Discrete-time bifurcation behavior of a prey-predator system with generalized predator

Singh, Harkaran; Dhar, Joydip; Bhatti, Harbax Singh

doi:10.1186/s13662-015-0546-z

Cited by 19 publications

(10 citation statements)

References 34 publications

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“…Lemma 2.2. [15,17,22,34] Assume F (λ) = λ 2 + Bλ + C, where B and C are two real constants and let F (1) > 0. Suppose λ 1 and λ 2 are two roots of F (λ) = 0.…”

Section: Stability Analysis and Fixed Points Of The Systemmentioning

confidence: 99%

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

Kangalgil

Işık

2020

Hacettepe Journal of Mathematics and Statistics

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This article is about a discrete-time predator-prey model obtained by the forward Euler method. The stability of the fixed point of the model and the existence conditions of the Neimark-Sacker bifurcation are investigated. In addition, the direction of the Neimark-Sacker bifurcation is given. Moreover, OGY control method is to implement to control chaos caused by the Neimark-Sacker bifurcation. Finally, Neimark-Sacker bifurcation, chaos control strategy, and asymptotic stability of the only positive fixed point are verified with the help of numerical simulations. The existence of chaotic behavior in the model is confirmed by computing of the maximum Lyapunov exponents.

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“…Lemma 2.2. [15,17,22,34] Assume F (λ) = λ 2 + Bλ + C, where B and C are two real constants and let F (1) > 0. Suppose λ 1 and λ 2 are two roots of F (λ) = 0.…”

Section: Stability Analysis and Fixed Points Of The Systemmentioning

confidence: 99%

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

Kangalgil

Işık

2020

Hacettepe Journal of Mathematics and Statistics

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“…where |c| << 1 represents a small perturbation in c 1 . Assume that (x * , y * ) = denotes fixed point for (14).…”

Section: Neimark-sacker Bifurcationmentioning

confidence: 99%

“…Zhao et al [13] reported flip bifurcation and Neimark-Sacker bifurcation for a class of discrete prey-predator interaction. Singh et al [14] explored local dynamics and bifurcation analysis for a class of Leslie-Gower type predator-prey model with predator partially dependent on prey. Asheghi [15] discussed plant-herbivore type predator-prey interaction for non-overlapping generations.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and Chaos Control for a Discrete-Time Lotka-Volterra Model

Din¹,

Rafaqat

Elsadany

et al. 2020

IEEE Access

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Bifurcation theory (center manifold and Ljapunov-Schmidt reduction, normal form theory, universal unfolding, calculation of bifurcation diagrams) has become an important and very useful means in the solution of nonlinear stability problems in many branches of engineering. The present study deals with qualitative behavior of a two-dimensional discrete-time system for interaction between prey and predator. The discrete-time model has more chaotic and rich dynamical behavior as compare to its continuous counterpart. We investigate the qualitative behavior of a discrete-time Lotka-Volterra model with linear functional response for prey. The local asymptotic behavior of equilibria is discussed for discrete-time Lotka-Volterra model. Furthermore, with the help of bifurcation theory and center manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two chaos control methods, that is, OGY feedback control and hybrid control strategy, are implemented. Numerical simulations are provided to illustrate theoretical discussion and their effectiveness. INDEX TERMS Lotka-Volterra model, stability, flip bifurcation, Hopf bifurcation, chaos control.

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“…Similarly, Liu and Xiao [5] presented complex dynamics for a discrete Lotka-Volterra system after implementation of Euler method. For a similar type of investigations related to predator-prey systems the interested reader is referred to [6][7][8][9][10][11][12][13][14][15][16][17]. All these studies reveal that the discrete predator-prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts.…”

Section: Introductionmentioning

confidence: 99%

A dynamically consistent nonstandard finite difference scheme for a predator–prey model

et al. 2019

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The interaction between prey and predator is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of predator and prey interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for a class of predator-prey systems with Holling type-III functional response. Positivity, boundedness, and persistence of solutions are investigated. Analysis of existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes a Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes a Neimark-Sacker bifurcation at its interior fixed point. A numerical simulation is provided to strengthen our theoretical discussion.

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Discrete-time bifurcation behavior of a prey-predator system with generalized predator

Cited by 19 publications

References 34 publications

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

Dynamics and Chaos Control for a Discrete-Time Lotka-Volterra Model

A dynamically consistent nonstandard finite difference scheme for a predator–prey model

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