Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations
A general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtain...
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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/10.1155/2012/521729 |
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| author | Xianning Liu Lansun Chen |
| author_facet | Xianning Liu Lansun Chen |
| author_sort | Xianning Liu |
| collection | DOAJ |
| description | A general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtained via the comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence of a positive periodic solution is also studied by the bifurcation theory. Application is given to a Lotka-Volterra predator-prey system with periodic impulsive immigration of the predator. It is shown that the results are quite different from the corresponding system without impulsive immigration, where extinction of the prey can never be achieved. The prey will be extinct or permanent independent of whether the system without impulsive effect immigration is permanent or not. The model and its results suggest an approach of pest control which proves more effective than the classical one. |
| format | Article |
| id | doaj-art-50ab87e456584210b70126cd5c8e74ac |
| 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-50ab87e456584210b70126cd5c8e74ac2025-02-03T01:28:48ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/521729521729Extinction and Permanence of a General Predator-Prey System with Impulsive PerturbationsXianning Liu0Lansun Chen1Key Laboratory of Eco-Environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, ChinaInstitute of Mathematics, Academy of Mathematics and System Sciences, Beijing 100080, ChinaA general predator-prey system is studied in a scheme where there is periodic impulsive perturbations. This scheme has the potential to protect the predator from extinction but under some conditions may also serve to lead to extinction of the prey. Conditions for extinction and permanence are obtained via the comparison methods involving monotone theory of impulsive systems and multiple Liapunov functions, which establish explicit bounds on solutions. The existence of a positive periodic solution is also studied by the bifurcation theory. Application is given to a Lotka-Volterra predator-prey system with periodic impulsive immigration of the predator. It is shown that the results are quite different from the corresponding system without impulsive immigration, where extinction of the prey can never be achieved. The prey will be extinct or permanent independent of whether the system without impulsive effect immigration is permanent or not. The model and its results suggest an approach of pest control which proves more effective than the classical one.http://dx.doi.org/10.1155/2012/521729 |
| spellingShingle | Xianning Liu Lansun Chen Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations Journal of Applied Mathematics |
| title | Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations |
| title_full | Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations |
| title_fullStr | Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations |
| title_full_unstemmed | Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations |
| title_short | Extinction and Permanence of a General Predator-Prey System with Impulsive Perturbations |
| title_sort | extinction and permanence of a general predator prey system with impulsive perturbations |
| url | http://dx.doi.org/10.1155/2012/521729 |
| work_keys_str_mv | AT xianningliu extinctionandpermanenceofageneralpredatorpreysystemwithimpulsiveperturbations AT lansunchen extinctionandpermanenceofageneralpredatorpreysystemwithimpulsiveperturbations |