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.

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.

Stability, bifurcation analysis and chaos control for a predator-prey system

2018 ◽  
Vol 25 (3) ◽  
pp. 612-626 ◽  
Author(s):  
Qamar Din
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Chaos Control ◽  
Period Doubling ◽  
Qualitative Behavior ◽  
Time Model ◽  
Center Manifold Theorem ◽  
Stability Of Equilibria ◽  
Local Asymptotic Stability ◽  
Predator Model

We study qualitative behavior of a modified prey–predator model by introducing density-dependent per capita growth rates and a Holling type II functional response. Positivity of solutions, boundedness and local asymptotic stability of equilibria were investigated for continuous type of the prey–predator system. In order to discuss the rich dynamics of the proposed model, a piecewise constant argument was implemented to obtain a discrete counterpart of the continuous system. Moreover, in the case of a discrete-time prey–predator model, the boundedness of solutions and local asymptotic stability of equilibria were investigated. With the help of the center manifold theorem and bifurcation theory, we investigated whether a discrete-time model undergoes period-doubling and Neimark–Sacker bifurcation at its positive steady-state. Finally, two novel generalized hybrid feedback control methods are presented for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. In order to illustrate the effectiveness of the proposed control strategies, numerical simulations are presented.

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yue Zhang ◽  
Xue Li ◽  
Xianghua Zhang ◽  
Guisheng Yin
Keyword(s):  
Fixed Point ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Theoretic Analysis ◽  
Simulation Results ◽  
The Stability ◽  
Positive Fixed Point

Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

scholarly journals On neutral-delay two-species Lotka-Volterra competitive systems

1991 ◽  
Vol 32 (3) ◽  
pp. 311-326 ◽  
Author(s):  
Y. Kuang
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Qualitative Behavior ◽  
Neutral Delay ◽  
Competitive System ◽  
Competitive Systems ◽  
Local Asymptotic Stability ◽  
Discrete Delays

AbstractThe qualitative behavior of positive solutions of the neutral-delay two-species Lotka-Volterra competitive system with several discrete delays is investigated. Sufficient conditions are obtained for the local asymptotic stability of the positive steady state. In fact, some of these sufficient conditions are also necessary except at those critical values. Results on the oscillatory and non-oscillatory characteristics of the positive solutions are also included.

scholarly journals Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 473
Author(s):  
Jehad Alzabut ◽  
A. George Maria Selvam ◽  
Rami Ahmad El-Nabulsi ◽  
D. Vignesh ◽  
Mohammad Esmael Samei
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Current Collector ◽  
Stability Of Solutions ◽  
Electric Bus ◽  
Pantograph Equation ◽  
Non Local

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ*β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

scholarly journals Zero Eigenvalue Analysis for the Determination of Multiple Steady States in Reaction Networks

1998 ◽  
Vol 53 (3-4) ◽  
pp. 171-177
Author(s):  
Hsing-Ya Li
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Reaction Network ◽  
Steady States ◽  
Eigenvalue Analysis ◽  
Zero Eigenvalue ◽  
Positive Steady State ◽  
Positive Steady States ◽  
Positive Rate ◽  
System Of Inequalities

Abstract A chemical reaction network can admit multiple positive steady states if and only if there exists a positive steady state having a zero eigenvalue with its eigenvector in the stoichiometric subspace. A zero eigenvalue analysis is proposed which provides a necessary and sufficient condition to determine the possibility of the existence of such a steady state. The condition forms a system of inequalities and equations. If a set of solutions for the system is found, then the network under study is able to admit multiple positive steady states for some positive rate constants. Otherwise, the network can exhibit at most one steady state, no matter what positive rate constants the system might have. The construction of a zero-eigenvalue positive steady state and a set of positive rate constants is also presented. The analysis is demonstrated by two examples.

scholarly journals Global Stability and Hopf-bifurcation Analysis of Biological Systems using Delayed Extended Rosenzweig-MacArthur Model

2018 ◽  
Vol 12 (2) ◽  
pp. 171
Author(s):  
Enobong E. Joshua ◽  
Cec Ekemini T. Akpan
Keyword(s):  
Experimental Data ◽  
Population Dynamics ◽  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Asymptotically Stable ◽  
Stability Switches ◽  
Large Delay ◽  
Local Hopf Bifurcation ◽  
Delay Margin ◽  
Theoretical Results

This paper investigates the global asymptotic stability of a Delayed Extended Rosenzweig-MacArthur Model via Lyapunov-Krasovskii functionals. Frequency sweeping technique ensures stability switches as the delay parameter increases and passes the critical bifurcating threshold.The model exhibits a local Hopf-bifurcation from asymptotically stable oscillatory behaviors to unstable strange chaotic behaviors dependent of the delay parameter values.Hyper-chaotic fluctuations were observed for large delay values far away from the critical delay margin. Numerical simulations of experimental data obtained via non-dimensionalization have shown the applications of theoretical results in ecological population dynamics.

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