Stabilization due to predator interference: comparison of different analysis approaches
We study the influence of the particular form of the functionalresponse in two-dimensional predator-prey models with respect to the stabilityof the nontrivial equilibrium. This equilibrium is stable between its appearanceat a transcritical bifurcation and its destabilization at a Hopf bifurcation,gi...
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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2008-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2008.5.567 |
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| author | G.A.K. van Voorn D. Stiefs T. Gross B. W. Kooi Ulrike Feudel S.A.L.M. Kooijman |
| author_facet | G.A.K. van Voorn D. Stiefs T. Gross B. W. Kooi Ulrike Feudel S.A.L.M. Kooijman |
| author_sort | G.A.K. van Voorn |
| collection | DOAJ |
| description | We study the influence of the particular form of the functionalresponse in two-dimensional predator-prey models with respect to the stabilityof the nontrivial equilibrium. This equilibrium is stable between its appearanceat a transcritical bifurcation and its destabilization at a Hopf bifurcation,giving rise to periodic behavior. Based on local bifurcation analysis, we introducea classification of stabilizing effects. The classical Rosenzweig-MacArthurmodel can be classified as weakly stabilizing, undergoing the paradox of enrichment,while the well known Beddington-DeAngelis model can be classifiedas strongly stabilizing. Under certain conditions we obtain a completestabilization, resulting in an avoidance of limit cycles. Both models, in theirconventional formulation, are compared to a generalized, steady-state independenttwo-dimensional version of these models, based on a previously developednormalization method. We show explicitly how conventional and generalizedmodels are related and how to interpret the results from the rather abstractstability analysis of generalized models. |
| format | Article |
| id | doaj-art-b916ad7e7b7345fb93b51a5e0009bd97 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2008-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-b916ad7e7b7345fb93b51a5e0009bd972025-01-24T01:58:23ZengAIMS PressMathematical Biosciences and Engineering1551-00182008-05-015356758310.3934/mbe.2008.5.567Stabilization due to predator interference: comparison of different analysis approachesG.A.K. van Voorn0D. Stiefs1T. Gross2B. W. Kooi3Ulrike Feudel4S.A.L.M. Kooijman5Dept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamDept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamDept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamDept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamDept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamDept. Theor. Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HV AmsterdamWe study the influence of the particular form of the functionalresponse in two-dimensional predator-prey models with respect to the stabilityof the nontrivial equilibrium. This equilibrium is stable between its appearanceat a transcritical bifurcation and its destabilization at a Hopf bifurcation,giving rise to periodic behavior. Based on local bifurcation analysis, we introducea classification of stabilizing effects. The classical Rosenzweig-MacArthurmodel can be classified as weakly stabilizing, undergoing the paradox of enrichment,while the well known Beddington-DeAngelis model can be classifiedas strongly stabilizing. Under certain conditions we obtain a completestabilization, resulting in an avoidance of limit cycles. Both models, in theirconventional formulation, are compared to a generalized, steady-state independenttwo-dimensional version of these models, based on a previously developednormalization method. We show explicitly how conventional and generalizedmodels are related and how to interpret the results from the rather abstractstability analysis of generalized models.https://www.aimspress.com/article/doi/10.3934/mbe.2008.5.567stability.bifurcation analysisgeneralized modelfunctional response |
| spellingShingle | 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 and Engineering stability. bifurcation analysis generalized model functional response |
| title | Stabilization due to predator interference: comparison of different analysis approaches |
| title_full | Stabilization due to predator interference: comparison of different analysis approaches |
| title_fullStr | Stabilization due to predator interference: comparison of different analysis approaches |
| title_full_unstemmed | Stabilization due to predator interference: comparison of different analysis approaches |
| title_short | Stabilization due to predator interference: comparison of different analysis approaches |
| title_sort | stabilization due to predator interference comparison of different analysis approaches |
| topic | stability. bifurcation analysis generalized model functional response |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2008.5.567 |
| work_keys_str_mv | AT gakvanvoorn stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches AT dstiefs stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches AT tgross stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches AT bwkooi stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches AT ulrikefeudel stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches AT salmkooijman stabilizationduetopredatorinterferencecomparisonofdifferentanalysisapproaches |