TURING PATTERNS IN GENERAL REACTION-DIFFUSION SYSTEMS OF BRUSSELATOR TYPE

2010 ◽  
Vol 12 (04) ◽  
pp. 661-679 ◽  
Author(s):  
MARIUS GHERGU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Chemical Reactions ◽  
Diffusion Coefficients ◽  
Crucial Role ◽  
Reaction Diffusion ◽  
Turing Patterns ◽  
General Reaction ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Superlinear Growth

We study the reaction-diffusion system [Formula: see text] Here Ω is a smooth and bounded domain in ℝN (N ≥ 1), a, b, d1, d2 > 0 and f ∈ C1[0, ∞) is a non-decreasing function. The case f(u) = u2 corresponds to the standard Brusselator model for autocatalytic oscillating chemical reactions. Our analysis points out the crucial role played by the nonlinearity f in the existence of Turing patterns. More precisely, we show that if f has a sublinear growth then no Turing patterns occur, while if f has a superlinear growth then existence of such patterns is strongly related to the inter-dependence between the parameters a, b and the diffusion coefficients d1, d2.

scholarly journals Analysis of stochastic sensitivity of Turing patterns in distributed reaction-diffusion systems

2020 ◽  
Vol 55 ◽  
pp. 155-163
Author(s):  
A.P. Kolinichenko ◽  
L.B. Ryashko
Keyword(s):  
Diffusion Coefficients ◽  
Stochastic Dynamics ◽  
Random Noise ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Stochastic Sensitivity ◽  
Diffusion Systems ◽  
The Mean

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed.

scholarly journals Stochastic simulation of the spatio-temporal dynamics of reaction-diffusion systems: the case for the bicoid gradient

2010 ◽  
Vol 7 (1) ◽  
Author(s):  
Paola Lecca ◽  
Adaoha E. C. Ihekwaba ◽  
Lorenzo Dematté ◽  
Corrado Priami
Keyword(s):  
Diffusion Coefficient ◽  
Stochastic Simulation ◽  
Diffusion Coefficients ◽  
Temporal Dynamics ◽  
Concentration Field ◽  
Reaction Diffusion ◽  
Local Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

SummaryReaction-diffusion systems are mathematical models that describe how the concentrations of substances distributed in space change under the influence of local chemical reactions, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose solution predicts how diffusion causes the concentration field to change with time. This change is proportional to the diffusion coefficient. If the solute moves in a homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and solute. However, in nonhomogeneous and structured media the assumption of constant intracellular diffusion coefficient is not necessarily valid, and, consequently, the diffusion coefficient is a function of the local concentration of solvent and solutes. In this paper we propose a stochastic model of reaction-diffusion systems, in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces. We then describe the software tool Redi (REaction-DIffusion simulator) which we have developed in order to implement this model into a Gillespie-like stochastic simulation algorithm. Finally, we show the ability of our model implemented in the Redi tool to reproduce the observed gradient of the bicoid protein in the Drosophila Melanogaster embryo. With Redi, we were able to simulate with an accuracy of 1% the experimental spatio-temporal dynamics of the bicoid protein, as recorded in time-lapse experiments obtained by direct measurements of transgenic bicoidenhanced green fluorescent protein.

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