Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

Khan, A. Q.

doi:10.1002/mma.4290

Cited by 10 publications

(4 citation statements)

References 33 publications

(45 reference statements)

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“…In order for (10) to undergo a Neimark-Sacker bifurcation, it is mandatory that the following discriminatory quantity, i.e., χ = 0 (see [22][23][24][25][26][27][28][29][30][31][32]),…”

Section: Neimark-sacker Bifurcation Aboutmentioning

confidence: 99%

Bifurcations of a two-dimensional discrete-time predator–prey model

Khan

2019

Adv Differ Equ

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We study the local dynamics and bifurcations of a two-dimensional discrete-time predator-prey model in the closed first quadrant R 2 +. It is proved that the model has two boundary equilibria: O(0, 0), A(α 1-1 α 1 , 0) and a unique positive equilibrium B(1 α 2 , α 1 α 2-α 1-α 2 α 2) under some restriction to the parameter. We study the local dynamics along their topological types by imposing the method of linearization. It is proved that a fold bifurcation occurs about the boundary equilibria: O(0, 0), A(α 1-1 α 1 , 0) and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium B(1 α 2 , α 1 α 2-α 1-α 2 α 2). It is also proved that the model undergoes a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium B(1 α 2 , α 1 α 2-α 1-α 2 α 2) and meanwhile a stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between predator and prey populations. Numerical simulations are presented to verify not only the theoretical results but also to exhibit the complex dynamical behavior such as the period-2,-4,-11,-13,-15 and-22 orbits. Further, we compute the maximum Lyapunov exponents and the fractal dimension numerically to justify the chaotic behaviors of the discrete-time model. Finally, the feedback control method is applied to stabilize chaos existing in the discrete-time model.

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“…In order for (10) to undergo a Neimark-Sacker bifurcation, it is mandatory that the following discriminatory quantity, i.e., χ = 0 (see [22][23][24][25][26][27][28][29][30][31][32]),…”

Section: Neimark-sacker Bifurcation Aboutmentioning

confidence: 99%

Bifurcations of a two-dimensional discrete-time predator–prey model

Khan

2019

Adv Differ Equ

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“…(see other studies [9][10][11][12] ) Let H( ) = 2 + B 1 + B 2 , where B 1 and B 2 are constants. Suppose H(1) > 0 and 1 and 2 are solutions of H( ) = 0.…”

Section: Lemmamentioning

confidence: 99%

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confidence: 99%

Supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey model

Khan

2018

Math Methods in App Sciences

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MSC Classification: 39A10; 40A05; 92D25; 70K50; 35B35 We study the local dynamics and supercritical Neimark-Sacker bifurcation of a discrete-time Nicholson-Bailey host-parasitoid model in the interior of R 2 + . It is proved that if > 1, then the model has a unique positive equi-, which is locally asymptotically focus, unstable focus and nonhyperbolic under certain parametric condition. Furthermore, it is proved that the model undergoes a supercritical Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium point, and meanwhile, the stable closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasiperiodic oscillations between host and parasitoid populations. Some numerical simulations are presented to verify theoretical results.

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“…In Section 3, choosing δ as the bifurcation parameter, Neimark-Sacker bifurcation analysis is studied. It is shown that the model (1.3) undergoes Neimark-Sacker bifurcation by using the bifurcation theory [23,33]. In Section 4, OGY control strategy is applied for chaos control due to occurrence of Neimark-Sacker bifurcation.…”

Section: Introductionmentioning

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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Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

Cited by 10 publications

References 33 publications

Bifurcations of a two-dimensional discrete-time predator–prey model

Bifurcations of a two-dimensional discrete-time predator–prey model

Supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey model

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

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