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

Khan, A. Q.

doi:10.1002/mma.4934

Cited by 3 publications

(2 citation statements)

References 20 publications

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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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“…In recent years, many scientists, ecologists, and biologists have studied and analyzed the relationship between hosts and parasitoids [18][19][20][21][22][23][24][25][26]. For some hosts of endangered species, specifc parasite reduction is likely to lead to the immediate extinction of the host species.…”

Section: Introductionmentioning

confidence: 99%

The Diffusion-Driven Instability for a General Time-Space Discrete Host-Parasitoid Model

Zhang

2023

Discrete Dynamics in Nature and Society

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In this paper, we consider a general time-space discrete host-parasitoid model with the periodic boundary conditions. We analyzed and obtained some usual conditions, such as Turing instability occurrence, Flip bifurcation occurrence, and Neimark-Sacker bifurcation occurrence. We also find several multiple bifurcation phenomena, such as 1 : 2 Resonance, Neimark-Sacker-Flip bifurcation, Neimark-Sacker-Neimark-Sacker bifurcation, Neimark-Sacker-Flip-Flip bifurcation, induced by diffusion. In a modified Nicholson-Bailey model, employing the mutual interference and diffusive interaction as bifurcation parameters, there appear route from standing Turing instability to multiple bifurcations by changing diffusive parameters. Some numerical simulations of the modified Nicholson-Bailey model support these corollaries.

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Bifurcations and chaotic behavior of a predator-prey model with discrete time

Hong¹,

Zhang

2023

MATH

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<abstract><p>In this paper, the dynamical behavior of a predator-prey model with discrete time is discussed in terms of both theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed. It is proved that the system undergoes Flip bifurcation and Hopf bifurcation around its unique positive equilibrium point using center manifold theorem and bifurcation theory. Additionally, by applying small perturbations to the bifurcation parameter, chaotic cases occur at some corresponding internal equilibria. Finally, numerical simulations are provided with the help of maximum Lyapunov exponent and phase diagrams, which reveal a complex dynamical behavior.</p></abstract>

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

Cited by 3 publications

References 20 publications

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

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

The Diffusion-Driven Instability for a General Time-Space Discrete Host-Parasitoid Model

Bifurcations and chaotic behavior of a predator-prey model with discrete time

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