scholarly journals Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

scholarly journals Dynamical Analysis in a Delayed Predator-Prey System with Stage-Structure for Both the Predator and the Prey

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

A predator-prey system with two delays and stage-structure for both the predator and the prey is considered. Sufficient conditions for the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. Specially, the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are determined by applying the normal form theory and center manifold argument. Some numerical simulations for justifying the theoretical analysis are also provided.

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 Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zizhen Zhang ◽  
Fangfang Yang ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Transmission Model ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Synthetic Drug ◽  
The Stability

This paper is concerned with the Hopf bifurcation of a synthetic drug transmission model with two delays. Firstly, some sufficient conditions of delay-induced bifurcation for such a model are captured by using different combinations of the two delays as the bifurcation parameter. Secondly, based on the center manifold theorem and normal form theory, some sufficient conditions determining properties of the Hopf bifurcation such as the direction and the stability are established. Finally, to underline the effectiveness of the obtained results, some numerical simulations are ultimately addressed.

scholarly journals The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

scholarly journals Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Fengying Wei ◽  
Lanqi Wu ◽  
Yuzhi Fang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma
Keyword(s):  
Hopf Bifurcation ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Explicit Formulas ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
The Stability ◽  
Theoretical Results ◽  
Positivity And Boundedness

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

2014 ◽  
Vol 55 (3) ◽  
pp. 214-231 ◽  
Author(s):  
E. KARAOGLU ◽  
H. MERDAN
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Discrete Delays ◽  
Form Theory ◽  
Two Delays ◽  
Ratio Dependent ◽  
Centre Manifold Theorem ◽  
Analysis Stability

AbstractThe aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

scholarly journals Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Long Li ◽  
Yanxia Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Stability Method ◽  
The Stability ◽  
Delay Differential

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

scholarly journals Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Recurrent Neural Network ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Manifold Theory

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

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