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

Rana, S. M. Sohel

doi:10.1186/s42787-019-0055-4

Cited by 20 publications

(10 citation statements)

References 22 publications

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“…Assume that u = x − x * , u = y − y * . Then we transform the fixed point p 1 ( ce (10) into the origin. Then we have where and h = h 1 .…”

Section: Flip Bifurcationmentioning

confidence: 99%

“…Assume that u = x − x * , u = y − y * . Then we transform the fixed point p 1 ( ce (10) into the origin. Then we have where Ê11 , Ê12 , Ê13 , Ê14 , Ê21 , Ê22 , Ê23 , Ê24 are given in ( 14) by substituting h for h 2 + h * .…”

Section: Flip Bifurcationmentioning

confidence: 99%

“…Furthermore, as compared to continuous models, these models give efficient computing results for numerical simulations as well as richer dynamical properties [1][2][3][4][5][6][7] . Many fascinating works on the stability, bifurcation and chaotic occurrences in discrete temporal models have appeared in the literature in recent years [8][9][10][11][12][13][14][15] . Because of its widespread occurrence and relevance, research into the dynamic connection between prey-predator has been and will continue to be a hot issue for a long time.…”

mentioning

confidence: 99%

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Qualitative analysis and phase of chaos control of the predator-prey model with Holling type-III

AL-Kaff

El-Metwally

Elabbasy

2022

Sci Rep

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In this study, we investigate the dynamics of a discrete-time with predator-prey system with a Holling-III type functional response model. The center manifold theorem and bifurcation theory are used to create existence conditions for flip bifurcations and Neimark-Sacker bifurcations. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits are examples of numerical simulations that not only show the soundness of theoretical analysis but also show complicated dynamical behaviors and biological processes. From the point of view of biology, this implies that the tiny integral step size can steady the system into locally stable coexistence. Yet, the large integral step size may lead to instability in the system, producing more intricate and richer dynamics. This also means that when the intrinsic death rate of the predator is high, this leads to a chaotic growth rate of the prey. The model has bifurcation features that are similar to those seen in logistic models. In addition, there is a bidirectional Neimark-Sacker bifurcation for both prey and predator, and therefore we obtain a direct correlation in symbiosis. This means that the higher the growth rate of the prey, the greater the growth rate of the predator. Therefore, the operation of predation has increased. The opposite is also true. Finally, the OGY approach is used to control chaos in the predator and prey model. which led to a new concept which we call bifurcation phase of control chaos.

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“…Assume that u = x − x * , u = y − y * . Then we transform the fixed point p 1 ( ce (10) into the origin. Then we have where and h = h 1 .…”

Section: Flip Bifurcationmentioning

confidence: 99%

Section: Flip Bifurcationmentioning

confidence: 99%

mentioning

confidence: 99%

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Qualitative analysis and phase of chaos control of the predator-prey model with Holling type-III

AL-Kaff

El-Metwally

Elabbasy

2022

Sci Rep

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“…Numerous discrete systems have aroused the interest of academics investigating the Neimark-Sacker and flip bifurcations, stable orbits, and chaotic attractors (see [24,25,36,37]). The center manifold theory and standard form can mathematically quantify these phenomena.…”

Section: Introductionmentioning

confidence: 99%

Chaotic dynamics of the fractional order Schnakenberg model and its control

Uddin

Rana

2023

Math Appl Sci Eng

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The Schnakenberg model is thought to be the Caputo fractional derivative. In order to create caputo fractional differential equations for the Schnakenberg model, a discretization process is first used. The fixed points in the model are categorized topologically. Then, we show analytically that, under certain parametric conditions, a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation are supported by a fractional order Schnakenberg model. Using central manifold and bifurcation theory, we demonstrate the presence and direction of NS and Flip bifurcations. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order Schnakenberg model. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. In order to support the system’s chaotic characteristics, we also compute the maximal Lyapunov exponents and fractal dimensions quantitatively. Finally, the chaotic trajectory of the system is stopped using the OGY approach, hybrid control method, and state feedback method.

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“…Discrete-time dynamical systems have complicated and diversified dynamical properties [25][26][27][28][29][30][31]. The system to be analyzed in this work is then obtained by using the forward Euler technique on the system (1.3) as follows:…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Ahmed,

Almatrafi

2023

Int. J. Anal. Appl.

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The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

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

Cited by 20 publications

References 22 publications

Qualitative analysis and phase of chaos control of the predator-prey model with Holling type-III

Qualitative analysis and phase of chaos control of the predator-prey model with Holling type-III

Chaotic dynamics of the fractional order Schnakenberg model and its control

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

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