scholarly journals Stationary localized structures and the effect of the delayed feedback in the Brusselator model

2018 ◽  
Vol 376 (2135) ◽  
pp. 20170385 ◽  
Author(s):  
B. Kostet ◽  
M. Tlidi ◽  
F. Tabbert ◽  
T. Frohoff-Hülsmann ◽  
S. V. Gurevich ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Delayed Feedback ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Delayed Feedback Control ◽  
Brusselator Model ◽  
Localized Structures ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions

The Brusselator reaction–diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

scholarly journals Spatially localized structures in the Gray–Scott model

2018 ◽  
Vol 376 (2135) ◽  
pp. 20170375 ◽  
Author(s):  
Punit Gandhi ◽  
Yuval R. Zelnik ◽  
Edgar Knobloch
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Dissipative Structures ◽  
Localized States ◽  
Numerical Continuation ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Localized Structures ◽  
Reaction Diffusion Model ◽  
The One

Spatially localized structures in the one-dimensional Gray–Scott reaction–diffusion model are studied using a combination of numerical continuation techniques and weakly nonlinear theory, focusing on the regime in which the activator and substrate diffusivities are different but comparable. Localized states arise in three different ways: in a subcritical Turing instability present in this regime, and from folds in the branch of spatially periodic Turing states. They also arise from the fold of spatially uniform states. These three solution branches interconnect in complex ways. We use numerical continuation techniques to explore their global behaviour within a formulation of the model that has been used to describe dryland vegetation patterns on a flat terrain. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

scholarly journals Time-delayed feedback control of breathing localized structures in a three-component reaction–diffusion system

2014 ◽  
Vol 372 (2027) ◽  
pp. 20140014 ◽  
Author(s):  
Svetlana V. Gurevich
Keyword(s):  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Delayed Feedback Control ◽  
Straightforward Manner ◽  
Localized Structures ◽  
Feedback Strength ◽  
Three Component Reaction ◽  
Time Delayed Feedback

The dynamics of a single breathing localized structure in a three-component reaction–diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov–Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.

scholarly journals Traveling Wave Solutions of a Four Dimensional Reaction-Diffusion Model for Porcine Reproductive and Respiratory Syndrome with Time Dependent Infection Rate

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 30
Author(s):  
Jeerawan Suksamran ◽  
Yongwimon Lenbury ◽  
Sanoe Koonprasert
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Growing Pigs ◽  
Respiratory Syndrome Virus ◽  
Swine Industry ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions ◽  
Tangent Method ◽  
Spatial Transmission

Porcine reproductive and respiratory syndrome virus (PRRSV) causes reproductive failure in sows and respiratory disease in piglets and growing pigs. The disease rapidly spreads in swine populations, making it a serious problem causing great financial losses to the swine industry. However, past mathematical models used to describe the spread of the disease have not yielded sufficient understanding of its spatial transmission. This work has been designed to investigate a mathematical model for the spread of PRRSV considering both time and spatial dimensions as well as the observed decline in infectiousness as time progresses. Moreover, our model incorporates into the dynamics the assumption that some members of the infected population may recover from the disease and become immune. Analytical solutions are derived by using the modified extended hyperbolic tangent method with the introduction of traveling wave coordinate. We also carry out a stability and phase analysis in order to obtain a clearer understanding of how PRRSV spreads spatially through time.

scholarly journals Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1516
Author(s):  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Stelios Bekiros ◽  
Jinping Liu ◽  
Hadi Jahanshahi ◽  
...  
Keyword(s):  
Diffusion Model ◽  
Reaction Diffusion ◽  
Linear Control ◽  
Control Law ◽  
Synchronization Problem ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions ◽  
Error System ◽  
The Stability ◽  
Complex Formations

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions.

scholarly journals Spatio-temporal Brusselator Model and Biological Pattern Formation

2021 ◽  
pp. 88-99
Author(s):  
Zakir Hossine ◽  
Oishi Khanam ◽  
Md. Mashih Ibn Yasin Adan ◽  
Md. Kamrujjaman
Keyword(s):  
Pattern Formation ◽  
Diffusion Model ◽  
Initial Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Brusselator Model ◽  
Reaction Diffusion Model ◽  
Biological Pattern Formation ◽  
Spatio Temporal

This paper explores a two-species non-homogeneous reaction-diffusion model for the study of pattern formation with the Brusselator model. We scrutinize the pattern formation with initial conditions and Neumann boundary conditions in a spatially heterogeneous environment. In the whole investigation, we assume the case for random diffusion strategy. The dynamics of model behaviors show that the nature of pattern formation with varying parameters and initial conditions thoroughly. The model also studies in the absence of diffusion terms. The theoretical and numerical observations explain pattern formation using the reaction-diffusion model in both one and two dimensions.

Dissipative structures and morphogenetic pattern in unicellular algae

1981 ◽  
Vol 294 (1074) ◽  
pp. 547-588 ◽  
Keyword(s):  
Cell Wall ◽  
Cell Division ◽  
Diffusion Mechanism ◽  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Two Dimensions ◽  
Unicellular Algae ◽  
Reaction Diffusion Model ◽  
Surface Ornamentation ◽  
Cell Wall Ingrowths

Patterns of cell wall growth and ornamentation in unicellular algae, mainly in desmids, are compared with patterns generated by Tyson’s Brusselator, a two-morphogen reaction-diffusion model. The model generates hexagonal arrays of points in two dimensions, according well with the observed patterns of surface ornamentation on desmid zygospores. Computed patterns in one dimension and of branching on a circular disc account both qualitatively and quantitatively for morphogenetic patterns that develop following cell division in several desmid genera. Cell wall ingrowths appear to be under similar pattern control to wall outgrowths during morphogenesis, which suggests the involvement of a reaction-diffusion mechanism in establishing and correctly positioning the cell division septum. The application of the model to morphogenesis in Acetabularia and diatoms is also discussed.

Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction–diffusion model of nuclear reactors

2010 ◽  
Vol 140 (1) ◽  
pp. 189-201 ◽  
Author(s):  
Rui Peng ◽  
Dong Wei ◽  
Guoying Yang
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Model ◽  
Blow Up ◽  
Nuclear Reactors ◽  
Reaction Diffusion ◽  
Fast Neutrons ◽  
Fuel Temperature ◽  
Reaction Diffusion Model ◽  
Coexistence States ◽  
Spatial Dimensions

We investigate a non-cooperative reaction-diffusion model arising in the theory of nuclear reactors and are concerned with the associated steady-state problem. We determine the asymptotic behaviour of the coexistence states near the point of bifurcation from infinity, which exhibits the following very interesting spatial blow-up pattern: when the fuel temperature reaches a certain value, the free fast neutrons undergoing nuclear reaction will blow up in each spatial point of the interior of the reactor. Without any restriction on spatial dimensions, we also discuss the uniqueness and stability of the coexistence states. Our results complement and sharpen those derived in two recent works by Arioli and Lóopez-Gómez.

STABILITY AND BIFURCATION ANALYSIS IN A DIFFUSIVE BRUSSELATOR SYSTEM WITH DELAYED FEEDBACK CONTROL

2012 ◽  
Vol 22 (02) ◽  
pp. 1250037 ◽  
Author(s):  
WENJIE ZUO ◽  
JUNJIE WEI
Keyword(s):  
Feedback Control ◽  
Functional Differential Equations ◽  
Turing Instability ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Brusselator Model ◽  
Constant Equilibrium ◽  
The Stability

A diffusive Brusselator model with delayed feedback control subject to Dirichlet boundary condition is considered. The stability of the unique constant equilibrium and the existence of a family of inhomogeneous periodic solutions are investigated in detail, exhibiting rich spatiotemporal patterns. Moreover, it shows that Turing instability occurs without delay. And under certain conditions, the constant equilibrium switches finite times from stability to instability to stability, and becomes unstable eventually, as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the analysis results.

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