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Modern Physics Letters BVol. 27, No. 1 (2013) 1350006 (9 pages)c! World Scientific Publishing CompanyDOI: 10.1142/S0217984913500061

TURING INSTABILITY AND PATTERN FORMATION IN ASEMI-DISCRETE BRUSSELATOR MODEL

L. XU!, L. J. ZHAO and Z. X. CHANG

School of Science, Tianjin University of Commerce, Tianjin 300134, China!beifang xl@163.com

J. T. FENG

School of Science, Shanghai University, Shanghai 200444, China

G. ZHANG

School of Science, Tianjin University of Commerce, Tianjin 300134, China

Received 11 August 2012Revised 15 October 2012Accepted 21 October 2012

Published 23 November 2012

In this paper, a semi-discrete Brusselator system is considered. The Turing instabilitytheory analysis will be given for the model, then Turing instability conditions can bededuced combining linearization method and inner product technique. A series of nu-merical simulations of the system are performed in the Turing instability region, variouspatterns such as square, labyrinthine, spotlike patterns, can be exhibited. The impact ofthe system parameters and di!usion coe"cients on patterns can also observed visually.

Keywords: Semi-discrete Brusselator system; Turing instability; pattern.

1. Introduction

The last few years has been a period of rapid advance in our understanding ofthe spatiotemporal dynamics of far-from-equilibrium reaction-di!usion systems.Reaction-di!usion systems can display rich and complex dynamics in both the tem-poral and spatial domains since the fundamental paper of Turing,1 which demon-strates that a system of two or more mutually interacting di!usible chemicals canundergo spontaneous spatial patterns within a specific region of parameter space. Atypical Turing system consists of at least two chemical reactants, usually referred toas activator and inhibitor, reacting in such a way that their steady state is stable to

!Corresponding author.

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small perturbations in the absence of di!usion, but becomes unstable when di!usionis present. Based on Turing’s seminal paper about di!usion instability a broad the-ory of pattern formation in reaction-di!usion systems has been developed.2,3 SuchTuring patterns have also been produced experimentally in chemical systems,4,5

such as the Belousov–Zhabotinsky (BZ) reaction, which has been found to be themost simple and prototypical example of pattern forming chemical reaction.

The Brusselator serves as a simple two-species model for the Belousov–Zhabotinsky reaction, a hypothetical oscillating chemical reaction system proposedby Prigogine in 1968 which is considered one of the simplest reaction–di!usion mod-els exhibiting Turing and Hopf instabilities. The spatiotemporal evolution of themain variables is given by the following partial di!erential equations:

!""#

""$

!u

!t= f(u, v) +Du!2u ,

!v

!t= g(u, v) +Dv!2v ,

(1)

where Du and Dv are di!usion coe"cients, u and v are concentrations of mor-phogens, f and g are nonlinear functions that represent the reaction kinetics, whichfor the Brusselator model are

%f(u, v) = a" (b + 1)u+ u2v ,

g(u, v) = bu" u2v ,(2)

here a and b are kept as the control parameters of the system.In the last decades, the Brusselator system has been extensively investigated

from both analytical and numerical point of view (see, for instance, Refs. 6–13 andreferences therein). For example, in Ref. 6, polynomial based di!erential quadraturemethod (DQM) is applied to the numerical solution of a class of two-dimensionalreaction–di!usion Brusselator system. Convergence and stability of the method arealso examined numerically. Steady states of the Brusselator system is developedto model morphogenesis and pattern formation in chemical reactions, by a properchange of variable x (see Ref. 7). The nonexistence and existence of positive non-constant steady states, as the system parameters are varied, are obtained in Ref. 9.The fascinating Turing instability and Turing pattern, such as stripes and spots,have been observed through this simple-looking coupled system.12,13

Note that the model (1) includes a basic assumption: the cells always live ina continuous patch environment. However, this may not be the case in reality,and the motion of individuals of given cells is random and isotropic, i.e. withoutany preferred direction, the cells are also absolute individuals. The cells or unitsare also absolute individuals in microscopic sense, and each isolated cell exchangesmaterials by di!usion with its neighbors.1,14 Thus, it is reasonable to considera 1D or 2D spatially discrete reaction–di!usion system in order to explain thechemical system. A semi-discrete Oregonator model has been discussed in detail,and some patterns di!erent from the corresponding continuous system are obtained

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Turing Instability and Pattern Formation in a Semi-Discrete Brusselator Model

by simulations.14 Similarly, a 2D spatially discrete reaction–di!usion Brusselatorsystem can be modeled as follows:

%u!ij(t) = a" (b+ 1)uij(t) + u2

ij(t)vij(t) +D1!2uij(t) ,

v!ij(t) = buij(t)" u2ij(t)vij(t) +D2!2vij(t) ,

(3)

where !2 is discrete Laplace operator defined as follows:

!2uij(t) = ui+1,j(t) + ui,j+1(t) + ui"1,j(t) + ui,j"1(t)" 4uij(t) (4)

and

!2vij(t) = vi+1,j(t) + vi,j+1(t) + vi"1,j(t) + vi,j"1(t)" 4vij(t) . (5)

It also indicates the coupling or di!usion from the cells to the left (i, j " 1) andright (i, j " 1), top (i+ 1, j) and bottom (i" 1, j), respectively.

In this paper, we shall be concerned with Turing instability and pattern for-mation in the above semi-discrete Brusselator model for autocatalytic oscillatingchemical reactions. The paper is organized as follows. The Turing instability theoryanalysis will be given for a semi-discrete Brusselator system, then Turing instabil-ity conditions can be deduced combining linearization method and inner producttechnique in Sec. 2. In Sec. 3, based on the results of Sec. 2, a series of numericalsimulations are performed, and di!erent patterns, including not only tripes andspotlike patterns but square, labyrinthine patterns, can be exhibited. The num-ber of the eigenvalues is illuminated by calculation and the unstable spaces can beclearly expressed. The impact of the system parameters and di!usion coe"cientson patterns can also observed visually. The final section is the conclusion.

2. Turing Instability

In this section, we will discuss Turing instability of the following system%u!ij(t) = a" (b+ 1)uij(t) + u2

ij(t)vij(t) +!2uij(t) ,

v!ij(t) = buij(t)" u2ij(t)vij(t) + d!2vij(t) ,

(6)

with the periodic boundary conditions

ui,0(t) = ui,m(t), ui,1(t) = ui,m+1(t), u0,j(t) = um,j(t), u1,j(t) = um+1,j(t) ,

(7)

and

vi,0(t) = vi,m(t), vi,1(t) = vi,m+1(t), v0,j(t) = vm,j(t), v1,j(t) = vm+1,j(t) ,

(8)

where i, j # {1, 2, . . . ,m} = [1,m], t # R+ = [0,$), m is a positive integer

!2uij(t) = ui+1,j(t) + ui,j+1(t) + ui"1,j(t) + ui,j"1(t)" 4uij(t) , (9)

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and

!2vij(t) = vi+1,j(t) + vi,j+1(t) + vi"1,j(t) + vi,j"1(t)" 4vij(t) . (10)

First of all, we show some properties of the following system:%u!ij(t) = a" (b+ 1)uij(t) + u2

ij(t)vij(t),

v!ij(t) = bu,j(t)" u2ij(t)vij(t).

(11)

In the sense of chemistry, only non-negative steady states of (11) are really ofinterests. It is obvious that (11) has a unique positive constant solution (u#, v#),which is (u#, v#) = (a, b

a ). Linear stability analysis around this steady state yieldsthe characteristic equation for the eigenvalues:

J =

&fu fv

gu gv

'

(u!,v!)

=

&b" 1 a2

"b "a2

'. (12)

From the Jacobian matrix, we observe that

trA = b" 1" a2 ,

detA = a2 > 0 .(13)

Thus, the steady state (u#, v#) is stable against homogeneous perturbations if

trA = b" 1" a2 < 0 , (14)

namely

b < 1 + a2 . (15)

So we can get the following fact.

Theorem 1. The system (11) at the positive steady state (u#, v#) is local asymp-totically stable when the condition (15) holds.

Next, we will point out that under certain conditions on the parameters, theuniform steady-state (u#, v#) can be unstable in the presence of di!usion.

We linearize the reaction di!usion system (6) about the steady state, and obtain

w!ij(t) = Awij(t) +D!2wij(t), D =

(1 00 d

), (16)

with the periodic boundary conditions

wi,0(t) = wi,m(t), wi,1(t) = wi,m+1(t), w0,j(t) = wm,j(t), w1,j(t) = wm+1,j(t) ,

(17)

where

wij(t) =

(uij(t)" u#

vij(t)" v#

)=

(xij(t)yij(t)

). (18)

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Turing Instability and Pattern Formation in a Semi-Discrete Brusselator Model

In order to study the instability of (16), we consider eigenvalues of the followingequation:

!2Xij + µXij = 0 , (19)

with the periodic boundary conditions:

Xi,0 = Xi,m, Xi,1 = Xi,m+1, X0,j = Xm,j, X1,j = Xm+1,j . (20)

In view of Ref. 15, the eigenvalue problem (19) and (20) has the eigenvalues

µl,s = 4

(sin2

(l " 1)"

m+ sin2

(s" 1)"

m

)= k2ls for l, s # [1,m] . (21)

Then respectively taking the inner product of (16) with the corresponding eigen-function X ij

ts of the eigenvalue µt,s, we obtain!"""""#

"""""$

m*

i,j=1

X ijlsx

!ij = fu

m*

i,j=1

X ijlsxij + fv

m*

i,j=1

X ijls yij +

m*

i,j=1

X ijls!

2xij ,

m*

i,j=1

X ijtsy

!ij = gu

m*

i,j=1

X ijlsxij + gv

m*

i,j=1

X ijls yij + d

m*

i,j=1

X ijls!

2yij .

(22)

Let U(t) =+m

i,j=1 Xijlsxij and V (t) =

+mi,j=1 X

ijtsyij , and use the periodic

boundary conditions (17) and (20), we have%U !(t) = fuU(t) + fvV (t)" k2lsU(t) ,

V !(t) = guU(t) + gvV (t)" dk2lsV (t) ,(23)

or%U !(t) = (fu " k2ls)U(t) + fvV (t) ,

V !(t) = guU(t) + (gv " dk2ls)V (t) .(24)

Which has the eigenvalue equation

#2" ["k2ls(1+d)+ (fu+ gu)]#+h(k2ls) = #2" ["k2ls(1+d)" 1]#+h(k2ls) = 0 , (25)

here

h(k2ls) = dk4ls " (dfu + gv)k2ls + |A| = dk4ls " (d(b " 1)" a2)k2ls + a2 . (26)

Obviously

"k2ls(1 + d)" 1 < 0 . (27)

Then, for the homogeneous steady state to be unstable to an inhomogeneousperturbation

h(k2ls) = dk4ls " (dfu + gv)k2ls + |A| = dk4ls " (d(b " 1)" a2)k2ls + a2 < 0 (28)

for k2ls # [0, 8].

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In this inequality, the first term is clearly positive and the last term is too. Thisleads to the necessary condition for a Turing instability,

b > 1 +a2 + k2lsa

2 + dk4lsdk2ls

, (29)

namely

a2 + 1 > b > 1 +a2 + k2lsa

2 + dk4lsdk2ls

. (30)

Thus, in the view of Theorem 1, we can obtain the fact as follows.

Theorem 2. There exist positive numbers d and the eigenvalues of the k2ls suchthat the condition (30) holds, then the system (6)–(8) at the positive homogeneoussteady state (u#, v#) is unstable.

Theorems 1 and 2 imply the system (6)–(8) is di!usion-driven unstable or Turingunstable.

3. Numerical Simulation

To provide some numerical evidence for the qualitative dynamic behavior of thetime continuous but spatial discrete system (6), the simulations are performed withperiodic boundary conditions in a square domain of size: 128%128 (grid: 128%128).To solve di!erential equations by computers, the time evolution should be discrete,i.e. the time goes in steps of #t. The time evolution can be solved by the Eulermethod, approximating the value of the concentration at the next time step basedon the change rate of the concentration at the previous time step.

Initial conditions u0 and v0 typically used in simulations of Turing systems arerandom or special perturbations around the stationary state (u#, v#) as follows:

%u0 = u# + $ ,

v0 = v# + % ,

where $ and % are small amplitude random perturbations 1% around the steadystate. Here spatial-temporal plots will be ones about v.

There are examples of numerical calculations, obtained by the proper choice ofparameters, which show either striped patterns or spotted patterns. These patternsare the same as ones found in the corresponding continuous Brusselator model. Inthe simulations di!erent types of dynamics are also observed (see Fig. 1). We givethe time series of the mean concentration of morphogens, which are respectivelycorresponding to the above patterns and can reveal their characters.

It is well known that Turing instability is di!usion-driven instability, thus thedi!usion rate of the cells is vital to the pattern formation. To investigate the e!ectof di!usion coe"cients on patterns, by keeping all the other parameters of the sys-tem fixed, we change the di!usion coe"cient. Figures 2(a)–2(i) exhibit in detail the

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Turing Instability and Pattern Formation in a Semi-Discrete Brusselator Model

Fig. 1. (Color online) Selective pattern and corresponding time series of the meanconcentration in the Turing instability region for the semi-discrete Brusselator model.(a) Labyrinthine pattern when a = 3.5, b = 12, d = 3, #t = 0.0025 and iterative time t =300,000; (b) time series of the mean concentration corresponding to (a); (c) square pattern whena = 3, b = 9, d = 2.4, #t = 0.0025 and iterative time t = 300,000; (d) time series of the meanconcentration corresponding to (c); (e) dot-like pattern when a = 3, b = 9, d = 10, !t = 0.0025and iterative time t = 100,000; (f) time series of the mean concentration corresponding to (e);(g) square-like pattern when a = 5, b = 11, d = 4.8, #t = 0.0025 and iterative time t = 300,000;(h) time series of the mean concentration corresponding to (g).

di!erent distribution of patterns with varying values of d. By increasing d, somepatterns of spot-stripes are given, and the number of strips are increasing gradu-ally [Fig. 2(a)&Fig. 2(d)&Fig. 2(g)]. According to the time series of the meanconcentration, we can find an interesting phenomena that a trough will appear atturn-left position, although each curve gets close to a certain value asymptotically.

It is a fact that the condition for a spatial mode to be unstable and thus growinto a pattern is Re(#) > 0. In Fig. 2, we also show the dispersion relation, fromwhich we can see that the spatial patterns can occur due to the positive real partsof #, and the unstable space resulted from # can be expressed clearly.

Similarly, we search for the impact of the system coe"cients on the patterns.We only change b with the other parameters fixed, and find that this can changethe emerging pattern dramatically, as depicted in Fig. 3 when a = 2.5, d = 5.3.Obviously, a spotted pattern is presented in Fig. 3(a) with b = 5.5. For b = 5.6, eye-like pattern will emerge. As b continues increasing, more strips are in the accordingpatterns [Figs. 3(c) and 3(d)]. But when b is further increased, more spots insteadof some or even for all strips are in the according patterns. As the parameter bcloses to the critical region of the Turing instability, the emergence possibility ofspot pattern will be high.

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Fig. 2. (Color online) Two-dimensional Turing structures and corresponding time series of themean concentration and linear stability spectrum of the model when a = 3, b = 9, #t = 0.0025and d = 2.4 in (a), d = 2.6 in (d), d = 2.9 in (g).

Fig. 3. (Color online) Plot of the almost steady state profiles in simulations of the semi-discreteBrusselator system at #t = 0.0025 and iterative time t = 300,000 on the domain [0, 128]" [0, 128],with the same parameters a = 2.5, d = 5.3 and di!erent parameter b, (a) b = 5.5; (b) b = 5.6;(c) b = 5.8; (d) b = 6.5; (e) b = 7.0; (f) b = 7.5.

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4. Conclusion

In conclusion, by modeling a semi-discrete Brusselator system, unreported behav-ior is obtained, di!erent patterns such as square, labyrinthine, chaotic structureare found in simulations. The impact of the system parameters and di!usion co-e"cients on patterns can also observed visually, and some interesting situationscan be observed according to the continuous Brusselator system. But the relation-ships between the adopted parameter values and the kinds of pattern which can beobtained should be studied in our further work, we will show the form of a statediagram related to the model parameter space. Another problem to be consideredis how di!erent initial values u0 and v0 to impact pattern formation. The furtherstudy along this line may lead to better prediction for patterns formation, and bet-ter control of pattern formation may be possible when the initial values are wellcharacterized. So it will drive us to extend our view as of what spatially explicitmodels are capable of achieving.

Acknowledgments

The authors thank the referees for their valuable comments and suggestions, andthank Dr. X. F. Li for language assistance. This work was financially supported byTianjin University of Commerce with the grant number of X0803 and YB201128.

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