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.

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.

Spatiotemporal Patterns in a Delayed Reaction–Diffusion Mussel–Algae Model

2019 ◽  
Vol 29 (12) ◽  
pp. 1950164 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Dimensionless Parameter ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Patterns ◽  
Delayed Reaction ◽  
Distribution Of Eigenvalues ◽  
Positivity Of Solutions ◽  
The Stability

We study the spatiotemporal patterns of a delayed reaction–diffusion mussel–algae system subject to Neumann boundary conditions. This paper is a continuation of our previous studies on the mussel–algae model. We prove the global existence and positivity of solutions. By analyzing the distribution of eigenvalues, we obtain the stability conditions for the positive constant steady state, the existence of Hopf bifurcation and the Turing instability. We show the dynamic classification near the Turing–Hopf singularity in the dimensionless parameter space and observe a transiently spatially nonhomogeneous periodic solution in simulations. Both theoretical and numerical results reveal that the Turing–Hopf bifurcation can enrich the diversity of the spatial distribution of populations.

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.

Effects of Delay and Diffusion on the Dynamics of a Leslie–Gower Type Predator–Prey Model

2014 ◽  
Vol 24 (04) ◽  
pp. 1450043
Author(s):  
Jia-Fang Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Periodic Solutions ◽  
Positive Constant ◽  
Neumann Boundary ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Delayed System ◽  
The Stability ◽  
And Diffusion

In this paper, we consider the effects of time delay and space diffusion on the dynamics of a Leslie–Gower type predator–prey system. It is shown that under homogeneous Neumann boundary condition the occurrence of space diffusion does not affect the stability of the positive constant equilibrium of the system. However, we find that the incorporation of a discrete delay representing the gestation of prey species can not only destabilize the positive constant equilibrium of the system but can also cause a Hopf bifurcation at the positive constant equilibrium as it crosses some critical values. In particular, we prove that these Hopf bifurcations' periodic solutions are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as periodic solutions of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system will generate spatially nonhomogeneous periodic solutions. The results in this work demonstrate that diffusion plays an important role in deriving complex spatiotemporal dynamics.

scholarly journals Hopf bifurcation in a delayed reaction–diffusion–advection equation with ideal free dispersal

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunfeng Liu ◽  
Yuanxian Hui
Keyword(s):  
Hopf Bifurcation ◽  
Principal Eigenvalue ◽  
Reaction Diffusion ◽  
Advection Equation ◽  
Delayed Reaction ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Form Theory ◽  
The Stability ◽  
Ideal Free

AbstractIn this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results.

Stability and Hopf Bifurcation of a Reaction–Diffusion Neutral Neuron System with Time Delay

2017 ◽  
Vol 27 (14) ◽  
pp. 1750214 ◽  
Author(s):  
Tao Dong ◽  
Linmao Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neuron System ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
The Stability ◽  
System With Time Delay ◽  
Zero Equilibrium

In this paper, a type of reaction–diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

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.

scholarly journals The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

2018 ◽  
Vol 23 (5) ◽  
pp. 749-770 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Arbitrary Order ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Positive Equilibrium ◽  
Distribution Of Eigenvalues ◽  
The Stability

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSION-COMPETITION SYSTEM WITH DELAYS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250049 ◽  
Author(s):  
JIA-FANG ZHANG ◽  
WAN-TONG LI ◽  
XIANG-PING YAN
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Interspecies Competition ◽  
Globally Asymptotically Stable ◽  
Two Delays ◽  
Delayed System ◽  
The Stability

A competitive Lotka–Volterra reaction-diffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive constant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies competition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters (delays) in the system cross some critical values. In particular, we prove that these Hopf bifurcations are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as Hopf bifurcation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.

A delayed reaction–diffusion viral infection model with nonlinear incidences and cell-to-cell transmission

2021 ◽  
Author(s):  
Qing Ge ◽  
Xia Wang ◽  
Libin Rong
Keyword(s):  
Viral Infection ◽  
Traveling Wave Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Infection Model ◽  
Threshold Dynamics ◽  
Delayed Reaction ◽  
Homogeneous Neumann Boundary Condition ◽  
Cell Transmission ◽  
Viral Infection Model

In this paper, we propose a reaction–diffusion viral infection model with nonlinear incidences, cell-to-cell transmission, and a time delay. We impose the homogeneous Neumann boundary condition. For the case where the domain is bounded, we first study the well-posedness. Then we analyze the local stability of homogeneous steady states. We establish a threshold dynamics which is completely characterized by the basic reproduction number. For the case where the domain is the whole Euclidean space, we consider the existence of traveling wave solutions by using the cross-iteration method and Schauder’s fixed point theorem. Finally, we study how the speed of spread in space affects the spread of cells and viruses. We obtain the existence of the wave speed, which is dependent on the diffusion coefficient.

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