Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response
A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone...
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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/10.1155/2013/152459 |
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| _version_ | 1832553176647073792 |
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| author | Jiao Jiang Yongli Song |
| author_facet | Jiao Jiang Yongli Song |
| author_sort | Jiao Jiang |
| collection | DOAJ |
| description | A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods. |
| format | Article |
| id | doaj-art-29109440d03e4d6f8742c2b48f1e03a1 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-29109440d03e4d6f8742c2b48f1e03a12025-02-03T05:54:38ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/152459152459Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional ResponseJiao Jiang0Yongli Song1Department of Mathematics, Shanghai Maritime University, Shanghai 201306, ChinaDepartment of Mathematics, Tongji University, Shanghai 200092, ChinaA delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.http://dx.doi.org/10.1155/2013/152459 |
| spellingShingle | Jiao Jiang Yongli Song Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response Abstract and Applied Analysis |
| title | Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response |
| title_full | Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response |
| title_fullStr | Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response |
| title_full_unstemmed | Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response |
| title_short | Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response |
| title_sort | stability and bifurcation analysis of a delayed leslie gower predator prey system with nonmonotonic functional response |
| url | http://dx.doi.org/10.1155/2013/152459 |
| work_keys_str_mv | AT jiaojiang stabilityandbifurcationanalysisofadelayedlesliegowerpredatorpreysystemwithnonmonotonicfunctionalresponse AT yonglisong stabilityandbifurcationanalysisofadelayedlesliegowerpredatorpreysystemwithnonmonotonicfunctionalresponse |