scholarly journals Noise-induced spatial pattern formation in stochastic reaction-diffusion systems

2012 ◽  
Author(s):  
Yutaka Hori ◽  
Shinji Hara
Keyword(s):  
Pattern Formation ◽  
Spatial Pattern ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatial Pattern Formation ◽  
Stochastic Reaction

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2011 ◽  
pp. 205-213
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this chapter, we establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

scholarly journals HIERARCHICAL MODELS FOR SPATIAL PATTERN FORMATION IN BIOLOGY

1995 ◽  
Vol 03 (04) ◽  
pp. 987-997 ◽  
Author(s):  
P. K. MAINI
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Spatial Pattern ◽  
Diffusion Model ◽  
Hierarchical Models ◽  
Reaction Diffusion ◽  
Skeletal Patterning ◽  
Reaction Diffusion Model ◽  
Spatial Pattern Formation ◽  
Set Up

We review some recent work investigating a hierarchy of patterning processes in which a reaction-diffusion model forms the top level. In one such hierarchy, it is assumed that the boundary is differentiated, and it is shown that this can greatly enhance the robustness of the patterns subsequently formed by the reaction-diffusion model. In the second, a spatial heterogeneity in background environment is first set-up by a simple gradient model. The resulting patterns produced by the reaction-diffusion system may be isolated to specific parts of the domain. The application of such hierarchical models to skeletal patterning in the tetrapod limb is considered.

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