STABILITY OF BIFURCATED PERIODIC SOLUTIONS IN A DELAYED COMPETITION SYSTEM WITH DIFFUSION EFFECTS

2009 ◽  
Vol 19 (03) ◽  
pp. 857-871 ◽  
Author(s):  
XIANG-PING YAN ◽  
WAN-TONG LI
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Theoretical Predictions ◽  
Form Theory ◽  
Functional Differential ◽  
The Stability

In this paper, a delayed Lotka—Volterra two species competition diffusion system with a single discrete delay and subject to the homogeneous Dirichlet boundary conditions is considered. By applying the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), the stability of bifurcated periodic solutions occurring through Hopf bifurcations is studied. It is shown that the bifurcated periodic solution occurring at the first bifurcation point is orbitally asymptotically stable on the center manifold while those occurring at other bifurcation points are unstable. Finally, some numerical simulations to a special example are included to verify our theoretical predictions.

HOPF BIFURCATION AND STABILITY OF PERIODIC SOLUTIONS IN THE DELAYED LIÉNARD EQUATION

2008 ◽  
Vol 18 (10) ◽  
pp. 3147-3157 ◽  
Author(s):  
GUANG-PING HU ◽  
WAN-TONG LI ◽  
XIANG-PING YAN
Keyword(s):  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Liénard Equation ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Trivial Equilibrium ◽  
Functional Differential ◽  
The Stability ◽  
Lienard Equation ◽  
Zero Equilibrium

In this paper, the classical Liénard equation with a discrete delay is considered. Under the assumption that the classical Liénard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for functional differential equations. Finally, in order to verify these theoretical conclusions, some numerical simulations are given.

LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL

2004 ◽  
Vol 14 (11) ◽  
pp. 3909-3919 ◽  
Author(s):  
YONGLI SONG ◽  
JUNJIE WEI ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Global Hopf Bifurcation ◽  
Local Hopf Bifurcation ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.

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.

scholarly journals Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Luca Guerrini ◽  
Giovanni Molica Bisci
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Production Function ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

scholarly journals Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay

2014 ◽  
Vol 19 (1) ◽  
pp. 132-153 ◽  
Author(s):  
Wenjie Zuo ◽  
Junjie Wei
Keyword(s):  
Functional Response ◽  
Functional Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
The Maximum Principle ◽  
Boundary Equilibrium ◽  
Ratio Dependent ◽  
Functional Differential ◽  
The Stability

A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xin-You Meng ◽  
Li Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Functional Differential

In this paper, a diffusion two-phytoplankton one-zooplankton model with time delay, Beddington–DeAnglis functional response, and Holling II functional response is proposed. First, the existence and local stability of all equilibria of such model are studied. Then, the existence of Hopf bifurcation of the corresponding model without diffusion is given by taking time delay as the bifurcation parameter. Next, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and center manifold theorem. Furthermore, due to the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the model is investigated by considering time delay as the bifurcation parameter. The explicit formulas to determine the properties of Hopf bifurcation are given by the method of the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are performed to check out our theoretical results.

scholarly journals Stability and Sensitive Analysis of a Model with Delay Quorum Sensing

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Zhonghua Zhang ◽  
Yaohong Suo ◽  
Juan Zhang
Keyword(s):  
Normal Form ◽  
Center Manifold ◽  
Reduction Method ◽  
Delay Model ◽  
Threshold Parameter ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

This paper formulates a delay model characterizing the competition between bacteria and immune system. The center manifold reduction method and the normal form theory due to Faria and Magalhaes are used to compute the normal form of the model, and the stability of two nonhyperbolic equilibria is discussed. Sensitivity analysis suggests that the growth rate of bacteria is the most sensitive parameter of the threshold parameterR0and should be targeted in the controlling strategies.

STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY

2009 ◽  
Vol 19 (07) ◽  
pp. 2283-2294 ◽  
Author(s):  
CUN-HUA ZHANG ◽  
XIANG-PING YAN
Keyword(s):  
Functional Differential Equations ◽  
Distributed Delay ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Functional Differential ◽  
The Stability

This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

BIFURCATION ANALYSIS FOR A PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND TIME DELAY

2008 ◽  
Vol 01 (02) ◽  
pp. 209-224 ◽  
Author(s):  
QINTAO GAN ◽  
RUI XU ◽  
PINGHUA YANG
Keyword(s):  
Time Delay ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

In this paper, a predator-prey model with prey dispersal and time delay is investigated. By analyzing the corresponding characteristic equation of a positive equilibrium, the local stability of the positive equilibrium and the existence of Hopf bifurcation are discussed. By using the normal form theory and center manifold reduction, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Numerical simulations are given to illustrate the theoretical predictions.

scholarly journals Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Computer Simulations ◽  
Center Manifold ◽  
Volterra Model ◽  
Normal Form Theory ◽  
Biological Meaning ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

This paper deals with a competitor-competitor-mutualist Lotka-Volterra model. A series of sufficient criteria guaranteeing the stability and the occurrence of Hopf bifurcation for the model are obtained. Several concrete formulae determine the properties of bifurcating periodic solutions by applying the normal form theory and the center manifold principle. Computer simulations are given to support the theoretical predictions. At last, biological meaning and a conclusion are presented.

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