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.

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.

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.

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 Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

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 of a Delayed Infection Model with General Incidence Function

2019 ◽  
Vol 2019 ◽  
pp. 1-16
Author(s):  
Suxia Zhang ◽  
Hongsen Dong ◽  
Jinhu Xu
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Infection Model ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Incidence Function ◽  
Stability And Instability ◽  
Different Types ◽  
The Stability ◽  
Theoretical Results

In this paper, an infection model with delay and general incidence function is formulated and analyzed. Theoretical results reveal that positive equilibrium may lose its stability, and Hopf bifurcation occurs when choosing delay as the bifurcation parameter. The direction of Hopf bifurcation and the stability of the periodic solutions are also discussed. Furthermore, to illustrate the numerous changes in the local stability and instability of the positive equilibrium, we conduct numerical simulations by using four different types of functional incidence, i.e., bilinear incidence, saturation incidence, Beddington–DeAngelis response, and Hattaf–Yousfi response. Rich dynamics of the model, such as Hopf bifurcations and chaotic solutions, are presented numerically.

Analysis of a Delayed Predator–Prey System with Harvesting

2018 ◽  
Vol 19 (3-4) ◽  
pp. 335-349
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
Center Manifold ◽  
Differential Algebra ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
The Stability

AbstractThis article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

Persistence, Stability and Hopf Bifurcation in a Diffusive Ratio-Dependent Predator–Prey Model with Delay

2014 ◽  
Vol 24 (07) ◽  
pp. 1450093 ◽  
Author(s):  
Yongli Song ◽  
Yahong Peng ◽  
Xingfu Zou
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Boundary Equilibrium ◽  
Ratio Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

scholarly journals Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaojian Zhou ◽  
Xin Chen ◽  
Yongzhong Song
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Bioeconomic Model ◽  
Numerical Tests ◽  
The Stability

We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2149-2157 ◽  
Author(s):  
JUNJIE WEI ◽  
DEJUN FAN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Higher Dimensional ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Equation With Delay ◽  
Bendixson Criterion

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

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