Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System

Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented amethod to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.


Introduction
The pattern formation was first investigated and interpreted by Turing sixty years ago [1].Othmer and Scriven [2] proposed that the Turing instability which is initially stable steady-state of a dynamical system can become unstable if we consider diffusion in the system.It is also possible in networkorganized systems which is important for understanding of multicellular morphogenesis.Recently Turing bifurcation, amplitude equation, and secondary bifurcation have become more significant to study the pattern formation [3][4][5]; Lee and Cho found that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing [6].Moreover, Peña and Pérez-García showed that slightly squeezed hexagons are locally stable in a full range of distorted angles [7].The domain coarsening process is strongly affected by the spatial separation between groups created by the Turing pattern formation process [8] and the robustness problem is also investigated [9].The effects of cross-diffusion, the phenomenon in which a gradient in the concentration of one species induces the change of other species, on pattern formation in Reaction-Diffusion Systems have been discussed in many theoretical papers [10].Fanelli et al. [11] showed that cross-diffusion can destabilize uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems.On the other hand, cross-diffusion can also stabilize a uniform equilibrium which is stable for the kinetic system but unstable for the self-diffusion reaction system [12].In conclusion, spatial patterns in Reaction-Diffusion Systems have attracted the interest of experimentalists and theorists during the last few decades.However, until now, no general theoretical analysis has been proposed for the possible role of noise in dissipative pattern formation.
Noise is a ubiquitous phenomenon in nature and is always deemed to play a very important role in natural synthetic system [13].Coherence resonance and stochastic resonance in a noise-driven gene network regulated by small RNA [14,15].Viney and Reece [16] treated noise as adaptive and suggested that applying evolutionary rigour to the study of noise is necessary to fully understand organismal phenotypes.Scarsoglio et al. [17] presented different stochastic mechanisms of spatial pattern formation with a variable as noise-induced phenomena.Hori and Hara provided a mechanistic basis of Turing pattern formation that is induced by intrinsic noise and derived an efficient computation tool to examine the spatial power spectrum of the intrinsic noise [18].Sun et al. [19] revealed that noise can make the regular circle pattern to be a target wave-like pattern by numerical simulations.A stochastic version of the Brusselator model is proposed and studied via the system size expansion [20] and the mesoscopic equations governing the dynamics were derived and used to special models [21].Many studies have been presented in these research areas [22][23][24][25][26][27][28], as practice shows that theory on Turing bifurcation and pattern formation in dynamical system was rarely studied.
It is known that amplitude equation is not only a promising tool to investigate the RDS but also the main focus of the pattern dynamics [29,30].However, the amplitude equation is a complex process [31], and only a few systems have been chosen in the past for amplitude equation [32][33][34][35].In this paper, we studied pattern selection of amplitude equation with a pair of wave vector by using the standard multiple scale analysis [36,37].Previously, the researchers did not take into account the effect of noise when deriving the amplitude equation but we will include it.
Besides the study of patterns, it can offer useful information on the underlying processes causing possible changes in the system.In order to better understand the reaction diffusion model, first, we proposed to study the pattern formation with noise based on the theory.In this paper, we obtained some interesting results explaining biological mechanism in a modified system.Moreover, we also investigated the relationship between the Reaction-Diffusion System and noise, revealing how the dynamics of the model regulation is affected by noise which provides a way to investigate the mechanism of pattern formation.
The paper is organized as follows.In Section 2, we present the general reaction diffusion with noise and derive the condition of Hopf bifurcation and Turing bifurcation.In Section 3, we derive the amplitude equation from Reaction-Diffusion System with noise.In Section 4, we utilize an example to illustrate the application of these ideas and using simulations validate theoretical results and present some interesting pattern dynamical phenomena.Finally, we summarize our results and conclude.

Turing Bifurcation with SRDS
Since we know that noise plays an important role in the nonlinear systems, some promising results have been presented [11,12,32].However, most people investigated noise by simulation and seldom put forward the theoretical conclusion, especially on pattern formation.In this paper, we study the effect of noise on pattern formation by deriving the Turing bifurcation, to know how it affects the pattern formation.The general diffusion form with noise is as follows: where  is the noise and ∇ 2 is the Laplace operator;  1 ,  3 and  2 ,  4 are diffusion parameters and noise magnitude, respectively.
For convenience, we just consider as random variable in this system.In order to obtain the stability of this spatially uniform solution, we consider a perturbation of the form in the following: In the convergence domain, we can obtain the linear system of stochastic system as (3) at where the matrix  is the partial derivative of (, V), (, V) at ( 0 , V 0 ) and For convenience, we can get the linearized system governing the dynamics of  is defined by where the coefficient matrix is given by where In the standard way, we assume that  take the form as and get the characteristic equation from system (4) as follows: Finally, we solve the characteristic equation and obtain the eigenvalues

Amplitude Equation with a Pair of Wave Vector
For a modified model [38] with the external stimulus V, the following is obtained: In this paper, we expanded (12) at equilibrium (0, 0) by using the Taylor expansion and then we truncated the expansion at third order; it is found that only third order  − 2 =  −  3 + ⋅ ⋅ ⋅ will be included and higher order will not affect the amplitude equation in the process.And it can be written as In the following, we use multiple scale analysis to derive the amplitude equations with a pair of wave vector when || =   .Denote  as the controlled parameters.When the controlled parameter is larger than the critical value of Turing point, the solutions of the systems ( 13) can be expanded as Close to onset  =   , one has that / =  + ().
Based on the center manifold near the Turing bifurcation point, it can be concluded that amplitude  satisfies / = (, ).
From the standard multiple scale analysis, up to the third order in the perturbations, the spatiotemporal evolution of the amplitudes can be described as Due to spatial translational symmetry, we have the following equation: Comparing ( 15) with ( 16) and from the center manifold theory, we know that amplitude equation does not include the amplitude with unstable mode.As a result, we have the following equations: In the following, we will give the expressions of  0 , , and .Let system (13) be written as where Advances in Mathematical Physics is the variable, is the linear operator, and is the nonlinear term, where  22 = (1/2 2 ) 21 and  42 = (1/2 2 ) 41 .We need to investigate the dynamical behavior when  is close to   , and then we expand  as where  is a small enough parameter.We expand  and  as the series form of : and  in the Appendix.Linear operator  can be expanded as and   and  in the Appendix.Let and   is a dependent variable.For the derivation of time, we have that The solutions of systems ( 13) have the following form: This expression implies that the bases of the solutions have nothing to do with time and the amplitude  is a variable that changes slowly.As a result, it can be written generally as the following equation: Substituting the above equations into (24) and expanding (24) according to different orders of , we can obtain three equations as follows: and  1 ,  2 in the Appendix.We first consider the case of the first order of .Since   is the linear operator of the system close to the onset, ( 1 , V 1 )  is the linear combination of the eigenvectors that corresponds to the zero eigenvalue since that Let  1 =  by assuming  1 = 1; then, where  in the Appendix and  is the amplitude of the mode     .Now, we consider the case of the second order of  2 .According to the Fredholm solubility condition [32], the vector function of the right hand of the above equation must be orthogonal with the zero eigenvectors of operator  +  .And the zero eigenvectors of adjoint operator  +  are and  in the Appendix.It can be obtained from the orthogonality condition that By using the same methods, we deduce and coefficients in the Appendix.
For the case of the third order of  3 , replace  1 , V 1 ,  2 , and V 2 by their expression and ,  in the Appendix.Using the Fredholm solubility condition again, we can obtain  And then we substitute system ( 28) and ( 33) into (24) to simplify [32]; we obtain the expressions of the coefficients of  0 , , and  in the Appendix. And So the equation of amplitude is as follows: Here, we will investigate the dynamics of amplitude equation by using the linear stability analysis [30,32] and study the different pattern.The dynamical systems (38) possess two kinds of solution as follows.
(i) The stationary solution  = 0 is stable for  < 0 and unstable for  > 0.

Simulation
As the examples of Reaction-Diffusion System with noise, we use the following: where The model is simulated numerically in two spatial dimensions and employ the zero-flux boundary conditions in (39).We set time step and space step as 0.02 and 1, respectively.The bifurcation space divide the space into four domains (Figure 1(a)).On bottom domain, locating below two bifurcation spaces, the system lies in the steady state (Figure 3(d)).The middle domain are regions of pure Turing and pure Hopf in stabilities (Figures 3(b) and 3(c)).On the top, two bifurcation spaces interact (Figure 3(a)).It is found that noise contribute to Turing bifurcation and Hopf bifurcation.

Conclusion
As we all know, noise could make a bistable system which switches and regulates relevant mechanism [39].Similarly, it was presented in [40] to understand the biological pattern formation and we presented the spatial pattern with different noise intensities, which gave results supporting that noise could make pattern formation switch (Figures 2(a) and 2(b)) for the Stochastic Reaction-Diffusion Systems.Later some special biological models [21] have been studied and its biological mechanism was explained by the type of pattern formation.For example, the system will exhibit a characteristic excursion in phase space before the variables  and V relax back to their rest values [38].The only spot pattern existing in Figure 2(b) means that the noise exceeds the maximum value that a biological system could bear and makes the biological system worse.Instead, the appropriate noise could keep a biological system working towards better development.
For the Turing instability, the different pattern formation occurs in different condition (Figure 3), especially, the pure Turing domain (Figure 3(b)) and the pure Hopf domain (Figure 3(c)), which shows that not only Turing bifurcation but also Hopf bifurcation makes the system vary and decides whether to relax back to their rest values, then keeping good condition.The patterns discussed above show that the distribution and interaction of ion density and electric potential are caused by noise and diffusion.As a result, we can control the distribution of ion by noise, diffusion, and so on.Our research may help cure some diseases caused by the conduction of electrical impulses along a nerve fiber in future.
To summarise, noise effects have been paid much attention due to its strong impact on pattern formation [19][20][21][22][23][24][25].In this article, we presented the theoretical, analytical, and numerical study of the Turing instability accompanied with noise.We examined the effects of noise on pattern formation and the interaction between Hopf bifurcation space and Turing bifurcation space.It is found that the systems with noise effect have rich spatial dynamics by performing a series of numerical simulations.Thus we know that noise plays an important role in Turing bifurcation and Hopf bifurcation.Moreover, we derive the amplitude equation with a pair of wave vector and analyze the stability.It should be noted that noise contribute to the type of pattern formation and the stability.For future study, we would use the theoretical concepts to solve some other problems and find out a general way to deal with it.