Pattern dynamics in a Gierer–Meinhardt model with a saturating term

Song, Yongli; Yang, Rui; Sun, Gui‐Quan

doi:10.1016/j.apm.2017.01.081

Cited by 38 publications

(9 citation statements)

References 36 publications

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“…Since then, the dynamics of pattern formation has attracted wide attention in biology [2][3][4], chemistry [5], aquatic ecosystem [6,7], embryogenesis [8][9][10], and so on. With the application of spatial patterns in multifarious ecosystems systems, the stability of pattern solutions [11,12], the stability of the positive equilibrium solutions, the Hopf bifurcation at the critical point, the Turing bifurcation in different reaction-diffusion systems, and the dynamical properties in Turing-Hopf bifurcation point have been getting a significant progress on the stability of pattern formation [13].…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

2021

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In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

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Section: Introductionmentioning

confidence: 99%

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

2021

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“…Turing structures arise when imbalances in diffusion rates make a stable steadystate system sensitive to small heterogeneous perturbations. The ideas have profoundly influenced theoretical understanding of pattern formation in nonlinear optics (see [1], [25] and [49]), fluid dynamic (see [7] and [29]), colloidal lithography [46], epidemiology (see [35] and [40]), cell biology [17], solid-liquid-vapour system [31], biological morphogenesis (see [44], [9], [16], [38] and [43]), nanostructure (see [4], [13], [19] and [41]), biochemical networks [42], phytoplankton dynamics [21], material science (see [5], [13], [39] and [6]), electrochemistry [23], chemical reaction (see [44], [32], [20], [19] and [41]), social science [2], etc.…”

Section: Introductionmentioning

confidence: 99%

The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

Zhang

Yan

2020

Applied Mathematics and Computation

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For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.

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“…In , Turing explained that the involvement of diffusion, under some circumstances, could lead to pattern forming instability, called “Turing instability”. Later developments in this direction may be found in references . A general one‐dimensional two species (chemicals) coupled reaction–diffusion model may be defined as:

\begin{matrix} u_{t} - d_{u} u_{xx} & = f ((),, u, v), ((),, t, x) \in ((],, 0, T) \times Ω, \end{matrix}

\begin{matrix} v_{t} - d_{v} v_{xx} & = g ((),, u, v), ((),, t, x) \in ((],, 0, T) \times Ω, \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Yadav

Jiwari

2018

Numerical Methods Partial

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In this article, we establish the existence and uniqueness of solutions to the coupled reaction-diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank-Nicolson and the predictor-corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second-order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme. KEYWORDS a priori error estimate, Brusselator, coupled reaction-diffusion models, Crank-Nicolson method, existence and uniqueness, Galerkin method, Gray-Scott, predictor-corrector method INTRODUCTIONThe coupled reaction-diffusion models frequently arise in the field of chemistry [1, 2], biology [3,4], sociology [5], physics, geology, ecology, and so forth. These models are naturally applied in chemistry, for example, the Brusselator model. However, such models can also describe dynamical processes of nonchemical nature, for instance, the predator-prey model. In 1952, in his pioneering work on morphogenesis, Turing [6] first proposed that the reaction-diffusion systems may be used to study the replicating patterns such as stripes, spots, and dappling, seen on the skin of many animals such as zebras, lions, and cats. In [6], Turing explained that the involvement of diffusion, Numer Methods Partial Differential Eq. 2019;35:830-850. wileyonlinelibrary.com/journal/num

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Pattern dynamics in a Gierer–Meinhardt model with a saturating term

Cited by 38 publications

References 36 publications

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

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