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Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.

In time or space, patterns have nonuniform macroscopic structure with regularity. From the thermodynamic point of view, the nature of the pattern formation can be divided into two categories. One is presenting in the thermodynamic equilibrium conditions, such as the crystal structure of inorganic chemistry, the self-organic pattern formation of organic polymers and so on. The other is the station far from the thermodynamic equilibrium conditions, such as the ripples of the sea, the surface patterns of the animal, the strip clouds in the sky and so on.

Reaction-diffusion system is one of the fundamental equations which describe the motion of the nature. It not only has a wide practical background, but also is used in many fields, for example, predator-prey model, spread of infectious diseases, migration of population and spread of forest fires. Its mathematical model is a special kind of parabolic partial differential equations. As for reaction-diffusion systems, the coupling of nonlinear dynamical and linear diffusion leads to spontaneously producing a variety of ordered or disordered pattern of the system. This is the pattern dynamics of the reaction-diffusion systems [

The classical method to study the Turing patterns of reaction-diffusion is the analysis of linear stability method [

The range of parameters of the Turing pattern can be obtained by the analysis of linear stability method. Based on the parameters which limit to the range, this article solves the reaction-diffusion model by the use of the combination of differential method and homotopy analysis method. Then the changes of the mechanism of Turing patterns can be under control by the simulation of experimental data and analysis of utilizing this method. Finally, the study of the concrete case shows the feasibility and effectiveness of this method.

The general mathematical representation of two-dimensional reaction-diffusion model is as follows,

Their definition are as follows: u and v are different reactant concentrations vector;

Discrete the reaction-diffusion model in discrete nodes

And,

According to the thought of homotopy analysis method, we take the value

and impose the initial guess solutions

The problem (8) shows that two cases. One is

are continuous from initial guess solutions

where

The two sides of zero-order Equations (8) are solved m-order derivative about q and divided

where

The solutions of problem (10) are,

Choose the auxiliary function

In order to make Taylor series are convergent at

In summary, the m-order homotopy approximate solutions of problem (1) are

The typical Brusselator reaction diffusion model is as follows,

where

el and the steady-state solution of the homogeneous perturbation analysis, the guess of initial solution can be as follows,

where

where

Brusselator model is a classical dissipative structure model, which has been studied by many researchers. The conditions for Turing bifurcation of Brusselator model is as follows,

Parameters

Parameters | The effective area about | The effective area about |
---|---|---|

[−3, 1] | [−0.7, 0.5] | |

[−2.5, 1] | [−0.5, 0.5] |

Parameter

The parameters

As time goes on, the final pattern structure will tend to a steady state. Turing patterns structure of homotopy series solutions are sensitive to the selection of initial guess solution and are affected by the wave number. In summary, in the range of parameter about Turing patterns, the system will appear striped patterns, point patterns and the coexistence of striped and point pattern with the time going on.

In this paper, a new method based on differential method and homotopy analysis method is used to solve the typical two-dimensional reaction diffusion model. Different shapes of Turing pattern can be obtained through Matlab mathematical software on experimental data simulation of the structure of the solution. And it is proved that the proposed method which to solving nonlinear reaction diffusion problems is feasible and effective. The new method not only reduces the dimension about differential in space, but also gets the analytical expression with physical parameters through the homotopy in time. It will facilitate the analysis of the influence of parameters variation on Turing pattern structure. The method which is differential in space and homotopy in time domain has enormous potential for solving definite solution of nonlinear partial differential, and it has great promotional value.

The innovation point of this article is making use of the new method to solve the two-dimensional reaction diffusion model. The new method is combining the differential method and homotopy analysis method.

This work is supported by the National Sciences Foundation of People’s Republic of China under Grant 1140011526.

GendaiGu,HongxiaoPeng, (2015) Numerical Simulation of Reaction-Diffusion Systems of Turing Pattern Formation. International Journal of Modern Nonlinear Theory and Application,04,215-225. doi: 10.4236/ijmnta.2015.44016