Stability Analysis of Gray-Scott Model in One-dimension

Abbas Y. Al-Bayati Saad A. Manaa Abdulghafor M. Al-Rozbayani abdulghafor_rozbayani@uomosul.edu.iq  College of Computer Sciences and Mathematics,  University of Mosul, Iraq  Received on: 27/03/2007 Accepted on: 28/06/2007 ABSTRACT  In this Paper, we studied the stability analysis of steady state solutions of Gray-Scott Model in one-dimension using Fourier mode and we showed that the solutions are conditionally stable.


Introduction:
A system of nature whatever that exists in stable state, in one sense or another, if small disturbance or change in the system, does not exist in time-dependent state in which the planets move about the sun in an orderly fashion. It is known that small additional celestial body is introduced into the system, and then the original state is stable to small disturbance. Similar equations of stability arise in every physical problem [5].
Saad and Bashar, studied the stability of a model of a fully developed laminar fluid flow in rectangular bend duct with secondary flow has been disturbed [9].
Sherratt [12] derived a condition for the wave train itself to be a stable solution, and present numerical evidence for a complex sequence of bifurcations in the unstable region of parameter space in reaction-diffusion equations.
Wan et. al [13] is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including on not previously considered. Gray  ) forms the basis for the simplest homogenous system to display "exotic" behavior, even under well-stirred, isothermal, open conditions (CSTR). They find multi stability, hysterias, extinction and anomalous relaxation times.
Scott [11] showed that the reactions that representations of nonlinear chemical feedback in an isothermal system are the proper type autocatalytic steps quadratic u v 2v +→ rat=k1 uv cubic u 2v 3v +→ rat=k1 uv 2 are coupled with the diffusion of the reactants through a permeable boundary form and an external reservoir when the concentrations are held constant.
In this paper, we study the steady state solution and disturbance cases, when the wave amplitudes are constants of the system parameters. The Gray-Scott scheme, which presents cubic-autocatalysis with linear catalyst decay, has been much considered, because of its multiple steadystate response and oscillatory solutions for review and descriptions of much of this work. The scheme is u+2v → 3v, rate = uv 2 , v → w, rate =v, …(1) Where the concentrations of the reactant and autocatalyst are u and v, respectively. The parameters  and  are rate constants. The catalyst is not stable, but undergoes a simple linear decay to a product w. This allows a much wider variety of behavior in the system, than does the cubic reaction alone ( [3], [4]). The nonlinear phenomena may be due to feedback through the detailed chemical mechanism or through departure from the isothermal state [1].

The Mathematical Model
The cubic-autocatalytic reaction with linear decay (1) is considered in a non dimensional reaction-diffusion cell are as follows: The system (2) is in non-dimensional form with the concentrations of the reactant and autocatalyst given by u and v, respectively. The reactor has a permeable boundary at x = 1. Joined to a reservoir which contains u and v at constant concentrations. The boundary condition at x = 0 is a symmetry condition an identical reservoir is located at x = -1. The system is characterized by three non-dimensional parameters. The ratio of the autocatalyst and reactant concentrations in the reservoir is v0. The parameter  is a measure of the importance of the reaction terms, compared with diffusion, while  is a measure of the importance of autocatalyst decay, compared with the cubic-reaction. The simplest way to adjust the nondimensional parameters experimentally is by changing the reservoir concentrations.
Other possibilities for varying the non-dimensional parameters include changing the diffusivity of the system or the length of the reactor. The diffusivity could be changed by adjusting the temperature or by the addition of otherwise inactive salts [7].

Stability of the model:
We study the analysis of non-dimensional Gray-Scott model in one dimension (2) using Fourier mode, and we assume that the value of concentrations of the reactant u and autocalyst v, has the following form [5]: where 1 u and 1 v denote the steady state case and 2 u and 2 v denote the disturbance case. If we substitute (3) in (2) and neglecting the nonlinear terms, we get the following two systems (steady state system): (4d) and the disturbance system has the form:

Steady state case solutions:
For treatment of stability of the model, first the whole solution of the steady state case , we solve equation (4b) for v1 we get: where S e e .

Disturbance case:
When the wave amplitudes are constants, stability analysis has been recently studied by numerous authors [6] and it is of great interest because of the growing industrial importance.
Assume that 2 u and 2 v has the following form [8]: where (c = c1 + ic2), is an eigenvalue represent the speed of the wave, y1 and y2 are constants and k is the wave number. The problem is stable if the linearized equation corresponds to eigenvalue c with negative part (c2 < 0) for presented configurations [5]. Now, if we substitute (7) in the equations (5a), we shall get:  if c2 < 0 then the system is stable.
if c2 = 0 then get the neutral curve.

CONCLUSION
From stability analysis of steady state solutions we conclude that the system is stable when c2 < 0.