Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System
A class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/260798 |
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| author | Shuang Guo Weihua Jiang |
| author_facet | Shuang Guo Weihua Jiang |
| author_sort | Shuang Guo |
| collection | DOAJ |
| description | A class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations. |
| format | Article |
| id | doaj-art-b658d09e28b84b7fa8fdb9b3e63490d9 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b658d09e28b84b7fa8fdb9b3e63490d92025-02-03T05:46:40ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/260798260798Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey SystemShuang Guo0Weihua Jiang1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaA class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations.http://dx.doi.org/10.1155/2012/260798 |
| spellingShingle | Shuang Guo Weihua Jiang Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System Journal of Applied Mathematics |
| title | Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System |
| title_full | Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System |
| title_fullStr | Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System |
| title_full_unstemmed | Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System |
| title_short | Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System |
| title_sort | global stability and hopf bifurcation for gause type predator prey system |
| url | http://dx.doi.org/10.1155/2012/260798 |
| work_keys_str_mv | AT shuangguo globalstabilityandhopfbifurcationforgausetypepredatorpreysystem AT weihuajiang globalstabilityandhopfbifurcationforgausetypepredatorpreysystem |