Bifurcation and chaos control in a discrete-time predator–prey model with nonlinear saturated incidence rate and parasite interaction

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

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Optimal Control Strategy for a Discrete Time Smoking Model with Specific Saturated Incidence Rate

Discrete Dynamics in Nature and Society ◽

10.1155/2018/5949303 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 5

Author(s):

Abderrahim Labzai ◽

Omar Balatif ◽

Mostafa Rachik

Keyword(s):

Optimal Control ◽

Incidence Rate ◽

Discrete Time ◽

Control Strategy ◽

Optimal Controls ◽

Optimization Strategy ◽

Optimal Control Strategy ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Heavy Smokers

The aim of this paper is to study and investigate the optimal control strategy of a discrete mathematical model of smoking with specific saturated incidence rate. The population that we are going to study is divided into five compartments: potential smokers, light smokers, heavy smokers, temporary quitters of smoking, and permanent quitters of smoking. Our objective is to find the best strategy to reduce the number of light smokers, heavy smokers, and temporary quitters of smoking. We use three control strategies which are awareness programs through media and education, treatment, and psychological support with follow-up. Pontryagins maximum principle in discrete time is used to characterize the optimal controls. The numerical simulation is carried out using MATLAB. Consequently, the obtained results confirm the performance of the optimization strategy.

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Discrete-Time Predator-Prey Model with Bifurcations and Chaos

Mathematical Problems in Engineering ◽

10.1155/2020/8845926 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

K. S. Al-Basyouni ◽

A. Q. Khan

Keyword(s):

Fixed Point ◽

Fixed Points ◽

Discrete Time ◽

Control Method ◽

Linear Stability Theory ◽

Flip Bifurcation ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

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