scholarly journals Global classical solutions to reaction-diffusion systems in one and two dimensions

2018 ◽  
Vol 16 (2) ◽  
pp. 411-423 ◽  
Author(s):  
Bao Quoc Tang
Keyword(s):  
Reaction Diffusion ◽  
Two Dimensions ◽  
Classical Solutions ◽  
Reaction Diffusion Systems ◽  
Global Classical Solutions ◽  
Diffusion Systems

The Chemical Basis of Morphogenesis (1952)

2004 ◽  
Author(s):  
Alan Turing
Keyword(s):  
Reaction System ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Stationary Waves ◽  
Chemical Substances ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Physical Laws ◽  
And Diffusion

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogenous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading. In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge. The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Replicating Spots in Reaction-Diffusion Systems

1997 ◽  
Vol 07 (05) ◽  
pp. 1149-1158 ◽  
Author(s):  
Kyoung J. Lee ◽  
Harry L. Swinney
Keyword(s):  
Cell Division ◽  
Numerical Simulations ◽  
Continuous Process ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Diffusion System ◽  
One Dimension ◽  
Reaction Diffusion Systems ◽  
Gel Layer ◽  
Diffusion Systems

We review the phenomenon of replicating spots in reaction-diffusion systems and discuss the mechanism of replication. This phenomenon was discovered in recent experiments on a ferrocyanide-iodate-sulfite reaction-diffusion system. Patterns form in a thin gel layer that is in contact with a continuously fed stirred reservoir. Patterns of spots are observed to undergo a continuous process of growth and multiplication through cell division and death through overcrowding. A similar phenomenon is also found in numerical simulations in one dimension on a four-species model of the ferrocyanide-iodate-sulfite reaction and in simulations in two dimensions of simpler two-species reaction-diffusion models: Gray–Scott model by J. Pearson and FitzHugh–Nagumo model by A. Hagberg and E. Meron.

PATTERN FORMATION IN CHEMOTAXIC REACTION-DIFFUSION SYSTEMS

2012 ◽  
Vol 05 (03) ◽  
pp. 1260013
Author(s):  
HIROTO SHOJI ◽  
KEITARO SAITOH
Keyword(s):  
Free Energy ◽  
Pattern Formation ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Volume Filling ◽  
Reaction Diffusion Systems ◽  
Receptor Aggregation ◽  
Diffusion Systems

In this study, we investigate two-dimensional patterns generated by chemotaxis reaction-diffusion systems. We numerically examine the Keller–Segel models with the volume-filling aggregation term and the receptor aggregation term in two dimensions. Spotted, striped and reversed spotted patterns are obtained as stable motionless equilibrium patterns. The relative stability of these patterns is studied numerically on the basis of the derived free energy. The intuitive understanding of these generated patterns and the relation with three-dimensional patterns are also discussed.

The chemical basis of morphogenesis

1952 ◽  
Vol 237 (641) ◽  
pp. 37-72 ◽  
Keyword(s):  
Reaction System ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Stationary Waves ◽  
Chemical Substances ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Physical Laws ◽  
And Diffusion

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

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