Bifurcation and chaotic behavior of a discrete-time predator–prey system

He, Zhen; Lai, Xin

doi:10.1016/j.nonrwa.2010.06.026

Cited by 163 publications

(105 citation statements)

References 29 publications

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“…Let 1 and 2 be the two roots of (8), which are called eigenvalues of the fixed point ( , ). We recall some definitions of topological types for a fixed point ( , ) [12,14,32] …”

Section: The Existence and Stability Of The Fixed Pointsmentioning

confidence: 99%

“…In recent years, discrete-time population systems also come to the fore due to the following reasons [9][10][11][12][13][14][15][16]: Firstly, discrete-time systems are more suitable than continuoustime systems to describe populations with nonoverlapping generations. Secondly, they can produce more complex and rich dynamical behaviors than continuous-time systems.…”

Section: Introductionmentioning

confidence: 99%

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Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Baek

2018

Mathematical Problems in Engineering

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The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R 2 + are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.

show abstract

“…Let 1 and 2 be the two roots of (8), which are called eigenvalues of the fixed point ( , ). We recall some definitions of topological types for a fixed point ( , ) [12,14,32] …”

Section: The Existence and Stability Of The Fixed Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Baek

2018

Mathematical Problems in Engineering

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“…Qualitative analyses of these works found many rich dynamics which include limit cycle, states of stability, codimension 1 subcritical Hopf bifurcation, codimension 2 Bogdanov-Takens bifurcation, and codimension 3 degenerate focus type Bogdanov-Takens bifurcation around positive equilibrium. But lots of exploratory works have suggested that the discretization of predator-prey models is more suitable compared to continuous ones when size of populations is small [3][4][5][6][7][8][9][10]. These researches have mainly focused on Gauss-type predator-prey interaction with monotonic functional responses.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

Rana

Kulsum

2017

Discrete Dynamics in Nature and Society

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The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R 2 + . Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system.

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“…For example, the transition between spiking and bursting in the model can be understood by homoclinic bifurcations [12,13]. More information on bifurcation can be found in [1,4,[8][9][10][11][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

2014

Journal of Applied Mathematics

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1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

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Bifurcation and chaotic behavior of a discrete-time predator–prey system

Cited by 163 publications

References 29 publications

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

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