Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model

2019 ◽  
Vol 12 (04) ◽  
pp. 1950044 ◽  
Author(s):  
Muhammad Aqib Abbasi ◽  
Qamar Din
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Technique ◽  
Qualitative Behavior ◽  
Flip Bifurcation ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Crowding Effects

The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

scholarly journals A Density-Dependent Host-Parasitoid Model with Stability, Bifurcation and Chaos Control

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 536 ◽  
Author(s):  
Xiaorong Ma ◽  
Qamar Din ◽  
Muhammad Rafaqat ◽  
Nasir Javaid ◽  
Yongliang Feng
Keyword(s):  
Fixed Point ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Chaos Control ◽  
Growth Function ◽  
Host Population ◽  
Qualitative Behavior ◽  
Positive Steady State ◽  
Positive Fixed Point

The aim of this article is to study the qualitative behavior of a host-parasitoid system with a Beverton-Holt growth function for a host population and Hassell-Varley framework. Furthermore, the existence and uniqueness of a positive fixed point, permanence of solutions, local asymptotic stability of a positive fixed point and its global stability are investigated. On the other hand, it is demonstrated that the model endures Hopf bifurcation about its positive steady-state when the growth rate of the consumer is selected as a bifurcation parameter. Bifurcating and chaotic behaviors are controlled through the implementation of chaos control strategies. In the end, all mathematical discussion, especially Hopf bifurcation, methods related to the control of chaos and global asymptotic stability for a positive steady-state, is supported with suitable numerical simulations.

Bifurcations and chaos control in a discrete-time biological model

2020 ◽  
Vol 13 (04) ◽  
pp. 2050022 ◽  
Author(s):  
A. Q. Khan ◽  
T. Khalique
Keyword(s):  
Discrete Time ◽  
Chaos Control ◽  
Control Method ◽  
Complex Dynamics ◽  
Small Neighborhood ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Flip Bifurcation ◽  
Time Model ◽  
Predator Prey Model

In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

2021 ◽  
pp. 107754632110564
Author(s):  
Waqas Ishaque ◽  
Qamar Din ◽  
Muhammad Taj
Keyword(s):  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Mutual Interference ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
Prey Interaction

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Control Design Applied to a Micro Electro Mechanical System: MEMS Comb Drive

2011 ◽  
Vol 217-218 ◽  
pp. 33-38 ◽  
Author(s):  
Alessandra Bonato Altran ◽  
Fábio Roverto Chavarette ◽  
Carlos Roberto Minussi ◽  
Nelson José Peruzzi ◽  
Mara Lúcia Marthins Lopes ◽  
...  
Keyword(s):  
Optimal Control ◽  
Periodic Orbit ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Micro Electro Mechanical System ◽  
Control Technique ◽  
Reference Trajectory ◽  
Comb Drive ◽  
Linear Optimal Control ◽  
Small Periodic

This paper presents the linear optimal control technique for reducing the chaotic movement of the micro-electro-mechanical Comb Drive system to a small periodic orbit. We analyze the non-linear dynamics in a micro-electro-mechanical Comb Drive and demonstrated that this model has a chaotic behavior. Chaos control problems consist of attempts to stabilize a chaotic system to an equilibrium point, a periodic orbit, or more general, about a given reference trajectory. This technique is applied in analyzes the nonlinear dynamics in an MEMS Comb drive. The simulation results show the identification by linear optimal control is very effective.

Complex Dynamics of a Discrete-Time Prey–Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos

2020 ◽  
Vol 30 (10) ◽  
pp. 2050149
Author(s):  
Pinar Baydemir ◽  
Huseyin Merdan ◽  
Esra Karaoglu ◽  
Gokce Sucu
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Positive Equilibrium ◽  
Flip Bifurcation ◽  
Step Size ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Predator System

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

CHAOS CONTROL OF A MODIFIED COUPLED DYNAMOS SYSTEM

2007 ◽  
Vol 21 (26) ◽  
pp. 4593-4610 ◽  
Author(s):  
XING-YUAN WANG ◽  
XIANG-JUN WU
Keyword(s):  
Numerical Simulations ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Direct Method ◽  
Equilibrium Points ◽  
Unstable Equilibrium ◽  
Unstable Periodic Orbits ◽  
Lyapunov Direct Method

This paper studies the problem of controlling the chaotic behavior of a modified coupled dynamos system. Two different methods, feedback and non-feedback methods, are used to control chaos in the modified coupled dynamos system. Based on the Lyapunov direct method and Routh–Hurwitz criterion, the conditions suppressing chaos to unstable equilibrium points or unstable periodic orbits (limit cycles) are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the two different methods.

scholarly journals Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Qamar Din ◽  
A. A. Elsadany ◽  
Hammad Khalil
Keyword(s):  
Numerical Simulations ◽  
Chaos Control ◽  
Hybrid Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Pole Placement ◽  
Positive Equilibrium ◽  
Local Asymptotic Stability ◽  
Positive Equilibrium Point ◽  
Plant Herbivore

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

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