Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay

2012 ◽  
Vol 23 (6) ◽  
pp. 777-796 ◽  
Author(s):  
RUI HU ◽  
YUAN YUAN
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Laboratory Data ◽  
Steady State Solution ◽  
Steady States ◽  
Nicholson’S Blowflies Equation ◽  
Non Local ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

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 Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. Jin ◽  
C. Z. Wang
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Financial System ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

scholarly journals EFFECT OF TIME DELAY ON SPATIAL PATTERNS IN A AIRAL INFECTION MODEL WITH DIFFUSION

2016 ◽  
Vol 21 (2) ◽  
pp. 143-158
Author(s):  
Jia Liu ◽  
Qunying Zhang ◽  
Canrong Tian
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Viral Infection ◽  
Spatial Patterns ◽  
Infection Model ◽  
The Stability ◽  
Manifold Theory ◽  
Viral Infection Model ◽  
Theoretical Results

This paper is concerned with the dynamics of a viral infection model with diffusion under the assumption that the immune response is retarded. A time delay is incorporated into the model described the delayed immune response after viral infection. Based upon a stability analysis, we demonstrate that the appearance, or the absence, of spatial patterns is determined by the delay under some conditions. Moreover, the spatial patterns occurs as a consequence of Hopf bifurcation. By applying the normal form and the center manifold theory, the direction as well as the stability of the Hopf bifurcation is explored. In addition, a series of numerical simulations are performed to illustrate our theoretical results.

scholarly journals Hopf Bifurcation of an SIQR Computer Virus Model with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Computer Virus ◽  
Center Manifold Theory ◽  
Virus Prevalence ◽  
Virus Model ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

A delayed SIQR computer virus model is considered. It has been observed that there exists a critical value of delay for the stability of virus prevalence by choosing the delay as a bifurcation parameter. Furthermore, the properties of the Hopf bifurcation such as direction and stability are investigated by using the normal form method and center manifold theory. Finally, some numerical simulations for supporting our theoretical results are also performed.

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 Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

Global Hopf Bifurcation in a Phytoplankton–Zooplankton System with Delay and Diffusion

2021 ◽  
Vol 31 (08) ◽  
pp. 2150114
Author(s):  
Ming Liu ◽  
Jun Cao ◽  
Xiaofeng Xu
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Upper And Lower Solutions ◽  
Positive Periodic Solutions ◽  
Global Hopf Bifurcation ◽  
System With Delay ◽  
Local Hopf Bifurcation ◽  
Distribution Of Eigenvalues ◽  
The Stability ◽  
And Diffusion

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.

DELAYED BIFURCATION IN FIRST-ORDER SINGULARLY PERTURBED PROBLEMS WITH A NONGENERIC TURNING POINT

2012 ◽  
Vol 22 (12) ◽  
pp. 1250302 ◽  
Author(s):  
JIANHE SHEN ◽  
MAOAN HAN
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Turning Point ◽  
Upper And Lower Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
First Order ◽  
Delayed Bifurcation ◽  
Dynamical Properties ◽  
Theoretical Results

In this paper, based on the method of upper and lower solutions, delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point is studied. The asymptotic behavior of the solutions is understood by constructing the upper and lower solutions with the desired dynamical properties. As an application of the obtained results, delayed phenomenon of degenerate Hopf bifurcation in a planar polynomial differential system with a slowly varying parameter is discussed in detail and the maximal delay is calculated. Numerical simulations are carried out to verify the theoretical results.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

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