scholarly journals Turing instability and spatially homogeneous Hopf bifurcation in a diffusive Brusselator system

2020 ◽  
Vol 25 (4) ◽  
Author(s):  
Xiang-Ping Yan ◽  
Pan Zhang ◽  
Cun-Huz Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatial Diffusion ◽  
Positive Equilibrium ◽  
Local Asymptotic Stability ◽  
Ode System ◽  
Homogeneous Neumann Boundary Condition ◽  
Spatially Homogeneous

The present paper deals with a reaction–diffusion Brusselator system subject to the homogeneous Neumann boundary condition. When the effect of spatial diffusion is neglected, the local asymptotic stability and the detailed Hopf bifurcation of the unique positive equilibrium of the associated ODE system are analyzed. In the stable domain of the ODE system, the effect of spatial diffusion is explored, and local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are demonstrated. In addition, the direction of spatially homogeneous Hopf bifurcation and the stability of the spatially homogeneous bifurcating periodic solutions are also investigated. Finally, numerical simulations are also provided to check the obtained theoretical results.

Dynamics Analysis in a Gierer–Meinhardt Reaction–Diffusion Model with Homogeneous Neumann Boundary Condition

2019 ◽  
Vol 29 (09) ◽  
pp. 1930025 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Ya-Jun Ding ◽  
Cun-Hua Zhang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Positive Equilibrium ◽  
Reaction Diffusion Model ◽  
Homogeneous Neumann Boundary Condition

A reaction–diffusion Gierer–Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction–diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.

SPATIOTEMPORAL DYNAMICS OF A DIFFUSIVE LESLIE-TYPE PREDATOR–PREY MODEL WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE

2020 ◽  
Vol 28 (03) ◽  
pp. 785-809
Author(s):  
YAN LI ◽  
LINYAN ZHANG ◽  
DAGEN LI ◽  
HONG-BO SHI
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Homogeneous Model ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Asymptotic Stability ◽  
Diffusive Model

In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.

Spatiotemporal Dynamics and Pattern Formations of an Activator-Substrate Model with Double Saturation Terms

2021 ◽  
Vol 31 (09) ◽  
pp. 2150129
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
Juandi Xia ◽  
You Li
Keyword(s):  
Hopf Bifurcation ◽  
Dimensional Space ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Linear Operators ◽  
Homogeneous Neumann Boundary Condition ◽  
Subcritical Hopf Bifurcation ◽  
Pattern Formations ◽  
Substrate Model

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

Stability and Bifurcation Analysis in the Photosensitive CDIMA System with Delayed Feedback Control

2017 ◽  
Vol 27 (11) ◽  
pp. 1750177 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Positive Equilibrium ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Theoretical Results

A diffusive photosensitive CDIMA system with delayed feedback subject to Neumann boundary conditions is considered. We derive the conditions of the occurrence of Turing instability. We also investigate the influence of delay on the stability of the positive equilibrium of the system, and prove that delay induces the occurrence of Hopf bifurcation. By computing the normal form on the center manifold, we give the formulas determining the properties of the Hopf bifurcation. Finally, we give some numerical simulations to support and strengthen the theoretical results. Our study shows that diffusion and delayed feedback can effect the stability of the equilibrium of the system.

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.

scholarly journals Bifurcation Analysis in a Diffusion Mussel-Algae Interaction System with Delays Considering the Half-Saturation Constant

2021 ◽  
Author(s):  
Zhichao Jiang ◽  
Weicong Zhang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Delayed Reaction ◽  
Half Saturation Constant ◽  
Two Delays ◽  
Steady State Bifurcation ◽  
The Stability

Abstract In this paper, the kinetics of a class of delayed reaction-diffusion musselalgae system under Neumann boundary conditions with the half- saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affect the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among parameters, the half-saturation constant and two delays drive the complexity of the system dynamics.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

scholarly journals Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kejun Zhuang ◽  
Gao Jia ◽  
Dezhi Liu
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Time Behavior ◽  
Delayed Reaction ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Long Time ◽  
The Stability

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

2014 ◽  
Vol 07 (04) ◽  
pp. 1450045 ◽  
Author(s):  
Qinglai Dong ◽  
Wanbiao Ma
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Positive Equilibrium ◽  
Substrate Uptake ◽  
Sufficient Condition ◽  
Chemostat Model ◽  
Local Asymptotic Stability

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

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