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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Bibliographic Details
Published in:Advances in difference equations Vol. 2019; no. 1; pp. 1 - 22
Main Authors: Liu, Weiyi, Cai, Donghan
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 17.01.2019
Springer Nature B.V
SpringerOpen
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1687-1847
1687-1839
1687-1847
DOI:10.1186/s13662-019-1950-6