Schnakenberg model

The Schnakenberg model is a modification of the Brusselator scheme, and its mechanism consists of the following four steps ... 

The Schnakenberg model is a cross activator-inhibitor system iib > a. The deter-minant of the Jacobian is also always positive, h. = a + b), and no stationary bifurcation can occur. The Hopf threshold b is given by the cubic equation... 

For the Schnakenberg model. Sect. 1.4.5, the loss rate of the activator is fy = 1 and the right-hand side of the Hopf condition (10.115) is given by... 

RD,c> see (10.178). In the following we consider the Lengyel-Epstein model and the Schnakenberg model. Sect. 1.4.5, and the Gray-Scott model. Sect. 1.4.7. We will not display the explicit expression (10.184) for 6 for each model, but rather illustrate... 

Fig. 10.4 Plot of 0 yi for the Schnakenberg model with a subdiffusing inhibitor. The parameters are a = 0.1 and b = 0.9. Reprinted with permission from [485]. Copyright 2008 by the American Physical Society...

For modeling Turing patterns in the CIMA reaction other groups [66, 67] have used the Brusselator and the Schnakenberg models. Although these models show some similarities in their bifurcation sequences and qualitative patterns, quantitative comparison or prediction of new features at specific parameter values of the real chemical system are impossible with these models. Moreover, recent numerical studies [68] suggest that there may be qualitative differences as well between these models and the model derived from Equations (l)-(3). For these reasons we have relied exclusively on the CDIMA reaction and its model for both our experiments and our model calculations. 

Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful. Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions). Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the or JE for these. FR... 

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