Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy
In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of...
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AIMS Press
2015-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.1065 |
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| author | Qizhen Xiao Binxiang Dai |
| author_facet | Qizhen Xiao Binxiang Dai |
| author_sort | Qizhen Xiao |
| collection | DOAJ |
| description | In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems. |
| format | Article |
| id | doaj-art-195f6f44ffd84189bc0937fd0e38ce01 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2015-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-195f6f44ffd84189bc0937fd0e38ce012025-01-24T02:33:19ZengAIMS PressMathematical Biosciences and Engineering1551-00182015-05-011251065108110.3934/mbe.2015.12.1065Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategyQizhen Xiao0Binxiang Dai1School of Mathematics and Statistics, Central South University, Changsha, 410083School of Mathematics and Statistics, Central South University, Changsha, 410083In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.1065allee effectheteroclinic bifurcation.predator-prey modelstate feedback controlpositive periodic solution |
| spellingShingle | Qizhen Xiao Binxiang Dai Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy Mathematical Biosciences and Engineering allee effect heteroclinic bifurcation. predator-prey model state feedback control positive periodic solution |
| title | Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy |
| title_full | Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy |
| title_fullStr | Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy |
| title_full_unstemmed | Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy |
| title_short | Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy |
| title_sort | heteroclinic bifurcation for a general predator prey model with allee effect and state feedback impulsive control strategy |
| topic | allee effect heteroclinic bifurcation. predator-prey model state feedback control positive periodic solution |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.1065 |
| work_keys_str_mv | AT qizhenxiao heteroclinicbifurcationforageneralpredatorpreymodelwithalleeeffectandstatefeedbackimpulsivecontrolstrategy AT binxiangdai heteroclinicbifurcationforageneralpredatorpreymodelwithalleeeffectandstatefeedbackimpulsivecontrolstrategy |