Stochastic Turing patterns in the Brusselator model

Tommaso BiancalaniDipartimento di Fisica, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Florence, Italy

Duccio FanelliDipartimento di Energetica, Università degli Studi di Firenze, via S. Marta 3, 50139 Florence, Italy

and INFN, Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Florence, Italy

Francesca Di PattiDipartimento di Fisica “Galileo Galilei,” Università degli Studi di Padova, via F. Marzolo 8, 35131 Padova, Italy

Received 26 October 2009; revised manuscript received 4 February 2010; published 27 April 2010

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. Themean-field equations are derived and shown to yield to organized Turing patterns within a specific parametersregion. When determining the Turing condition for instability, we pay particular attention to the role ofcross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochasticfluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic ofthe mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameteryielding to the stochastic self-organization is wider than that determined via the conventional Turing approach,suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

DOI: 10.1103/PhysRevE.81.046215 PACS numbers: 89.75.Kd, 87.23.Cc, 87.10.Mn, 02.50.Ey

I. INTRODUCTION

Turing instability constitutes a universal paradigm for thespontaneous generation of spatially organized patterns 1. Itformally applies to a wide category of phenomena, whichcan be modeled via the so called reaction-diffusion schemes.These are mathematical models that describe the coupledevolution of spatially distributed species, as driven by micro-scopic reactions and freely diffusing in the embedding me-dium. Diffusion can potentially seed the instability by per-turbing the mean-field homogeneous state, through anactivator-inhibitor mechanism, and so yielding to the emer-gence of patched, spatially inhomogeneous, density distribu-tion 2. The realm of application of the Turing ideas encom-passes different fields, ranging from chemistry to biology,from ecology to physics. The most astonishing examples, asalready evidenced in Turing original paper, are perhaps en-countered in the context of morphogenesis, the branch ofembryology devoted to investigating the development of pat-terns and forms in biology 3.

Beyond the qualitative agreement, one difficulty in estab-lishing a quantitative link between theory and empirical ob-servations has to do with the strict conditions for which theorganized Turing patterns are predicted to occur. In particu-lar, and with reference to simple predator-prey competingpopulations, the relative degree of diffusivity of the interact-ing species has to be large according to the theory prescrip-tions, and at variance with the direct experimental evidence.Moreover, patterns formation appears to be rather robust innature, as opposed to the Turing predictive scenario, where afine tuning of the parameters is often necessary.

In 4 it was demonstrated that collective temporal oscil-lations can spontaneously emerge in a model of populationdynamics, as due to a resonance mechanism that amplifiesthe unavoidable intrinsic noise, originating from the discrete-ness of the system. Later on an extension of the model was

proposed 5 so to explicitly account for the notion of space.It was in particularly shown that the number density of theinteracting species oscillates both in time and space, a mac-roscopic effect resulting from the amplification of the sto-chastic fluctuations about the time independent solution ofthe deterministic equations. More recently, Butler and Gold-enfeld 6 proved that persistent spatial patterns and temporaloscillations induced by demographic noise, can develop in asimple predator-prey model of plankton-herbivore dynamics.The model considered by the authors of 6 exhibits a Turingorder in the mean field theory. The effect of intrinsic noisetranslates however into an enlargement of the parameter re-gion yielding to the Turing mechanism, when compared toits homologous domain predicted within the conventionallinear stability analysis. This is an individual based effect,which has to be accommodated for in any sensible model ofnatural phenomena, and which lacks in the Turing interpre-tative scenario, that formally applies to the idealized con-tinuum limit. Interestingly, as reported in 7, the discrete-ness of the scrutinized medium can yield to robust spatio-temporal structures, also when the system does not undergoTuring order in its mean-field, deterministic version.

Starting from this setting, we present the results of ourinvestigations carried out for a spatial version of the Bruss-elator model, which we shall be introducing in the forthcom-ing section. As opposed to the analysis in 6, we will operatewithin the so called urn protocol, where individual elementsbelonging to the inspected species are assumed to populatean assigned container. The proposed formulation of the Brus-selator differs from the one customarily reported in the lit-erature. Additional terms are in fact obtained building on theunderlying microscopic picture. These latter contributionsare generally omitted, an assumption that we interpret asworking in a diluted limit. Interestingly, the phase diagram ofthe homogeneous aspatial version of the model, is remark-

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ably different from its diluted analogue, a fact we will sub-stantiate in the following.

Furthermore, when constructing the mean-field dynamicsfrom the assigned microscopic rules, one recovers nontrivialcross-diffusive terms, as previously remarked in 5. Theseare derived within a self-consistent analysis and ultimatelystands from assuming a finite carrying capacity in a givenspatial patch. As such, they potentially bear an importantphysical meaning, which deserves to be further elucidated.This scheme, alternative to conventional reaction-diffusionmodels, calls for an extension of the original Turing condi-tion, that we will derive in the following. This is the secondresult of the paper.

Finally, by making use of the van Kampen system sizeexpansion, we are able to estimate the power spectrum offluctuations analytically and so delineate the boundaries ofthe spatially ordered domain in the relevant parameters’space. As already remarked in 6, the role played by theintimate graininess can impact dramatically the Turing vi-sion, returning a generalized scenario that holds promise tobridge the gap with observations.

II. SPATIAL VERSION OF THE BRUSSELATOR MODEL

The Brusselator model represents a paradigmatic exampleof an autocatalytic chemical reaction. This scheme was origi-nally devised in 1971 by Prigogine and Glandsdorff 8 andquickly gained its reputation as the prototype model for os-cillating chemical reaction of the Belosouv-Zhabotinsky9,10 type. In the following we shall consider a slightlymodified version of the original formulation, where the num-ber of reactants X and Y is conserved and totals in N, includ-ing the empties, here called E 4. Moreover we will considera spatially extended system composed of cells, each ofsize l, where reactions are supposed to occur 13. Practicallyeach cell hosts a replica of the Brusselator system: the mol-ecules are however allowed to migrate between adjacentcells, which in turn implies an effective spatial coupling im-puted to the microscopic molecular diffusion. A periodic ge-ometry is also assumed so to restore the translational invari-ance. Although the calculation can be carried out in anyspace dimension D see 5,7 we shall here mainly refer tothe D=1 case study, so to privilege the clarity of the messageover technical complications. It should be however remarkedthat our conclusions are general and remain unchanged inextended spatial settings. Mathematically, the model can becast in the form

A + Ei→a

A + Xi,

Xi + B→b

Yi + B ,

2Xi + Yi→c

3Xi,

Xi→d

Ei,

where the index i runs from 1 to and identifies the cellwhere the molecules are located. The autocatalytic species of

interest to us are Xi and Yi. The elements A and B work asenzymatic activators, and keep constant in number. As such,they can be straightforwardly absorbed into the definition ofthe reaction rates. Let us label with ni respectively, mi andoi the number of element of type Xi respectively, Yi and Eipopulating cell i. Then, assuming N to identify the maximumnumber of available cases within each cell, one gets N=ni+mi+oi a condition, which can be exploited to reduce theactual number of dynamical variables to two. The dynamicsof the model is ultimately related to studying the coupledinteraction between the discrete species ni and mi. The mi-gration between neighbors cells is specified through

Xi + Ej→

Ei + Xj ,

Yi + Ej→

Ei + Y j .

Assuming a perfect mixing in each individual cell i, the tran-sition probabilities T· · read 4

Tni + 1,mini,mi = aN − ni − mi

N,

Tni − 1,mi + 1ni,mi = bni

N,

Tni + 1,mi − 1ni,mi = dni

2mi

N3,

Tni − 1,mini,mi = cni

N, 1

where according to the standard convention, the rightmostinput specifies the original state and the other entry stems forthe final one. In addition, the migration mechanism betweenneighbors cell is controlled by

Tni − 1,nj + 1ni,nj = ni

N

N − nj − mj

Nz,

Tmi − 1,mj + 1mi,mj = mi

N

N − nj − mj

Nz, 2

where the positive constants and quantify the diffusionability of the two species and z stands for the number of firstneighbors. To complete the notation setting, we introduce the-dimensional vectors n and m to identify the state of thesystem. Their i-th components, respectively, read ni and mi.

The aforementioned system is intrinsically stochastic. Attime t there exists a finite probability to observe the systemin the state characterized by n and m. Let us label Pn ,m , tsuch a probability. One can then write down the so-calledmaster equation, a differential equation, which governs thedynamical evolution of the quantity Pn ,m , t. The masterequation for the case at hand takes the form

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tPn,m,t =

i

Xi

− − 1Tni + 1,mini,mi

+ Xi

+ − 1Tni − 1,mi, ni,mi

+ Xi

+ Y,i− − 1Tni − 1,mi + 1ni,mi

+ Xi

− Yi

+ − 1Tni + 1,mi − 1ni,mi

+ ji

Xi

+ Xj

− − 1Tni − 1,nj + 1ni,nj

+ Yi

+ Yj

− − 1Tmi − 1,mj + 1mi,mj

Pn,m,t , 3

where use has been made of the definition of the step opera-tors

Xi

f. . . ,ni, . . . ,m = f. . . ,ni 1, . . . ,m ,

Yi

fn, . . . ,mi, . . . = fn, . . . ,mi 1, . . . , 4

and where the second sum in Eq. 3 runs on first neighborsj i. Equation 3 is exact, the underlying dynamics being aMarkov process. The master equation 3 contains informa-tion on both the ideal mean-field dynamics formally recov-ered in the limit of diverging system size and the finite Ncorrections. To bring into evidence those two components,one can proceed according to the prescriptions of van Ka-mpen 11, and write the normalized concentration relative tothe interacting species as

ni

N= it +

i

N,

mi

N= it +

i

N, 5

where i and i stand for the stochastic contribution. Theansatz 5 is motivated by the central limit theorem and holdsprovided the dynamics evolve far from the absorbing bound-aries extinction condition. Here 1 /N plays the role of asmall parameter and paves the way to a perturbative analysisof the master equation. At the leading order one recovers themean-field equations, while the fluctuations are characterizedas next to leading corrections. The following section is de-voted to discussing the system dynamics, according to itsmean-field approximation.

III. MEAN-FIELD APPROXIMATIONAND THE TURING INSTABILITY

Performing the perturbative calculation one ends up withthe following system of partial differential equations for theit and it in cell i:

i = − bi − di + a1 − i − i + ci2i

+ i + ii − ii ,

i = bi − ci2i + i + ii − ii , 6

where = t / N. In Eq. 6 we have introduced the discreteLaplacian operator acting on a generic function f i as

f i =2

zji

f j − f i . 7

The above sum runs on the first neighbors, which we assumeto total in z. The limit → corresponds to shrinking thelattice spacing l to zero and so obtaining the continuummean-field description. In this limit the system 6 convergesto

= − b − d + a1 − − + c2i

+ 2 + 2 − 2 ,

= b − c2 + 2 + 2 − 2 , 8

where the population fractions go over the population densi-ties and the rescaling →l2 and →l2 have been per-formed 14. As also remarked in 5, cross-diffusive termsof the type 2−2 appear in the mean-field equations,as a relic of the microscopic rules of interaction. More pre-cisely, they arise as a direct consequence of the finite carry-ing capacity hypothesis. This observation materializes in acrucial difference with respect to the heuristically proposedreaction-diffusion schemes and points to the need for an ex-tended Turing analysis, i.e., derive the generalized conditionsyielding to organized spatial structures. Before addressingthis specific issue, we start by discussing the homogeneousfixed point of Eq. 8. From hereon we will set a=d=1, achoice already made in 12 for the original Brusselatormodel and which will make possible to visualize our conclu-sion in the reference plan b ,c for any fixed ratio of thediffusivity amount and .

A. Homogeneous fixed points

Plugging a=d=1 into Eqs. 8 and looking for homoge-neous solutions implies

I

II

III

1 2 3 4 5b

10

20

30

40

50c

FIG. 1. The nature of the fixed point 3 , 3 as a function ofthe parameters b ,c: in region I the fixed point is a stable node,while in II it is a stable spiral. The dashed line c=8b delineatesregion III where the fixed point disappears, following a saddle-nodebifurcation.

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= 1 − − − 1 + b + c2 ,

= b − c2 . 9

As an important remark we notice that in the diluted limit ,1, the system tends to the mean-field of the originalBrusselator model, as, e.g., reported in 12. For a propertuning of the chemical parameters assigning significantlydifferent strength one can prove that, if initialized so toverify the diluted limit, the system stays diluted all along its

subsequent evolution. As a corollary, the diluted solution iscontained in the formulation of the Brusselator here consid-ered, this latter being therefore regarded as a sound generali-zation of the former. Notice that when operating under di-luted conditions, the cross-diffusive terms appearing in thespatial equations 8 can be neglected and the standardreaction-diffusion scheme recovered. Again, let us emphasizethat it is the finite carrying capacity assumption that modifiesthe mean-field description.

System 9 admits three fixed points, namely,

1 = 0,

1 = 1, 2 =

c − − 8bc + c2

4c,

2 =1

2+

− 8bc + c2

2c, 3 =

c + − 8bc + c2

4c,

3 =1

2−

− 8bc + c2

2c. 10

The first of the above corresponds to the extinction of thespecies X: it exists for any choice of the freely changingparameters b ,c and corresponds to a stable attractor for thedynamics. This solution is not present in the original Bruss-elator mean-field equations and its origin can be traced backto the role played by the limited capacity of the container.

The remaining two fixed points are, respectively, a saddlepoint and a nontrivial stable attractor. They both manifestwhen the condition −8b+c0 is fulfilled. The saddle pointpartitions the available phase space into two domains, eachdefining the basin of attraction of the stable points. When c=8b a saddle-node bifurcation occurs: the fixed points

2 , 2 and 3 , 3 collide to eventually disappear. The fi-nite carrying capacity that follows the “urn representation” ofthe Brusselator dynamics destroys the limit cycle solution,which is instead found in its celebrated classical analogue12: no periodic solutions are found in the mean-field ap-proximation, unless the diluted limit is considered. The spe-

cific nature of fixed point 3 , 3 changes as a function ofthe parameters b ,c, as illustrated in Fig. 1. In region I, it isa stable node, while in II it is a stable spiral. Region IIIidentifies the parameters values for which it disappears. Weare here concerned with the stability of the homogeneous

fixed point 3 , 3 to inhomogeneous perturbation: can aninstability develop and yield to organized Turing-like spatialpatterns?

B. Extending the Turing instability mechanism:The effect of cross-diffusion

To answer the previous question one has to perform alinear stability analysis around the selected homogeneous so-

lution 3 , 3, hereafter termed , , accounting for theeffect of nonhomogeneous perturbation. The formal develop-ment is closely inspired by the original Turing calculation,but now the effects of cross-diffusive terms need to be ac-

commodated for. The technical details of the derivation areenclosed in the annexed Appendix, where the most generalproblem is defined and inspected. We will here take advan-tage from the conclusion therein reached and present the re-sults relative to the Brusselator model. In this specific casea=d=1, the dispersion relation A7 reads in particular

k2 = A + k2B +1

16cC + Dk2 + Ek41/2, 11

where

A = − 1 +3b

4−

c

16−

1

16c− 8b + c ,

B = −3

8+

c− 8b + c8c

−

4−

c− 8b + c4c

,

C = 256c2 + 128bc2 + 144b2c2 − 32c3 − 32bc3 + 2c4

+ c− 8b + c− 32c2 − 24bc2 + 2c3 ,

D = − 64c2 + 16bc2 + 8c3 + 128c2 + 96bc2

− 16c3 + c− 8b + c · − 64c − 80bc

+ 8c2 + 128c + 32bc − 16c2 ,

E = − 32bc2 + 40c22 + 128bc − 32c2 − 128bc2

+ 32c22 + c− 8b + c− 24c2 − 32c + 32c2 .

12

The perturbation gets amplified if k20 over a finite in-terval of k values. The edges, k1 and k2, of the interval areidentified by imposing k2=0, which returns the followingresults:

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k1,2 =1

22b2cb − 1 + 8b2 − − bc + c− 8b + c

2 − 2b − b F + c− 8b + cG1/21/2, 13

where

F = 24c22 − 8bc2 + c2 + 32b4 − 2

− 4b3c82 − 2 + 32 + b2c412 + c2 + c2 ,

G = 2− 4c2 + 8bc2 + 8b322 − − 2

+ b2− 44 + c2 − 16 + c2 . 14

By making use of conditions Eq. A9 as derived in theAppendix, the predicted region for the Turing instability istraced in the b ,c parameter space for an assigned ratio ofmutual diffusivity, see Fig. 2. When compared to the conven-tional Turing analysis, based on the heuristically hypoth-esized reaction-diffusion scheme i.e., removing the crossterms in Eq. 8, the region of inhomogeneous instabilityshrinks, the condition for spatial self-organization becomingeven more peculiar than so far believed.

Up to now we have carried out the analysis in the con-tinuous mean-field scenario neglecting the role of finite size

corrections. Stochastic fluctuations can however amplify dueto an inherent resonant mechanism and consequently giverise to a large scale spatial order, similar to that predictedwithin the Turing interpretative picture 6. The followingsection is entirely devoted to clarifying this important point.

IV. ROLE OF STOCHASTIC NOISE AND THE POWERSPECTRUM OF FLUCTUATIONS

At the next to leading order in the van Kampen perturba-tive development one ends up with a Fokker-Planck equationfor the probability distribution of the fluctuations , , t= Pn ,m , t where the vectors and have dimensions and respective components i and i. The derivation islengthy but straightforward and the reader can refer to, e.g.,5,7. for a detailed account on the mathematical technicali-ties. We will here report on the outcome of the calculation interms of the predicted power spectra of the fluctuations closeto equilibrium. In complete analogy with, e.g., Eqs. 36 and37 of 5 the latter can be written as

Pk,X = k2 ,

Pk,Y = k2 ,

where the and k stand, respectively, for Fourier temporaland spatial frequencies 15. Exploiting the formal analogybetween the governing Fokker-Planck equation and theequivalent Langevin equation one eventually obtains seeEqs. 38 and 39 in 5

Pk,X =C1 + Bk,11

2

2 − k,02 2 + k

22 ,

Pk,Y =C2 + Bk,22

2

2 − k,02 2 + k

22 ,

where

Ck,X = Bk,11Mk,222 − 2Bk,12Mk,12Mk,22 + Bk,22Mk,12

2 ,

15

Ck,Y = Bk,22Mk,112 − 2Bk,12Mk,21Mk,11 + Bk,11Mk,21

2 .

16

The 22 matrix Mk of elements Mk,ij reads

Mk = − 2 − b + 2c + 1 − k − 1 + c2 + k

b − 2c + k − c2 + 1 − k

,

17

and the elements Bk,ij, respectively, are

Bk,11 = 1 + b − + c2 − 2k + 22k + 2k,

18

Bk,12 = − b − c2 , 19

I II

2 4 6 8 10b

20

40

60

80

100c

0.5 1.0 1.5 2.0 2.5k

6

4

2

Λk

(b)

(a)

FIG. 2. Upper panel: the domain corresponding to the Turinginstability are displayed in the b ,c plan for =15 and =1. Thesolid line zone 2 delineates the region calculated when accountingfor the cross-diffusive contribution, the dashed one zone 1 standsfor the conventional Turing analysis where diffusion is modeled viaLaplacian operators i.e., disregarding cross-diffusion. Lowerpanel: dispersion relation k vs k. Here c=70 and b=3,5 ,6 ,7starting from the dotted bottom curve.

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Bk,22 = b + c2 − 2k + 2k + 22k, 20

with the additional condition Bk,12=Bk,21. Finally, k,02

=det Mk, k=−tr Mk, and the Fourier transform of the La-placian k=2coskl−1. In the continuum limit, the cellsize goes to zero and k scales as kk2 5.

We are now in a position to represent the power spectrumof the fluctuations as predicted by the system size expansion.In Fig. 3 we display a selection of power spectra relative toone population and for different values of the parameter b, atfixed c and diffusivity amount see caption. The snapshotsrefer to a region of the parameter space for which we do notexpect spatial order to appear, based on the Turing paradigm.However, and beyond the simplified mean-field viewpoint, aclear spatial peak is displayed, which gains in potency as theboundary of the Turing domain is being approached. Physi-cally, it seems plausible to assume that the demographicnoise can modify the dispersion relation. As a consequence,the curve k, negative defined beyond the region of Turinginstability, can locally cross the zero line, taking positivevalues in correspondence of specific k. These latter modesget therefore destabilized, yielding to quasi-Turing struc-tures. Clearly, the k candidate to drive the stochastic insta-bilities are the ones close to kmax, the wave number thatidentifies the position of the maximum of the dispersion re-lation. If the above scenario is correct, kmax should then bereasonably similar to the k value that locates the peak posi-tion in the power spectra Pk , ·0. In Fig. 4 such compari-son is drawn, for both species and over a window of b val-ues, confirming the adequacy of the proposed interpretation.Based on the above, we can convincingly argue that the spa-tial modes here predicted represent an ideal continuation ofthe Turing structures beyond the region of parameters de-

puted to the mean-field instability and, for this reason, wesuggest to refer to them as to stochastic Turing patterns.

The fact that spatial order appears for a wider range of thecontrol parameter is further confirmed by visual inspectionof Fig. 5, where the region of stochastic induced spatial or-ganization, delineated from the power spectra calculatedabove, is depicted and compared to the corresponding mean-field prediction. Notice that the domains yielding to stochas-tic Turing patterns structures are different depending on theconsidered species. This observation marks yet another dif-ference with respect to the standard mean-field theory. Wecan in fact expect to observe spatially ordered structures forjust one of the two species, while the other is homogenouslydistributed.

As an additional point, we consider the case ==1. Thischoice cannot yield to the genuine Turing order, a pro-nounced difference in the relative diffusivity rates being anunavoidable prerequisite. As opposed to the classical picture,stochastic Turing patterns can still emerge as demonstrated inFig. 6.

From the above expressions for the power spectra, onecan also show that for ==0 the aspatial limit quasiperi-odic time oscillations manifest in the concentrations. This isan effect of inner stochastic noise, which restores the oscil-

0.00.5

1.0

1.5

2.0

Ω

0

2

4

k

0.1

0.2

0.3

0.00.5

1.01.5

2.0

Ω

0

2

4

k

0.51.01.52.0

1 2 3 4 5k

0.1

0.2

0.3

PXΩ0, k

1 2 3 4 5k

0.5

1.0

1.5

2.0

PXΩ0, k

(b)(a)

(c) (d)

FIG. 3. Color online Upper panels: the power spectra Pk,Xfor c=70 and b=3,6 going from left to right. Here =15 and =1. The selected values of b and c fall outside the region of Turinginstability, as follows the mean-field linear stability calculation seeFig. 2. Lower panels: the power spectra Pk,X0 plotted vs k so toenable one to appreciating the range of excited spatial wavelengths.This latter approximately matches that found in the realm of theTuring mean-field calculation see Fig. 2, implying similar charac-teristic of the spatially ordered structures.

4.0 4.5 5.0 5.5 6.0 6.5 7.0b

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kmax

4.0 4.5 5.0 5.5 6.0 6.5 7.0b

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kmax

(b)(a)

FIG. 4. Color online Left panel: symbols refer to the values ink that identify the maximum of the function Pk,X0, here plotted asa function of b. The solid line stands for the position of the maxi-mum of the mean-field dispersion relation k. Parameters are setas in Fig. 3. Right panel: as in the left panel, but now the symbolsrefer to species Y.

2 4 6 8 10b

20

40

60

80

100c

FIG. 5. The extended domain of Turing-like instability predictedby the stochastic based analysis: the dashed line refers to species X,while the dot-dashed to species Y. The stochastic Turing region isconfronted to the corresponding mean-field solution solid line, alsodepicted in Fig. 3.

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lations destroyed by the introduction of the carrying capacity.In general terms, self-oscillatory dynamics could be possiblyunderstood, as resulting from the discrete nature of the me-dium, a scenario alternative to any ad hoc mean-field inter-pretation.

V. CONCLUSION

Spatially organized patterns are reported to occur in alarge gallery of widespread applications and are currentlyinterpreted by resorting to the paradigmatic Turing picture.This vision, though successful, relies on a mean-field de-scription of the relevant reaction-diffusion schemes, oftenguessed on purely heuristic basis. An alternative scenariowould require accounting for the intimate microscopic dy-namics and so encapsulating the effect of the small scalegraininess. This latter translates into stochastic fluctuationsthat, under specific conditions, can amplify and so give riseto organized spatio-temporal patterns. In this paper we haveconsidered a microscopic model of Brusselator, and shownthat Turing-like patterns can indeed emerge beyond the pa-rameter region predicted by the conventional Turing theoryand due to the role of finite size corrections to the mean-fieldidealized dynamics. This result is obtained via a system sizeexpansion, which enables us to return closed analytical ex-pressions for the power spectra of fluctuations. Our resultsagree with the conclusion reached in 6 for another modeland employing different analytical tools. Organized patternscan therefore occur more easily than expected, an observa-tion that can potentially help reconciling theory and obser-vations. In particular we find stochastic Turing patterns toemerge for , a condition for which Turing order is pre-vented to occur.

ACKNOWLEDGMENTS

D.F. wishes to thank Alan J. McKane for interesting dis-cussions and for pointing out reference 6. F.D.P. wishes tothank Javier Buceta for stimulating discussion. F.D.P thanksfinancial support from the ESF Short Visit Grant within theframework of the ESF activity entitled “Functional Dynam-ics in Complex Chemical and Biological Systems.”

APPENDIX

Let us start by the generic set of partial differential equa-tions

t = f, + 2 + 2 − 2 ,

t = g, + 2 + 2 − 2 , A1

which explicitly allow for cross-diffusive contributions. Wechoose to deal with zero flux boundary conditions.

Assume that in absence of diffusion homogeneous set-ting the model tends towards a fixed point specified by

, . In other words, , , do not depend over space vari-ables. Allowing for diffusion, and imposing , can makethe system unstable to spatial perturbation. To clarify thispoint, let us start by considering the Jacobian matrix for ==0

J = f f

g g . A2

The fixed point , is linearly stable if J has positivedeterminant and negative trace

det J = fg − gf 0, tr J = f + g 0. A3

Assuming Eq. A3 to hold one can proceed with a linear-

ization of Eq. A1 with ,0 around , and solooking for the sought condition of instability. Define

x= −

−, then Eq. A1 becomes

tx = Jx + D2x, D = 1 −

1 − . A4

Define the eigenfunctions of the Laplacian operator as

2 + k2Wkr = 0, r H ,

and write the solution to Eq. A4 in the form

xt,r = k

etakWkr . A5

Assume =1, or equivalently label with the ratio of thetwo diffusivities after proper rescaling of the rate coeffi-cients. Substituting ansatz A5 into Eq. A4 yields

etJ − k2D − 1Wk = 0,

which implies that the system admits a solution iff the matrixJ−k2D−1 is singular, i.e.,

detJ − k2D − 1 = 0. A6

Label the solutions of Eq. A6 as 1k2 and 2k2: they canbe interpreted as dispersion relation, specifying the timescale of departure or convergence of the k-th mode towardsthe deputed fixed point. If at least one of the two solutionsdisplays a positive real part, the mode is unstable, and drivesthe system dynamics towards a nonhomogeneous configura-tion in response to the initial perturbation. Manipulating thedeterminant, Eq. A6 takes the form

2 + qk2 + hk2 = 0, A7

where

qk2 = k2 + k21 − − − f − g,

0.00.5

1.01.5

2.0Ω

0

2

4

k

1234

1 2 3 4 5k

0.5

1.0

1.5

2.0PXΩ0, k

(b)(a)

FIG. 6. Color online Left panel: the power spectra Pk,X fora=d=1, c=70, and b=8. Spatial and temporal peaks appear now tobe decoupled. Right panel: the corresponding power spectra Pk,X0plotted vs k. A clear peak is displayed.

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hk2 = k4 − k4 − k4 − k2f + k2f + k2f

+ k2g − fg − k2g + k2g + fg.

1k2 and 2k2 are obtained as roots of Eq. A7. Theactual dispersion relation, k2 in the main body of the pa-per, is the one that displays the largest real part. We notice

that i qk2 is always positive as 0 by definition; ii1− − 0 as and are concentrations; iii −f−g0 due to Eq. A3. Left-hand side of Eq. A7 is hence aparabola with the concavity pointing upward and the mini-mum positioned in the half-plane of negative abscissa. Inconclusion a necessary and sufficient condition for the exis-tence of real roots is h0 where

hk2 = ak4 + bk2 + c, a = 1 − − ,

b = − f − g + f + g + f + g ,

c = det J , A8

with a0, c0. The condition for h0 corresponds to b0 and b2−4ac0. Summing up the condition for the generalizedTuring instability reads

f + g − f + g − f + g 0,

f + g − f + g − f + g2 41 − − det J , A9

together with Eq. A3.

1 A. M. Turing, Philos. Trans. R. Soc. London, Ser. B 237, 371952.

2 J. Buceta and K. Lindenberg, Phys. Rev. E 66, 046202 2002.3 J. D. Murray, Mathematical Biology, 3rd ed. Springer, New

York, 2003.4 A. J. McKane and T. J. Newman, Phys. Rev. Lett. 94, 218102

2005.5 C. A. Lugo and A. J. McKane, Phys. Rev. E 78, 051911

2008.6 T. Butler and N. Goldenfeld, Phys. Rev. E 80, 030902R

2009.7 P. de Anna, F. Di Patti, D. Fanelli, A. McKane, and T. Dauxois,

e-print arXiv:1001.4908.8 P. Glandsdorff and I. Prigogine, Thermodynamics Theory of

Structure Stability and Fluctuations Wiley, New York, 1971.9 B. P. Belousov, Collection of Short Papers on Radiation Medi-

cine for 1958 Med. Publ., Moschow, 1959; A. M. Zhabotin-sky, Biofizika 9, 306 1964.

10 S. Strogatz, Non Linear Dynamics and Chaos: With Applica-tions to Physics, Biology, Chemistry and Engineering PerseusBook Group, Cambridge, MA, 2001.

11 N. G. van Kampen, Stochastic Processes in Physics andChemistry, 3rd ed. Elsevier, Amsterdam, 2007.

12 R. P. Boland, T. Galla, and A. J. McKane, Phys. Rev. E 79,051131 2009.

13 Working with the empties E implies imposing a finite carryingcapacity in each microcell. This procedure defines the so called“urn protocol,” and allows for a technically easier implemen-tation of the van Kampen expansion as presented in Ref. 4.On the other hand, and besides technical reasons, assuming alimited capacity in space is a reasonable physical request. It istherefore interesting to explore how such a choice reflects onthe subsequent analysis. Let us anticipate that the presence ofthe empties E will sensibly modify the mean-field phase dia-gram of the Brusselator model, as concerns both the homoge-neous and the spatial dynamics. Conversely, the observationthat the Turing region gets enlarged by demographic noiseholds true, irrespectively of the specific urn representation hereinvoked Ref. 6.

14 The original parameters and quantify the ability of themolecules to exit/enter a given patch of size l. When l getsreduced, resp. should correspondingly increase, so thatthe quantity l2 resp. l2 converges to a finite value thephysical diffusivity when the limit → is taken. This lattervalue is in turn the one that enters the continuous equations8.

15 We define the Fourier transform fk of a function f j

defined on a one dimensional lattice with spacing l asfk= l j exp−ik · ljf j, see Ref. 5.

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