Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect

In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly,...
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Autor*in:	Wu, Daiyong [verfasserIn] Zhao, Hongyong

Format:	Artikel
Sprache:	Englisch

Erschienen:	2017

Rechteinformationen:	Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017

Schlagwörter:	Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents

Übergeordnetes Werk:	Enthalten in: Journal of difference equations and applications - Amsterdam : Gordon and Breach, 1995, 23(2017), 11, Seite 1765
Übergeordnetes Werk:	volume:23 ; year:2017 ; number:11 ; pages:1765

Links:	Volltext Link aufrufen Link aufrufen

DOI / URN:	10.1080/10236198.2017.1367389

Katalog-ID:	OLC1999617029

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520			\|a In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.
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650		4	\|a Bifurcation theory
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allfields	10.1080/10236198.2017.1367389 doi PQ20171228 (DE-627)OLC1999617029 (DE-599)GBVOLC1999617029 (PRQ)i1134-82ee90f713db798108a2ea3796d7133d89bc6124a51b4819a4f87f807ca7cda30 (KEY)0276061420170000023001101765complexdynamicsofadiscretepredatorpreymodelwiththe DE-627 ger DE-627 rakwb eng 510 ZDB Wu, Daiyong verfasserin aut Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour. Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017 Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents Zhao, Hongyong oth Enthalten in Journal of difference equations and applications Amsterdam : Gordon and Breach, 1995 23(2017), 11, Seite 1765 (DE-627)192176633 (DE-600)1306548-8 (DE-576)9192176631 1023-6198 nnns volume:23 year:2017 number:11 pages:1765 http://dx.doi.org/10.1080/10236198.2017.1367389 Volltext http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 23 2017 11 1765
spelling	10.1080/10236198.2017.1367389 doi PQ20171228 (DE-627)OLC1999617029 (DE-599)GBVOLC1999617029 (PRQ)i1134-82ee90f713db798108a2ea3796d7133d89bc6124a51b4819a4f87f807ca7cda30 (KEY)0276061420170000023001101765complexdynamicsofadiscretepredatorpreymodelwiththe DE-627 ger DE-627 rakwb eng 510 ZDB Wu, Daiyong verfasserin aut Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour. Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017 Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents Zhao, Hongyong oth Enthalten in Journal of difference equations and applications Amsterdam : Gordon and Breach, 1995 23(2017), 11, Seite 1765 (DE-627)192176633 (DE-600)1306548-8 (DE-576)9192176631 1023-6198 nnns volume:23 year:2017 number:11 pages:1765 http://dx.doi.org/10.1080/10236198.2017.1367389 Volltext http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 23 2017 11 1765
allfields_unstemmed	10.1080/10236198.2017.1367389 doi PQ20171228 (DE-627)OLC1999617029 (DE-599)GBVOLC1999617029 (PRQ)i1134-82ee90f713db798108a2ea3796d7133d89bc6124a51b4819a4f87f807ca7cda30 (KEY)0276061420170000023001101765complexdynamicsofadiscretepredatorpreymodelwiththe DE-627 ger DE-627 rakwb eng 510 ZDB Wu, Daiyong verfasserin aut Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour. Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017 Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents Zhao, Hongyong oth Enthalten in Journal of difference equations and applications Amsterdam : Gordon and Breach, 1995 23(2017), 11, Seite 1765 (DE-627)192176633 (DE-600)1306548-8 (DE-576)9192176631 1023-6198 nnns volume:23 year:2017 number:11 pages:1765 http://dx.doi.org/10.1080/10236198.2017.1367389 Volltext http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 23 2017 11 1765
allfieldsGer	10.1080/10236198.2017.1367389 doi PQ20171228 (DE-627)OLC1999617029 (DE-599)GBVOLC1999617029 (PRQ)i1134-82ee90f713db798108a2ea3796d7133d89bc6124a51b4819a4f87f807ca7cda30 (KEY)0276061420170000023001101765complexdynamicsofadiscretepredatorpreymodelwiththe DE-627 ger DE-627 rakwb eng 510 ZDB Wu, Daiyong verfasserin aut Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour. Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017 Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents Zhao, Hongyong oth Enthalten in Journal of difference equations and applications Amsterdam : Gordon and Breach, 1995 23(2017), 11, Seite 1765 (DE-627)192176633 (DE-600)1306548-8 (DE-576)9192176631 1023-6198 nnns volume:23 year:2017 number:11 pages:1765 http://dx.doi.org/10.1080/10236198.2017.1367389 Volltext http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 23 2017 11 1765
allfieldsSound	10.1080/10236198.2017.1367389 doi PQ20171228 (DE-627)OLC1999617029 (DE-599)GBVOLC1999617029 (PRQ)i1134-82ee90f713db798108a2ea3796d7133d89bc6124a51b4819a4f87f807ca7cda30 (KEY)0276061420170000023001101765complexdynamicsofadiscretepredatorpreymodelwiththe DE-627 ger DE-627 rakwb eng 510 ZDB Wu, Daiyong verfasserin aut Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour. Nutzungsrecht: © 2017 Informa UK Limited, trading as Taylor & Francis Group 2017 Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents Zhao, Hongyong oth Enthalten in Journal of difference equations and applications Amsterdam : Gordon and Breach, 1995 23(2017), 11, Seite 1765 (DE-627)192176633 (DE-600)1306548-8 (DE-576)9192176631 1023-6198 nnns volume:23 year:2017 number:11 pages:1765 http://dx.doi.org/10.1080/10236198.2017.1367389 Volltext http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 23 2017 11 1765
language	English
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topic_facet	Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents
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topic_title	510 ZDB Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect Marotto's Chaos flip bifurcation Allee effect predator-prey model 92D25 codimension-two bifurcation Neimark-Sacker bifurcation 39A30 Computer simulation Mathematical models Equilibrium Sensitivity analysis Bifurcation theory Numerical methods Liapunov exponents
topic	ddc 510 misc Marotto's Chaos misc flip bifurcation misc Allee effect misc predator-prey model misc 92D25 misc codimension-two bifurcation misc Neimark-Sacker bifurcation misc 39A30 misc Computer simulation misc Mathematical models misc Equilibrium misc Sensitivity analysis misc Bifurcation theory misc Numerical methods misc Liapunov exponents
topic_unstemmed	ddc 510 misc Marotto's Chaos misc flip bifurcation misc Allee effect misc predator-prey model misc 92D25 misc codimension-two bifurcation misc Neimark-Sacker bifurcation misc 39A30 misc Computer simulation misc Mathematical models misc Equilibrium misc Sensitivity analysis misc Bifurcation theory misc Numerical methods misc Liapunov exponents
topic_browse	ddc 510 misc Marotto's Chaos misc flip bifurcation misc Allee effect misc predator-prey model misc 92D25 misc codimension-two bifurcation misc Neimark-Sacker bifurcation misc 39A30 misc Computer simulation misc Mathematical models misc Equilibrium misc Sensitivity analysis misc Bifurcation theory misc Numerical methods misc Liapunov exponents
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title	Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect
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title_full	Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect
author_sort	Wu, Daiyong
journal	Journal of difference equations and applications
journalStr	Journal of difference equations and applications
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author_browse	Wu, Daiyong
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doi_str_mv	10.1080/10236198.2017.1367389
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title_sort	complex dynamics of a discrete predator-prey model with the prey subject to the allee effect
title_auth	Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect
abstract	In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.
abstractGer	In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.
abstract_unstemmed	In this paper, complex dynamics of the discrete predator-prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark-Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark-Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.
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title_short	Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect
url	http://dx.doi.org/10.1080/10236198.2017.1367389 http://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1367389 https://search.proquest.com/docview/1969003074
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author2	Zhao, Hongyong
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doi_str	10.1080/10236198.2017.1367389
up_date	2024-07-03T14:48:24.815Z
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