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Mathematical Biosciences and Engineering (MBE)
 

Stabilization due to predator interference: comparison of different analysis approaches

Pages: 567 - 583, Volume 5, Issue 3, July 2008      doi:10.3934/mbe.2008.5.567

 
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G.A.K. van Voorn - Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, Netherlands (email)
D. Stiefs - ICBM, Carl von Ossietzky Universit├Ąt, PF 2503, 26111 Oldenburg, Germany (email)
T. Gross - Dept. of Chem. Eng., Princeton University, Engineering Quadrangle, Princeton, NJ 08540, United States (email)
B. W. Kooi - Department of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, Netherlands (email)
Ulrike Feudel - Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky Universit├Ąt Oldenburg, PF 2503, 26111 Oldenburg, Germany (email)
S.A.L.M. Kooijman - Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam, Netherlands (email)

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.

Keywords:  bifurcation analysis, functional response, generalized model, stability.
Mathematics Subject Classification:  Primary: 92D25; Secondary: none.

Received: December 2007;      Accepted: March 2008;      Available Online: June 2008.