scholarly journals The Spatially Homogeneous Hopf Bifurcation Induced Jointly by Memory and General Delays in a Diffusive System

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Spatially Homogeneous ◽  
Diffusive System

scholarly journals The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Gestation Delay ◽  
Predator Prey Model ◽  
Spatially Homogeneous ◽  
Type 3 ◽  
With Memory ◽  
Diffusive System

Abstract In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in a diffusive system with memory and general delays. The developed algorithm for calculating the normal form can be used to investigate the direction and stability of Hopf bifurcation. Then, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without considering gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. Furthermore, by using the developed algorithm for calculating the normal form, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found theoretically. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.

A DIRECT COMPUTATION OF THE PHASE STABILITY CRITERION FOR SPATIALLY HOMOGENEOUS TIME PERIODIC SOLUTIONS CLOSE TO A HOPF BIFURCATION

2001 ◽  
Vol 11 (08) ◽  
pp. 2097-2103
Author(s):  
EMMANUEL RISLER
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Phase Stability ◽  
Stability Criterion ◽  
Linear Phase ◽  
Direct Computation ◽  
Classical Criterion ◽  
Time Periodic Solutions ◽  
Spatially Homogeneous ◽  
Time Periodic

We provide an elementary direct computation of the classical criterion governing the (linear) phase stability under inhomogeneous perturbations of a spatially homogeneous periodic orbit close to a spatially homogeneous supercritical Hopf bifurcation.

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.

Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

2017 ◽  
Vol 27 (13) ◽  
pp. 1750194 ◽  
Author(s):  
Chengxian Li ◽  
Haihong Liu ◽  
Tonghua Zhang ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Time Delays ◽  
Noncoding Rna ◽  
Small Noncoding Rna ◽  
Spatially Inhomogeneous ◽  
Gene Regulatory ◽  
Spatially Homogeneous ◽  
The Stability ◽  
And Diffusion

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

ON TURING-HOPF INSTABILITIES IN REACTION-DIFFUSION SYSTEMS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 257-274 ◽  
Author(s):  
MARIANO RODRÍGUEZ RICARD
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Hopf Bifurcations ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Spatially Homogeneous

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

scholarly journals Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping

2020 ◽  
Vol 15 ◽  
pp. 23 ◽  
Author(s):  
Fethi Souna ◽  
Salih Djilali ◽  
Fayssal Charif
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
State Density ◽  
Spatial Diffusion ◽  
Herd Behavior ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusive System

In this paper, we consider a new approach of prey escaping from herd in a predator-prey model with the presence of spatial diffusion. First, the sensitivity of the equilibrium state density with respect to the escaping rate has been studied. Then, the analysis of the non diffusive system was investigated where boundedness, local, global stability, Hopf bifurcation are obtained. Besides, for the diffusive system, we proved the occurrence of Hopf bifurcation and the non existence of diffusion driven instability. Furthermore, the direction of Hopf bifurcation has been proved using the normal form on the center manifold. Some numerical simulations have been used to illustrate the obtained results.

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.

Bifurcations in Twinkling Patterns for the Lengyel–Epstein Reaction–Diffusion Model

2021 ◽  
Vol 31 (11) ◽  
pp. 2150164
Author(s):  
J. Sarría-González ◽  
Ivonne Sgura ◽  
M. R. Ricard
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Diffusion Model ◽  
Reaction System ◽  
Reaction Diffusion ◽  
Spatially Inhomogeneous ◽  
Reaction Diffusion Model ◽  
Spatially Homogeneous ◽  
Time Periodic ◽  
Theoretical Results

Conditions for the emergence of strong Turing–Hopf instabilities in the Lengyel–Epstein CIMA reaction–diffusion model are found. Under these conditions, time periodic spatially inhomogeneous solutions can be induced by diffusive instability of the spatially homogeneous limit cycle emerging at a supercritical Bautin–Hopf bifurcation about the unstable steady state of the reaction system. We report numerical simulations by an Alternating Directions Implicit (ADI) method that show the formation of twinkling patterns for a chosen parameter value, thus confirming our theoretical results.

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