The stability and bifurcation analysis of a discrete Holling-Tanner model

Cao, Hui; Yue, Zongmin; Zhou, Yicang

doi:10.1186/1687-1847-2013-330

Cited by 13 publications

(8 citation statements)

References 11 publications

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“…2e) confirms that growth of parameter δ causes a chaotic dynamics for the discrete-time ratiodependent Holling-Tanner system. Let λ 1 and λ 2 be the roots of (17). Then, λ 1 + λ 2 = j 11 + j 22 − k 1 (18) and…”

Section: Fractal Dimensionmentioning

confidence: 99%

“…Recently, a little works in literature studied discrete-time Holling-Tanner models [16][17][18] and its chaotic behaviors. For instance, a discrete-time Holling and Leslie type predator-prey system with constant-yield prey harvesting analyzed in [16], in [17] the authors investigated a discrete Holling-Tanner model and a discrete predator-prey model with modified Holling-Tanner functional response discussed in [18]. These studies paid their attention to determine the stability and directions of flip and Neimark-Sacker bifurcations via use of center manifold theory.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

Rana

2019

J Egypt Math Soc

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A discrete-time Holling-Tanner model with ratio-dependent functional response is examined. We show that the system experiences a flip bifurcation and Neimark-Sacker bifurcation or both together at positive fixed point in the interior of R 2 + when one of the model parameter crosses its threshold value. We concentrate our attention to determine the existence conditions and direction of bifurcations via center manifold theory. To validate analytical results, numerical simulations are employed which include bifurcations, phase portraits, stable orbits, invariant closed circle, and attracting chaotic sets. In addition, the existence of chaos in the system is justified numerically by the sign of maximum Lyapunov exponents and fractal dimension. Finally, we control chaotic trajectories exists in the system by feedback control strategy.

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Section: Fractal Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

Rana

2019

J Egypt Math Soc

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“…Bifurcation occurs when the stability of an equilibrium point changes [48]. In general, the dynamics of a discrete SI model with integer-order has been examined by Hu et al [42].…”

Section: Numerical Examplesmentioning

confidence: 99%

Bifurcations and chaos in a discrete SI epidemic model with fractional order

et al. 2018

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In this study, a new discrete SI epidemic model is proposed and established from SI fractional-order epidemic model. The existence conditions, the stability of the equilibrium points and the occurrence of bifurcation are analyzed. By using the center manifold theorem and bifurcation theory, it is shown that the model undergoes flip and Neimark-Sacker bifurcation. The effects of step size and fractional-order parameters on the dynamics of the model are studied. The bifurcation analysis is also conducted and our numerical results are in agreement with theoretical results.

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“…In recent times, a few number of articles in literature discussed the dynamics of discrete-time predator-prey systems of Leslie type [11,12]. For example, a discretetime predator-prey system of Holling and Leslie type with constant-yield prey harvesting was investigated in [11], and a discrete predator-prey model with Holling-Tanner functional response was studied in [12].…”

Section: Introductionmentioning

confidence: 99%

“…For example, a discretetime predator-prey system of Holling and Leslie type with constant-yield prey harvesting was investigated in [11], and a discrete predator-prey model with Holling-Tanner functional response was studied in [12]. In their studies, the authors paid their attention to drive the existence of flip and NeimarkSacker bifurcations by using center manifold theory.…”

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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The stability and bifurcation analysis of a discrete Holling-Tanner model

Cited by 13 publications

References 11 publications

Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

Bifurcations and chaos in a discrete SI epidemic model with fractional order

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

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