STABILITY AND HOPF BIFURCATION FOR A DELAYED COOPERATIVE SYSTEM WITH DIFFUSION EFFECTS

2008 ◽  
Vol 18 (02) ◽  
pp. 441-453 ◽  
Author(s):  
XIANG-PING YAN ◽  
WAN-TONG LI
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Critical Values ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Spatially Homogeneous ◽  
Functional Differential

The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

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.

scholarly journals Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Yuzhen Bai ◽  
Xiaopeng Zhang
Keyword(s):  
Functional Response ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Small Diffusion ◽  
Predator Prey ◽  
Prey System ◽  
Functional Differential

This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.

scholarly journals Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

2019 ◽  
Vol 29 (11) ◽  
pp. 1950144 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Center Manifold Reduction ◽  
Existence Of Positive Solutions ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
The Stability

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

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 Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

scholarly journals Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
Dejun Fan ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
Functional Differential ◽  
Delay Differential ◽  
Equivariant Hopf Bifurcation

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

scholarly journals Hopf bifurcations in a three-species food chain system with multiple delays

2017 ◽  
Vol 15 (1) ◽  
pp. 508-519 ◽  
Author(s):  
Xiaoliang Xie ◽  
Wen Zhang
Keyword(s):  
Food Chain ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Chain System ◽  
Functional Differential ◽  
The Stability

Abstract This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Multiple Bifurcation Analysis in a Diffusive Eco-Epidemiological Model with Time Delay

2019 ◽  
Vol 29 (03) ◽  
pp. 1950033
Author(s):  
Nayyereh Babakordi ◽  
Hamid R. Z. Zangeneh ◽  
Mojtaba Mostafavi Ghahfarokhi
Keyword(s):  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Steady States ◽  
Epidemiological Model ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Functional Differential

In this paper, a delayed eco-epidemiological model with diffusion effects and homogeneous Neumann boundary conditions is proposed. Sufficient conditions for the occurrence of the Hopf-zero, Takens–Bogdanov and saddle-node bifurcations at several steady states are derived. By taking the delay as the bifurcation parameter, it was shown that spatially homogeneous and nonhomogeneous Hopf bifurcations occur at several steady states for a sequence of critical values of the delay parameter. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we present the explicit formula for determining the properties of spatial Hopf bifurcations. Some numerical simulations are carried out.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

2020 ◽  
Vol 30 (09) ◽  
pp. 2050130 ◽  
Author(s):  
Shangzhi Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Banach Spaces ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delay Effect ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Equivariant Hopf Bifurcation ◽  
Manifold Reduction

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

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