scholarly journals Global Hopf bifurcation for two zooplankton-phytoplankton model with two delays

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Renxiang Shi ◽  
Wenguo Yang
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Global Existence ◽  
The Other ◽  
Global Hopf Bifurcation ◽  
Two Delays ◽  
Bifurcating Periodic Solution

Abstract In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017).

Hopf bifurcation analysis of predator–prey model with two delays and disease transmission

2020 ◽  
Vol 13 (07) ◽  
pp. 2050068
Author(s):  
Renxiang Shi
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Global Existence ◽  
Bifurcation Analysis ◽  
Disease Transmission ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Two Delays

In this paper, we study the Hopf bifurcation of predator–prey system with two delays and disease transmission. Furthermore, the global existence of bifurcated periodic solution was studied, the influence of disease transmission is given. At last, some simulations are given to support our result.

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

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.

STABILITY AND HOPF BIFURCATION FOR A THREE-SPECIES REACTION–DIFFUSION PREDATOR–PREY SYSTEM WITH TWO DELAYS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350194
Author(s):  
GAO-XIANG YANG ◽  
JIAN XU
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Bifurcating Periodic Solution

In this paper, a three-species predator–prey system with diffusion and two delays is investigated. By taking the sum of two delays as a bifurcation parameter, it is found that the spatially homogeneous Hopf bifurcation can occur as the sum of two delays crosses a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are obtained by employing the center manifold theorem and the normal form theory. In addition, some numerical simulations are also given to illustrate the theoretical analysis.

scholarly journals Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. 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. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

scholarly journals Global Existence of Periodic Solutions in a Delayed Kaldor-Kalecki Model

2009 ◽  
Vol 14 (4) ◽  
pp. 463-472 ◽  
Author(s):  
A. Kaddar ◽  
H. Talibi Alaoui
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Business Cycle ◽  
Periodic Solutions ◽  
Cycle Model ◽  
Global Hopf Bifurcation ◽  
Numerical Examples ◽  
Existence Of Periodic Solutions ◽  
Business Cycle Model ◽  
Theoretical Results

This paper is concerned with a delayed Kaldor-Kalecki non-linear business cycle model in income. By applying a global Hopf bifurcation result due to Wu, the global existence of periodic solutions is investigated. Numerical examples will be given in the end, to illustrate our theoretical results.

Dynamics of Delayed Phytoplankton–Zooplankton System with Disease Spread Among Zooplankton

2021 ◽  
Vol 31 (12) ◽  
pp. 2150184
Author(s):  
Renxiang Shi
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Global Existence ◽  
Center Manifold ◽  
Disease Spread ◽  
Boundedness Of Solutions ◽  
System With Delay ◽  
Positivity And Boundedness

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JINGNAN WANG ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Local Hopf Bifurcation ◽  
Sunflower Equation ◽  
Nonlinear Relation ◽  
Bifurcation Direction ◽  
The Stability ◽  
Bifurcating Periodic Solution ◽  
Modified Equation

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

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