Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III
| Autor: | Remy Magloire Etoua, Christiane Rousseau |
|---|---|
| Rok vydání: | 2010 |
| Předmět: |
Pure mathematics
Applied Mathematics 010102 general mathematics Mathematical analysis Heteroclinic bifurcation Saddle-node bifurcation Nilpotent saddle bifurcation Bifurcation diagram 01 natural sciences Nonlinear Sciences::Chaotic Dynamics 010101 applied mathematics Pitchfork bifurcation Transcritical bifurcation Bifurcation theory Homoclinic bifurcation Bogdanov–Takens bifurcation Hopf bifurcation 0101 mathematics Generalized Gause model with prey harvesting Nonlinear Sciences::Pattern Formation and Solitons Analysis Generalized Holling response function of type III Mathematics |
| Zdroj: | Journal of Differential Equations. 249:2316-2356 |
| ISSN: | 0022-0396 |
| DOI: | 10.1016/j.jde.2010.06.021 |
| Popis: | In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: p ( x ) = m x 2 a x 2 + b x + 1 . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Lienard system, and for b = 0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b = 0 , the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species. |
| Databáze: | OpenAIRE |
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