Three-variable reversible Gray–Scott model

Abstract

Even though the field of nonequilibrium thermodynamics has been popular and its importance has been suggested by Demirel and Sandler ͓J. Phys. Chem. B 108, 31 ͑2004͔͒, there are only a few investigations of reaction-diffusion systems from the aspect of thermodynamics. A possible reason is that model equations are complicated and difficult to analyze because the corresponding chemical reactions need to be reversible for thermodynamical calculations. Here, we introduce a simple model for calculation of entropy production rate: a three-variable reversible Gray-Scott model. The rate of entropy production in self-replicating pattern formation is calculated, and the results are compared with those reported based on the Brusselator model in the context of biological cell division.

Structural stability and stabilization of solutions of the reversible three‐component Gray‐Scott system

“…By experimental and numerical investigations, as a reaction‐diffusion system, the pattern formation behavior of the model was explored in Pearson across a range of parameter values. As a modification of the irreversible Gray‐Scott model, the three‐component reversible Gray‐Scott model was first introduced in Mahara et al This model is based on the following set of two reversible chemical or biochemical reactions: where A and B are the reactants and P is the product of the reactions, k 1 and k 2 are the reaction rate constants of the first and second forward reactions, respectively, and k −1 and k −2 are the rate constants of the first and second backward reactions, respectively. In this case, since both the reactions are reversible, the system has three components A , B , and P .…”

“…under the Neumann and Dirichlet boundary conditions. Here, Ω is a bounded domain in R n (n ≤ 3) with smooth boundary Ω; u = u(x, t), v = v(x, t), and w = w(x, t) are the concentrations of a two step chemical reactions (see (4) below), and d i for i = 1, 2, 3 and F, G, k and N are all positive dimensionless constants. Introduced in Gray and Scott, 1,2 the original Gray-Scott model describes the dynamics of a nondiffusive, isothermal, cubic autocatalytic reactions in a continuously flowing, well-stirred gel reactor, which corresponds to the following scheme of two irreversible chemical reactions…”

Calculation of the Entropy Balance Equation in a Non-equilibrium Reaction-diffusion System

“…Then this principle can be applied for many phenomena. However, some numerical studies have failed to find a unique relation between entropy production and dissipative structures [26][27][28][29][30][31]. For example, the value of entropy production averaged over one cycle of oscillations in the oscillatory Oregonator can be both lower or higher than that of the corresponding unstable steady state, depending on the bifurcation parameter [26].…”

“…This is the reason why reversible chemical reactions are introduced. This model is called the reversible Gray-Scott model, which is modified from the original Gray-Scott model [29]. The model consists of two chemical reaction steps and three chemical spices U, W and P:…”

“…In this article, we use the reversible Gray-Scott (r-GS) model as a reaction-diffusion system [29]. First, we calculate the entropy production and show the relations between this quantity and pattern formation processes.…”