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 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 Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Mohammed Rizwan Sadiq Iqbal
Keyword(s):  
Magnetic Field ◽  
Inertial Force ◽  
Transverse Magnetic ◽  
Nonlinear Analyses ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Strength ◽  
Long Time ◽  
The Stability ◽  
The One

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.

scholarly journals Pressure-driven wrinkling of soft inner-lined tubes

2021 ◽  
Author(s):  
Benjamin Lee Foster ◽  
Nicolás Verschueren ◽  
Edgar Knobloch ◽  
Leonardo Gordillo
Keyword(s):  
Mixed Mode ◽  
Numerical Continuation ◽  
System Response ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Response Measures ◽  
Plane Compression ◽  
Pressure Changes ◽  
Secondary Bifurcations ◽  
Arterial Endothelium

Abstract A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly.

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.

Weakly Nonlinear Stability Analyses of Prototype Reaction-Diffusion Model Equations

SIAM Review ◽  
1994 ◽  
Vol 36 (2) ◽  
pp. 176-214 ◽  
Author(s):  
David J. Wollkind ◽  
Valipuram S. Manoranjan ◽  
Limin Zhang
Keyword(s):  
Diffusion Model ◽  
Nonlinear Stability ◽  
Reaction Diffusion ◽  
Weakly Nonlinear ◽  
Stability Analyses ◽  
Reaction Diffusion Model ◽  
Model Equations

scholarly journals Distributed Parameter State Estimation for the Gray–Scott Reaction-Diffusion Model

Systems ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 71
Author(s):  
Petro Feketa ◽  
Alexander Schaum ◽  
Thomas Meurer
Keyword(s):  
Finite Number ◽  
Exponential Convergence ◽  
Reaction Diffusion ◽  
Distributed Parameter ◽  
System State ◽  
State Information ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Complete State ◽  
The One

A constructive approach is provided for the reconstruction of stationary and non-stationary patterns in the one-dimensional Gray-Scott model, utilizing measurements of the system state at a finite number of locations. Relations between the parameters of the model and the density of the sensor locations are derived that ensure the exponential convergence of the estimated state to the original one. The designed observer is capable of tracking a variety of complex spatiotemporal behaviors and self-replicating patterns. The theoretical findings are illustrated in particular numerical case studies. The results of the paper can be used for the synchronization analysis of the master–slave configuration of two identical Gray–Scott models coupled via a finite number of spatial points and can also be exploited for the purposes of feedback control applications in which the complete state information is required.

On a reaction–diffusion model for sterile insect release method on a bounded domain

2014 ◽  
Vol 07 (03) ◽  
pp. 1450030 ◽  
Author(s):  
Weihua Jiang ◽  
Xin Li ◽  
Xingfu Zou
Keyword(s):  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Whole Space ◽  
The One ◽  
Sterile Insect Release ◽  
Release Method ◽  
Dimensional Domain ◽  
Sterile Insect Release Method

We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method (SIRM). Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci.5 (1992) 221–247] where the habitat is assumed to be the one-dimensional whole space ℝ, we consider this system in a bounded one-dimensional domain (interval). Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that: (i) when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; (ii) when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.

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.

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