Stabilization due to predator interference: comparison of different analysis approaches
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1.
Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam
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2.
ICBM, Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg
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3.
Dept. of Chem. Eng., Princeton University, Engineering Quadrangle, Princeton, NJ 08540
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4.
Department of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam
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5.
Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky Universität Oldenburg, PF 2503, 26111 Oldenburg
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Received:
01 December 2007
Accepted:
29 June 2018
Published:
01 June 2008
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MSC :
Primary: 92D25; Secondary: none.
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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.
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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Abstract
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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