scholarly journals Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model

2017 ◽  
Vol 10 (5) ◽  
pp. 1149-1164
Author(s):  
Hongyan Zhang ◽  
◽  
Siyu Liu ◽  
Yue Zhang ◽  
◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Spatiotemporal Pattern ◽  
Thomas Model ◽  
Homogeneous Reaction ◽  
Pattern Formations

scholarly journals Mode selection mechanism in traveling and standing waves revealed by Min wave reconstituted in artificial cells

2022 ◽  
Author(s):  
Sakura Takada ◽  
Natsuhiko Yoshinaga ◽  
Nobuhide Doi ◽  
Kei Fujiwara
Keyword(s):  
Mode Selection ◽  
Standing Waves ◽  
Reaction Diffusion ◽  
Wave Mode ◽  
Spatiotemporal Pattern ◽  
Selection Mechanism ◽  
Artificial Cells ◽  
Pattern Formations ◽  
Selection Of ◽  
Wave Modes

Reaction-diffusion coupling (RDc) generates spatiotemporal patterns, including two dynamic wave modes: traveling and standing waves. Although mode selection plays a significant role in the spatiotemporal organization of living cell molecules, the mechanism for selecting each wave mode remains elusive. Here, we investigated a wave mode selection mechanism using Min waves reconstituted in artificial cells, emerged by the RDc of MinD and MinE. Our experiments and theoretical analysis revealed that the balance of membrane binding and dissociation from the membrane of MinD determines the mode selection of the Min wave. We successfully demonstrated that the transition of the wave modes can be regulated by controlling this balance and found hysteresis characteristics in the wave mode transition. These findings highlight a novel role of the balance between activators and inhibitors as a determinant of the mode selection of waves by RDc and depict a novel mechanism in intracellular spatiotemporal pattern formations.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

LEEM and MEM Studies of Spatiotemporal Pattern Formation

1998 ◽  
Vol 05 (06) ◽  
pp. 1233-1239 ◽  
Author(s):  
K. C. Rose ◽  
W. Engel ◽  
F. Meißen ◽  
A. J. Patchett ◽  
A. M. Bradshaw ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Pattern Formation ◽  
Reaction Diffusion ◽  
Low Energy ◽  
Spatiotemporal Pattern ◽  
Oxidation Of Co ◽  
Spatiotemporal Pattern Formation ◽  
Low Energy Electron ◽  
Stationary Conditions ◽  
Global Coupling

Spatiotemporal pattern formation has been investigated with low energy electron microscopy (LEEM) and mirror electron microscopy (MEM) during the catalytic oxidation of CO on Pt{110} under both oscillatory and stationary conditions. Due to higher resolution and richer contrast than in previous PEEM studies, it has been possible to investigate the details of the nucleation and propagation of reaction–diffusion fronts which make up the patterns, Further, new and more complex patterns have been discovered which demonstrate the importance of both global coupling and reaction-induced roughening.

scholarly journals Spongiosa Primary Development: A Biochemical Hypothesis by Turing Patterns Formations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Oscar Rodrigo López-Vaca ◽  
Diego Alexander Garzón-Alvarado
Keyword(s):  
Endochondral Ossification ◽  
Reaction Diffusion ◽  
Cartilage Degradation ◽  
Reaction Diffusion Equations ◽  
Spatiotemporal Pattern ◽  
Vascular Endothelial ◽  
The Finite Element Method ◽  
Spongy Tissue ◽  
Endothelial Growth Factor ◽  
Raphson Method

We propose a biochemical model describing the formation of primary spongiosa architecture through a bioregulatory model by metalloproteinase 13 (MMP13) and vascular endothelial growth factor (VEGF). It is assumed that MMP13 regulates cartilage degradation and the VEGF allows vascularization and advances in the ossification front through the presence of osteoblasts. The coupling of this set of molecules is represented by reaction-diffusion equations with parameters in the Turing space, creating a stable spatiotemporal pattern that leads to the formation of the trabeculae present in the spongy tissue. Experimental evidence has shown that the MMP13 regulates VEGF formation, and it is assumed that VEGF negatively regulates MMP13 formation. Thus, the patterns obtained by ossification may represent the primary spongiosa formation during endochondral ossification. Moreover, for the numerical solution, we used the finite element method with the Newton-Raphson method to approximate partial differential nonlinear equations. Ossification patterns obtained may represent the primary spongiosa formation during endochondral ossification.

scholarly journals On the Interplay of Geometrical Shapes and the Analysis of a Dispersal Model for Pattern Formations

2019 ◽  
pp. 1-10
Author(s):  
Zakir Hossine ◽  
Md. Kamrujjaman
Keyword(s):  
Reaction Mechanism ◽  
Finite Difference Method ◽  
Reaction Diffusion ◽  
Diffusion Kinetics ◽  
Dispersal Model ◽  
Reaction Diffusion Model ◽  
Chemical Reaction Mechanism ◽  
Geometrical Shapes ◽  
Pattern Formations ◽  
Selection Of

A reaction-diffusion model is put forward which is capable of generating chemical maps whose concentration contours are similar to the patterns seen on the flanks of zebras, leopards and other mammals. Initially, this type of reaction diffusion kinetics model was introduced by Turing and later Murray applied it to animal coat patterns. Among many chemical reaction mechanism, we consider Schnackenberg reaction mechanism in details and show that the geometry and scale of the domain, the relevant part of the integument, during the time of laying down plays a crucial role in the structural patterns which result. Patterns which exhibit a limited randomness are obtained for a selection of geometries. Finally the system was solved numerically using finite difference method.

scholarly journals Correspondences between parameters in a reaction-diffusion model and connexin functions during zebrafish stripe formation

2021 ◽  
Author(s):  
Akiko Nakamasu
Keyword(s):  
Molecular Mechanisms ◽  
Reaction Diffusion ◽  
The Body ◽  
Pigment Cells ◽  
Turing Pattern ◽  
Reaction Diffusion Model ◽  
Spatial Periodicity ◽  
Pattern Formations ◽  
Mathematical Analyses ◽  
Spotted Pattern

Abstract Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.

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