Bifurcation, chaos analysis and control in a discrete-time predator–prey system

The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed...

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Vydané v:	Advances in difference equations Ročník 2019; číslo 1; s. 1 - 22
Hlavní autori:	Liu, Weiyi, Cai, Donghan
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 17.01.2019 Springer Nature B.V SpringerOpen
Predmet:	Analysis Bifurcation control Bifurcation theory Chaos control Chaos theory Computer simulation Control systems Difference and Functional Equations Discrete time systems Functional Analysis Local stability Marotto’s chaos Mathematical models Mathematics Mathematics and Statistics Neimark–Sacker bifurcation Numerical methods Ordinary Differential Equations Partial Differential Equations Predator-prey simulation Predator–prey model Neimark–Sacker bifurcation 34H10 34H20 37N25 39A28 Marotto’s chaos 34H15 Predator–prey model Chaos control Local stability Bifurcation control 37M20
ISSN:	1687-1847, 1687-1839, 1687-1847
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Abstract	The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed that under certain conditions, the system undergoes a Neimark–Sacker bifurcation when bifurcation parameter passes a critical value, and a closed invariant curve arises from a fixed point. Chaos in the sense of Marotto is also verified by both analytical and numerical methods. Furthermore, to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system, two control strategies are designed, respectively. Numerical simulations are presented not only to validate analytical results but also to show the complicated dynamical behavior.
AbstractList	The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed that under certain conditions, the system undergoes a Neimark–Sacker bifurcation when bifurcation parameter passes a critical value, and a closed invariant curve arises from a fixed point. Chaos in the sense of Marotto is also verified by both analytical and numerical methods. Furthermore, to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system, two control strategies are designed, respectively. Numerical simulations are presented not only to validate analytical results but also to show the complicated dynamical behavior. Abstract The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed that under certain conditions, the system undergoes a Neimark–Sacker bifurcation when bifurcation parameter passes a critical value, and a closed invariant curve arises from a fixed point. Chaos in the sense of Marotto is also verified by both analytical and numerical methods. Furthermore, to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system, two control strategies are designed, respectively. Numerical simulations are presented not only to validate analytical results but also to show the complicated dynamical behavior.
ArticleNumber	11
Author	Liu, Weiyi Cai, Donghan
Author_xml	– sequence: 1 givenname: Weiyi surname: Liu fullname: Liu, Weiyi email: weiyiliu@whu.edu.cn organization: School of Mathematics and Statistics, Wuhan University – sequence: 2 givenname: Donghan surname: Cai fullname: Cai, Donghan organization: School of Mathematics and Statistics, Wuhan University
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Snippet	The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the... Abstract The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of...
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SubjectTerms	Analysis Bifurcation control Bifurcation theory Chaos control Chaos theory Computer simulation Control systems Difference and Functional Equations Discrete time systems Functional Analysis Local stability Marotto’s chaos Mathematical models Mathematics Mathematics and Statistics Neimark–Sacker bifurcation Numerical methods Ordinary Differential Equations Partial Differential Equations Predator-prey simulation Predator–prey model
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Title	Bifurcation, chaos analysis and control in a discrete-time predator–prey system
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