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Mathematical Biosciences and Engineering

2008, Volume 5, Issue 3: 567-583. doi: 10.3934/mbe.2008.5.567
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Stabilization due to predator interference: comparison of different analysis approaches

  • 1.
    Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam
  • 2.
    ICBM, Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg
  • 3.
    Dept. of Chem. Eng., Princeton University, Engineering Quadrangle, Princeton, NJ 08540
  • 4.
    Department of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam
  • 5.
    Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky Universität Oldenburg, PF 2503, 26111 Oldenburg
  • Received: 01 December 2007 Accepted: 29 June 2018 Published: 01 June 2008

  • MSC : Primary: 92D25; Secondary: none.

  • We study the influence of the particular form of the functional response in two-dimensional predator-prey models with respect to the stability of the nontrivial equilibrium. This equilibrium is stable between its appearance at a transcritical bifurcation and its destabilization at a Hopf bifurcation, giving rise to periodic behavior. Based on local bifurcation analysis, we introduce a classification of stabilizing effects. The classical Rosenzweig-MacArthur model can be classified as weakly stabilizing, undergoing the paradox of enrichment, while the well known Beddington-DeAngelis model can be classified as strongly stabilizing. Under certain conditions we obtain a complete stabilization, resulting in an avoidance of limit cycles. Both models, in their conventional formulation, are compared to a generalized, steady-state independent two-dimensional version of these models, based on a previously developed normalization method. We show explicitly how conventional and generalized models are related and how to interpret the results from the rather abstract stability analysis of generalized models.

    Citation: G.A.K. van Voorn, D. Stiefs, T. Gross, B. W. Kooi, Ulrike Feudel, S.A.L.M. Kooijman. Stabilization due to predator interference: comparison of different analysis approaches[J]. Mathematical Biosciences and Engineering, 2008, 5(3): 567-583. doi: 10.3934/mbe.2008.5.567

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  • We study the influence of the particular form of the functional response in two-dimensional predator-prey models with respect to the stability of the nontrivial equilibrium. This equilibrium is stable between its appearance at a transcritical bifurcation and its destabilization at a Hopf bifurcation, giving rise to periodic behavior. Based on local bifurcation analysis, we introduce a classification of stabilizing effects. The classical Rosenzweig-MacArthur model can be classified as weakly stabilizing, undergoing the paradox of enrichment, while the well known Beddington-DeAngelis model can be classified as strongly stabilizing. Under certain conditions we obtain a complete stabilization, resulting in an avoidance of limit cycles. Both models, in their conventional formulation, are compared to a generalized, steady-state independent two-dimensional version of these models, based on a previously developed normalization method. We show explicitly how conventional and generalized models are related and how to interpret the results from the rather abstract stability analysis of generalized models.


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  • © 2008 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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