez E, Gonzalez-Olivares E. Dynamics on a Predator-prey Model, SIAM J. of Applied Math. 1999; 59(5):1867-1878.
  • Stephens PA, Sutherland WJ. Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 1999;...
  • Stephens PA, Sutherland WJ, Freckleton RP. What is the Allee effect?, Oikos. 1999; 87:185-190.
  • Turchin P. Complex population dynamics. A theoretical/empirical synthesis, Monographs in Population Biology 35. Princeton:
  • Princeton University Press, 2003.
  • van Voorn GAK, Hemerik L, Boer MP, Kooi BW. Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee. Math....
  • Volterra V. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie della R. Accademia dei Lincei, S.VI, IT....
  • Wang J, Shi J, Wei J, Predator-prey system with strong Allee effect in prey. J. of Mathematical Biology; 2011; 62:291-331.