American Institute of Mathematical Sciences

Advanced
2008, 5(3): 567-583. doi: 10.3934/mbe.2008.5.567

Stabilization due to predator interference: comparison of different analysis approaches

G.A.K. van Voorn 1, , D. Stiefs 2, , T. Gross 3, , B. W. Kooi 4, , Ulrike Feudel 5, and  S.A.L.M. Kooijman 1,

1. 

Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, Netherlands, Netherlands
2. 

ICBM, Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg, Germany
3. 

Dept. of Chem. Eng., Princeton University, Engineering Quadrangle, Princeton, NJ 08540, United States
4. 

Department of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, Netherlands
5. 

Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky Universität Oldenburg, PF 2503, 26111 Oldenburg

Received  December 2007 Revised  March 2008 Published  June 2008

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.
Keywords: stability., bifurcation analysis, generalized model, functional response.
Mathematics Subject Classification: Primary: 92D25; Secondary: non.
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. Mathematical Biosciences & Engineering, 2008, 5 (3) : 567-583. doi: 10.3934/mbe.2008.5.567
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