Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay
We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, b...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/454097 |
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| _version_ | 1850231933178478592 |
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| author | Ruiqing Shi Junmei Qi Sanyi Tang |
| author_facet | Ruiqing Shi Junmei Qi Sanyi Tang |
| author_sort | Ruiqing Shi |
| collection | DOAJ |
| description | We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay τ passes through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented. |
| format | Article |
| id | doaj-art-08ca0a39bc26478ebb50d0bd0e4bd118 |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-08ca0a39bc26478ebb50d0bd0e4bd1182025-08-20T02:03:20ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/454097454097Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed DelayRuiqing Shi0Junmei Qi1Sanyi Tang2College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, ChinaSchool of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, ChinaCollege of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, ChinaWe propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay τ passes through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.http://dx.doi.org/10.1155/2013/454097 |
| spellingShingle | Ruiqing Shi Junmei Qi Sanyi Tang Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay Abstract and Applied Analysis |
| title | Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay |
| title_full | Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay |
| title_fullStr | Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay |
| title_full_unstemmed | Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay |
| title_short | Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay |
| title_sort | stability and bifurcation analysis for a predator prey model with discrete and distributed delay |
| url | http://dx.doi.org/10.1155/2013/454097 |
| work_keys_str_mv | AT ruiqingshi stabilityandbifurcationanalysisforapredatorpreymodelwithdiscreteanddistributeddelay AT junmeiqi stabilityandbifurcationanalysisforapredatorpreymodelwithdiscreteanddistributeddelay AT sanyitang stabilityandbifurcationanalysisforapredatorpreymodelwithdiscreteanddistributeddelay |