Stability and Bifurcation Analysis in the Photosensitive CDIMA System with Delayed Feedback Control

2017 ◽  
Vol 27 (11) ◽  
pp. 1750177 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Positive Equilibrium ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Theoretical Results

A diffusive photosensitive CDIMA system with delayed feedback subject to Neumann boundary conditions is considered. We derive the conditions of the occurrence of Turing instability. We also investigate the influence of delay on the stability of the positive equilibrium of the system, and prove that delay induces the occurrence of Hopf bifurcation. By computing the normal form on the center manifold, we give the formulas determining the properties of the Hopf bifurcation. Finally, we give some numerical simulations to support and strengthen the theoretical results. Our study shows that diffusion and delayed feedback can effect the stability of the equilibrium of the system.

Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure

2016 ◽  
Vol 26 (03) ◽  
pp. 1650047 ◽  
Author(s):  
Jiantao Zhao ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
System With Delay ◽  
The Stability ◽  
Theoretical Results

A reaction–diffusion plankton system with delay and quadratic closure term is investigated to study the interactions between phytoplankton and zooplankton. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system. Our conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations to occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.

scholarly journals The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

2018 ◽  
Vol 23 (5) ◽  
pp. 749-770 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Arbitrary Order ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Positive Equilibrium ◽  
Distribution Of Eigenvalues ◽  
The Stability

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

Delayed Feedback Control and Bifurcation Analysis in a Chaotic Chemostat System

2015 ◽  
Vol 25 (06) ◽  
pp. 1550087 ◽  
Author(s):  
Zhichao Jiang ◽  
Wanbiao Ma
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Stable Steady State ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability

In this paper, the effect of delay on a nonlinear chaotic chemostat system with delayed feedback is investigated by regarding delay as a parameter. At first, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulation examples are given, which indicate that the chaotic oscillation can be converted into a stable steady state or a stable periodic orbit when delay passes through certain critical values.

scholarly journals Bifurcation Analysis in a Diffusion Mussel-Algae Interaction System with Delays Considering the Half-Saturation Constant

2021 ◽  
Author(s):  
Zhichao Jiang ◽  
Weicong Zhang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Delayed Reaction ◽  
Half Saturation Constant ◽  
Two Delays ◽  
Steady State Bifurcation ◽  
The Stability

Abstract In this paper, the kinetics of a class of delayed reaction-diffusion musselalgae system under Neumann boundary conditions with the half- saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affect the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among parameters, the half-saturation constant and two delays drive the complexity of the system dynamics.

Oscillations Induced by Quiescent Adult Female in a Reaction–Diffusion Model of Wild Aedes Aegypti Mosquitoes

2019 ◽  
Vol 29 (13) ◽  
pp. 1950189 ◽  
Author(s):  
A. Aghriche ◽  
R. Yafia ◽  
M. A. Aziz Alaoui ◽  
A. Tridane ◽  
F. A. Rihan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Aedes Aegypti ◽  
Center Manifold ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Reaction Diffusion Model ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

This paper takes the reaction–diffusion approach to deal with the quiescent females phase, so as to describe the dynamics of invasion of aedes aegypti mosquitoes, which are divided into three subpopulations: eggs, pupae and female. We mainly investigate whether the time of quiescence (delay) in the females phase can induce Hopf bifurcation. By means of analyzing the eigenvalue spectrum, we show that the persistent positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay crosses some critical value. Using normal form and center manifold theory, we investigate the stability of the bifurcating branches of periodic solutions and the direction of the Hopf bifurcation. Numerical simulations are carried out to support our theoretical results.

Stability and Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

scholarly journals Local Hopf Bifurcation in a Competitive Model of Market Structure with Consumptive Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xuhui Li
Keyword(s):  
Hopf Bifurcation ◽  
Market Structure ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Local Hopf Bifurcation ◽  
Competitive Model ◽  
Form Theory ◽  
The Stability

A competitive model of market structure with consumptive delays is considered. The local stability of the positive equilibrium and the existence of local Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equation. The explicit formulas determining the stability and other properties of bifurcating periodic solutions are derived by using normal form theory and center manifold argument. Finally, numerical simulations are given to support the analytical results.

scholarly journals Dynamics in a delayed diffusive cell cycle model

2018 ◽  
Vol 23 (5) ◽  
pp. 691-709
Author(s):  
Yanqin Wang ◽  
Ling Yang ◽  
Jie Yan
Keyword(s):  
Cell Cycle ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Spatial Dynamics ◽  
Positive Equilibrium ◽  
Cycle Model ◽  
Center Manifold Theorem ◽  
Diffusive Model ◽  
The Stability ◽  
Theoretical Results

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

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