Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control

In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré m...

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Main Authors: Jin Yang, Guangyao Tang, Sanyi Tang
Format: Article
Language:English
Published: Wiley 2018-01-01
Series:Discrete Dynamics in Nature and Society
Online Access:http://dx.doi.org/10.1155/2018/3467405
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author Jin Yang
Guangyao Tang
Sanyi Tang
author_facet Jin Yang
Guangyao Tang
Sanyi Tang
author_sort Jin Yang
collection DOAJ
description In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-k(k≥2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.
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issn 1026-0226
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publishDate 2018-01-01
publisher Wiley
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series Discrete Dynamics in Nature and Society
spelling doaj-art-6861c4116cf1471d849edee12bad02a62025-02-03T05:50:48ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/34674053467405Holling-Tanner Predator-Prey Model with State-Dependent Feedback ControlJin Yang0Guangyao Tang1Sanyi Tang2Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, ChinaKey Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, Hubei 445000, ChinaCollege of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, ChinaIn this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-k(k≥2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.http://dx.doi.org/10.1155/2018/3467405
spellingShingle Jin Yang
Guangyao Tang
Sanyi Tang
Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
Discrete Dynamics in Nature and Society
title Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
title_full Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
title_fullStr Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
title_full_unstemmed Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
title_short Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control
title_sort holling tanner predator prey model with state dependent feedback control
url http://dx.doi.org/10.1155/2018/3467405
work_keys_str_mv AT jinyang hollingtannerpredatorpreymodelwithstatedependentfeedbackcontrol
AT guangyaotang hollingtannerpredatorpreymodelwithstatedependentfeedbackcontrol
AT sanyitang hollingtannerpredatorpreymodelwithstatedependentfeedbackcontrol