Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay
This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic rev...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/107565 |
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| author | Xue Zhang Qing-ling Zhang |
| author_facet | Xue Zhang Qing-ling Zhang |
| author_sort | Xue Zhang |
| collection | DOAJ |
| description | This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results. |
| format | Article |
| id | doaj-art-e7ed64d48522439d9e9aeedebdbdb8a9 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-e7ed64d48522439d9e9aeedebdbdb8a92025-02-03T06:13:38ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/107565107565Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time DelayXue Zhang0Qing-ling Zhang1Department of Science, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang, Liaoning 110004, ChinaDepartment of Science, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang, Liaoning 110004, ChinaThis paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.http://dx.doi.org/10.1155/2014/107565 |
| spellingShingle | Xue Zhang Qing-ling Zhang Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay Journal of Applied Mathematics |
| title | Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay |
| title_full | Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay |
| title_fullStr | Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay |
| title_full_unstemmed | Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay |
| title_short | Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay |
| title_sort | bifurcation analysis and control of a differential algebraic predator prey model with allee effect and time delay |
| url | http://dx.doi.org/10.1155/2014/107565 |
| work_keys_str_mv | AT xuezhang bifurcationanalysisandcontrolofadifferentialalgebraicpredatorpreymodelwithalleeeffectandtimedelay AT qinglingzhang bifurcationanalysisandcontrolofadifferentialalgebraicpredatorpreymodelwithalleeeffectandtimedelay |