Turing Instability of Brusselator in the Reaction-Diffusion Network

Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

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Delay-Induced Self-Organization Dynamics in a Prey-Predator Network with Diffusion

10.21203/rs.3.rs-1079664/v1 ◽

2021 ◽

Author(s):

Qing Hu ◽

Jianwei Shen

Keyword(s):

Time Delays ◽

Random Network ◽

Turing Instability ◽

Self Organization ◽

Network System ◽

Turing Pattern ◽

Network Connection ◽

Pattern Dynamics ◽

Connection Probability ◽

The Stability

Abstract Time delays can induce the loss of stability and degradation of performance. In this paper, the pattern dynamics of a prey-predator network with diffusion and delay are investigated, where the inhomogeneous distribution of species in space can be viewed as a random network, and delay can affect the stability of the network system. Our results show that time delay can induce the emergence of Hopf and Turing bifurcations, which are independent of the network, and the conditions of bifurcation are derived by linear stability analysis. Moreover, we find that the Turing pattern can be related to the network connection probability. The Turing instability region involving delay and network connection probability is obtained. Finally, the numerical simulation verifies our results.

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Numerical investigation on oscillatory Turing patterns in a two-layer coupled reaction–diffusion system

Modern Physics Letters B ◽

10.1142/s0217984916500858 ◽

2016 ◽

Vol 30 (07) ◽

pp. 1650085 ◽

Cited By ~ 1

Author(s):

Xin-Zheng Li ◽

Zhan-Guo Bai ◽

Yan Li ◽

Kun Zhao

Keyword(s):

Dimensional Space ◽

Reaction Diffusion ◽

Matching Condition ◽

Turing Instability ◽

Reaction Diffusion System ◽

Diffusion System ◽

Square Lattice ◽

Stripe Pattern ◽

Brusselator Model ◽

Dynamic Patterns

In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction–diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.

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Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415500996 ◽

2015 ◽

Vol 25 (08) ◽

pp. 1550099 ◽

Cited By ~ 4

Author(s):

Yongli Cai ◽

Shuling Yan ◽

Hailing Wang ◽

Xinze Lian ◽

Weiming Wang

Keyword(s):

Time Delay ◽

Epidemic Model ◽

Spiral Wave ◽

Reaction Diffusion ◽

Turing Instability ◽

Spatiotemporal Dynamics ◽

Strong Impact ◽

Delayed Reaction ◽

Wave Patterns ◽

And Diffusion

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

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TURING INSTABILITY AND PATTERN FORMATION IN A SEMI-DISCRETE BRUSSELATOR MODEL

Modern Physics Letters B ◽

10.1142/s0217984913500061 ◽

2012 ◽

Vol 27 (01) ◽

pp. 1350006 ◽

Cited By ~ 4

Author(s):

L. XU ◽

L. J. ZHAO ◽

Z. X. CHANG ◽

J. T. FENG ◽

G. ZHANG

Keyword(s):

Turing Instability ◽

Linearization Method ◽

Inner Product ◽

Theory Analysis ◽

System Parameters ◽

Brusselator Model ◽

Instability Theory ◽

Product Technique ◽

The Impact ◽

And Diffusion

In this paper, a semi-discrete Brusselator system is considered. The Turing instability theory analysis will be given for the model, then Turing instability conditions can be deduced combining linearization method and inner product technique. A series of numerical simulations of the system are performed in the Turing instability region, various patterns such as square, labyrinthine, spotlike patterns, can be exhibited. The impact of the system parameters and diffusion coefficients on patterns can also observed visually.

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Pattern formation in a reaction-diffusion rumor propagation system with Allee effect and time delay

10.21203/rs.3.rs-239109/v1 ◽

2021 ◽

Author(s):

Linhe Zhu ◽

Le He

Keyword(s):

Diffusion Coefficient ◽

Time Delay ◽

Allee Effect ◽

Reaction Diffusion ◽

Turing Instability ◽

Small Time ◽

Diffusion Behavior ◽

Rumor Propagation ◽

Reaction Diffusion Model ◽

Stability And Instability

Abstract This paper analyzes the diffusion behavior of the suspicious and the infected cabins in cyberspace by establishing a rumor propagation reaction diffusion model with Allee effect and time delay. The Turing instability conditions of the system under various conditions are emphatically studied. After considering the delay effect of rumor propagation systems, we have studied the correlation between the stability of the system under the influence of small time delay and the homogeneous system near the equilibrium point, and the critical condition of the delay-induced spatial instability is given. Further considering the possibility of diffusion coefficient changing with time, the critical parameter curves of stability and instability of approximate systems are given by means of Floquet theory, and the necessary conditions of Turing-instability of periodic coefficient are studied. In the numerical simulations, we find that the variation of diffusion coefficient will change the pattern type, and the periodical diffusion behavior will affect the arrangement of the crowd gathering area in the pattern.

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Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

Mathematical Problems in Engineering ◽

10.1155/2020/1381095 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Qianqian Zheng ◽

Jianwei Shen

Keyword(s):

Reaction Diffusion ◽

Turing Instability ◽

Amplitude Equation ◽

Diffusion System ◽

Matrix Analysis ◽

Turing Bifurcation ◽

Pattern Dynamics ◽

Novel Approach ◽

Multivariate Pattern ◽

Theoretical Results

In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.

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Multistability and Stochastic Phenomena in the Distributed Brusselator Model

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4045405 ◽

2019 ◽

Vol 15 (1) ◽

Cited By ~ 2

Author(s):

Alexander Kolinichenko ◽

Lev Ryashko

Keyword(s):

Reaction Diffusion ◽

Turing Instability ◽

Spatial Structures ◽

Reaction Diffusion Systems ◽

Corporate Influence ◽

Basic Model ◽

Brusselator Model ◽

Diffusion Systems ◽

Random Disturbances ◽

Stochastic Phenomena

Abstract An influence of random disturbances on the pattern formation in reaction–diffusion systems is studied. As a basic model, we consider the distributed Brusselator with one spatial variable. A coexistence of the stationary nonhomogeneous spatial structures in the zone of Turing instability is demonstrated. A numerical parametric analysis of shapes, sizes of deterministic pattern–attractors, and their bifurcations is presented. Investigating the corporate influence of the multistability and stochasticity, we study phenomena of noise-induced transformation and generation of patterns.

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Analysis of stochastic sensitivity of Turing patterns in distributed reaction-diffusion systems

10.35634/2226-3594-2020-55-10 ◽

2020 ◽

Vol 55 ◽

pp. 155-163

Author(s):

A.P. Kolinichenko ◽

L.B. Ryashko

Keyword(s):

Diffusion Coefficients ◽

Stochastic Dynamics ◽

Random Noise ◽

Reaction Diffusion ◽

Turing Instability ◽

Reaction Diffusion Systems ◽

Brusselator Model ◽

Stochastic Sensitivity ◽

Diffusion Systems ◽

The Mean

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed.

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