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

Liu, Weiyi; Cai, Dongdong

doi:10.1186/s13662-019-1950-6

Cited by 33 publications

(11 citation statements)

References 33 publications

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“…Chakraborty et al [9] investigated stability analysis of a prey-predator model with optimal harvesting and square-root functional response. Liu and Cai [10] Table 1: Biological interpretations of chemostat parameters along with the corresponding dimensions.…”

Section: Motivation and Mathematical Modelling Of The Discretementioning

confidence: 99%

Bifurcation Analysis of a Discrete‐Time Chemostat Model

Al-Basyouni

Khan

2023

Mathematical Problems in Engineering

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In this article, a discretized two-dimensional chemostat model is investigated. By using the algebraic technique, it is proved that the discrete chemostat model has semitrivial equilibrium solution for all involved parametric values, but it has an interior equilibrium solution under certain parametric conditions. We have studied the local dynamics with topological classifications about the semitrivial and interior equilibrium solution on the basis of the theory of the discrete dynamical system. By bifurcation theory, we studied the existence of bifurcations for the understudied discrete chemostat model and proved that at a semitrivial equilibrium solution, it did not undergo flip bifurcation, but it undergoes flip bifurcation at interior equilibrium solution, and no other bifurcation occurs at this point. It has also explored the existence of periodic points and chaos control by the state feedback method. Finally, with the help of MATLAB software, flip bifurcation diagrams and corresponding maximum Lyapunov exponents were also presented. These numerical simulations not only show the correctness of theoretical results but also reveal the complex dynamics such as the orbits of period − 7 , − 8 , − 13 , and − 16 .

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Section: Motivation and Mathematical Modelling Of The Discretementioning

confidence: 99%

Bifurcation Analysis of a Discrete‐Time Chemostat Model

Al-Basyouni

Khan

2023

Mathematical Problems in Engineering

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“…) will be investigated based on theoretical studies in Sect. 3 by bifurcation theory [8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

Section: Bifurcation Analysismentioning

confidence: 99%

Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

Khan

Javaid

2021

Adv Differ Equ

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The local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.

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“…Because of this, compared to continuous forms, discrete-time model formulation and simulation are typically more straightforward, practical, and accurate. In recent years, the predator-prey relationship expressed in discretetime form has attracted much attention [23][24][25][26][27][28][29][30][31][32][33][34]. Tese studies examined both monotonic and nonmonotonic functional responses as well as the conventional logistic-type of prey growth rate.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Complexities of a Discrete Ivlev-Type Predator-Prey System

Rana

2023

Discrete Dynamics in Nature and Society

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In this study, we examine a discrete predator-prey system from the following two perspectives: (i) the functional response is of the Ivlev type and (ii) the prey growth rate is of the Gompertz type. We define the stability requirement for feasible fixed points. We demonstrate algebraically that if the bifurcation (control) parameter rises over its threshold value, the system encounters flip and Neimark–Sacker (NS) bifurcations in the vicinity of the interior fixed point. We explicitly establish the existence requirements and direction of bifurcations via the center manifold theory. Analytical findings are validated by numerical simulations, which are used to highlight the occurrence of instability and chaotic dynamics in the system. In order to regulate the chaotic trajectories that exist in the system, we adopt a feedback control approach.

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

Cited by 33 publications

References 33 publications

Bifurcation Analysis of a Discrete‐Time Chemostat Model

Bifurcation Analysis of a Discrete‐Time Chemostat Model

Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

Dynamical Complexities of a Discrete Ivlev-Type Predator-Prey System

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