patterns in reaction–diffusion systems generated by global alternation of dynamics

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Physica A 325 (2003) 230–242www.elsevier.com/locate/physa

Patterns in reaction–di�usion systems generatedby global alternation of dynamics

J. Buceta, Katja Lindenberg∗

Department of Chemistry and Biochemistry and Institute for Nonlinear Science,University of California San Diego, La Jolla, CA 92093-0340, USA

Received 30 October 2002

Abstract

We recently proposed a mechanism for inducing a Turing instability by alternation of reaction–di�usion dynamics each of which is pattern-free. Herein we shed light on the question of whichparticular switching schemes produce that instability and which do not. We also consider aparticular reaction–di�usion model to illustrate the mechanism.c© 2003 Elsevier Science B.V. All rights reserved.

PACS: 47.54.+r; 05.45.−a; 89.75.Kd

Keywords: Pattern formation; Reaction–di�usion; Global alternation

1. Introduction

Reaction–di�usion systems are common modeling tools in many 9elds of sciencesuch as chemistry [1], biology, and ecology [2]. In these models, the dynamics is pre-sented in terms of a competition between two or more species, activators and inhibitors,that di�use in a medium. When the di�usion, a homogenizing process, presents char-acteristic length scales that are di�erent for the two species, a morphological instabilitymay trigger pattern formation [3]. The mechanism is the celebrated Turing instability[4].Due to the omnipresence of Turing patterns in nature, the discovery of new mech-

anisms inducing these spatio-temporal structures is a subject of interest. We recentlyproposed a mechanism for pattern formation in reaction–di�usion systems based on

∗ Corresponding author. Tel.: +1-858-534-3285; fax: +1-858-534-7244.E-mail address: [email protected] (K. Lindenberg).

0378-4371/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-4371(03)00202-4

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J. Buceta, K. Lindenberg / Physica A 325 (2003) 230–242 231

global alternation of dynamics [5]. We showed that the alternation between dynam-ics that share the same pattern-free (homogeneous) stable state may induce a Turinginstability. This mechanism is based on a switching-induced ampli9cation of a shorttime instability, and is di�erent from the mechanism in Swift–Hohenberg-like modelsalso introduced recently [6], which is based on a switching-induced interruption of arelaxation process.In the context of noise-induced phenomena, a short time instability ampli9cation can

also lead to pattern formation [7,8]. However, in that case the Guctuations that triggerpattern formation are local, while herein we consider global alternation processes. Thatis, at each time the dynamical equations that drive the system are the same everywhere.Yet, not all global switching schemes lead to pattern formation. In this paper we showin some detail for a class of two-species models which alternation schemes can inducea Turing instability and which cannot.The paper is organized as follows. In Section 2 we introduce the reaction–di�usion

systems and the Turing instability. In Section 3 we present the global alternation mech-anism that may lead to pattern formation and show what type of switching schemescan trigger a Turing instability. In Section 4 we quantify the time scales that de9ne theparameter that controls the appearance of patterns. In Section 5 we illustrate the pro-posed mechanism by means of numerical simulations of a particular reaction–di�usionmodel. Finally, in Section 6 we summarize our main conclusions.

2. Reaction–di�usion systems

Consider the simplest reaction–di�usion system involving two interacting species uand v. The spatiotemporal evolution of the concentrations of the species u(r; t) andv(r; t) is given by

9u9t = f(u; v) + Du∇

2u;9v9t = g(u; v) + Dv∇

2v ; (1)

where the reaction terms f and g are in general nonlinear functions. In the steadystate, the concentrations us and vs satisfy the conditions

f(us; vs) = g(us; vs) = 0 :

The stability of the solution (us; vs) can be investigated by means of a linear pertur-bation analysis. Denoting small perturbations around the equilibrium concentrations us

and vs by u and v, so that u = us + u and v = vs + v, and linearizing f and garound the equilibrium point, we obtain

f(u; v) = fuu+ fvv+ O(u2; v2; uv) ;

g(u; v) = guu+ gvv+ O(u2; v2; uv) ;

where fz ≡ 9zf(u; v)|(us; vs), gz ≡ 9zg(u; v)|(us; vs), and z stands for u and v. We decom-pose u and v into Fourier modes,

z(r; t) =∑q

zq(t)cos(q · r) ;

232 J. Buceta, K. Lindenberg / Physica A 325 (2003) 230–242

and 9nd the evolution equation for the amplitudes of a mode q to be given by

99tZq = JqZq ; (2)

where

Zq =

(uq

vq

); Jq =

(fu − q2Du fv

gu gv − q2Dv

):

The formal solution of Eq. (2) is

Zq(t) = Zq(0)exp(Jqt) : (3)

Therefore, the real parts of the eigenvalues (q) of the operator Jq determine if a per-turbation diverges (Re( (q))¿ 0) or decays to zero (Re( (q))¡ 0). The eigenvaluessatisfy the equation

2(q)− (q)tr(Jq) + det(Jq) = 0 :

From general properties of polynomial equations we know that if, and only if, tr(Jq)¡ 0and det(Jq)¿ 0 then Re( (q))¡ 0 for all q and no perturbation diverges. Therefore,the 9xed point (us; vs) is stable with respect to homogeneous perturbations (q= 0) if

tr(J0)¡ 0 ⇒ fu + gv ¡ 0 ; (4)

det(J0)¿ 0 ⇒ fugv − fvgu ¿ 0 : (5)

Moreover, an inhomogeneous perturbation (q �= 0) will decay if:

tr(Jq)¡ 0 ⇒ fu + gv − q2(Du + Dv)¡ 0 (6)

det(Jq)¿ 0 ⇒ (fu − q2Du)(gv − q2Dv)− fvgu ¿ 0 : (7)

Since the di�usion coeLcients are positive, condition (4) ensures the condition (6).Note, however, that (5) does not ensure (7). The function det(Jq) has an absoluteminimum at q̃= [(Dugv+Dvfu)=(2DuDv)]1=2. Thus, if det(Jq̃)¿ 0 the condition (7) isautomatically satis9ed, that is,

Dugv + Dvfu¡ 2[DuDv(fugv − fvgu)]1=2 : (8)

Furthermore, since DuDv(fugv − fvgu)¿ 0, condition (7) is automatically satis9ed ifDugv + Dvfu¡ 0.Summing up, for a given reaction–di�usion system Eq. (1) a 9xed point (us; vs)

is stable with respect to both homogeneous and inhomogeneous perturbations if thefollowing suLcient (but not necessary) conditions are satis9ed:

fu + gv ¡ 0 ; (9)

fugv − fvgu ¿ 0 ; (10)

Dugv + Dvfu¡ 0 : (11)


On the other hand a steady state will be unstable with respect to inhomogeneousperturbations if Eqs. (9) and (10) hold but (8) is reversed (so that (11) no longerholds):

Dugv + Dvfu¿ 2[DuDv(fugv − fvgu)]1=2 : (12)

The latter case is known as the Turing instability [4] and leads to pattern formation.The wave vector q∗ of the spatial structure that develops is the one that maximizes (q), i.e., the solution of the equations

9 (q)9q

∣∣∣∣q=q∗

= 0;92 (q)9q2

∣∣∣∣q=q∗

¡ 0 : (13)

We point out the following. If a homogeneous state is stable, Eqs. (9) and (11) for-bid fu and gv to both be positive. However, those conditions can still be satis9ed iffugv ¡ 0. Moreover, if fugv ¡ 0 then, by Eq. (10), fvgu ¡ 0. Therefore, it is possiblethat even when a homogeneous state is stable, one 9eld, or even both, may presentan instability at short times. Independently of the initial condition, this instability willdrive the system up to time scales of order O(1=max{fu; fv; gu; gv}). As we will seebelow, the non-equilibrium process of dynamics alternation can amplify and sustainsuch an instability and lead to a destabilization of the equilibrium point.Note that in the case of single 9eld dynamics, as in the Swift–Hohenberg model

[9], if a homogeneous state is stable it is impossible for there to be such a shorttime instability. As a consequence the mechanism of pattern formation proposed hereinapplies only to multi2eld systems.

3. The name of the game: dynamics alternation

The idea underlying the alternation mechanism is as follows. Consider a dynamicalsystem whose behavior depends on a control parameter A that alternates dichotomouslybetween two values A1 and A2,

A(t) = A1�(t) + A2[1− �(t)] :Here �(t) is a dichotomous variable that takes on the values 0 or 1 with equal prob-ability. When switching is “fast” we can, by means of an adiabatic transformation,replace �(t) by its average value, �(t)〈�(t)〉 = 1=2, and therefore in that limitA(t) QA = 1

2(A1 + A2). It may then happen that although neither of the two valuesof the control parameter A leads to interesting dynamics, the average value QA does.Suppose that we globally alternate two reaction–di�usion dynamics in time. Moreover,consider that each of the two dynamics by itself leads to a single stable homogeneousequilibrium state. We argue that it is possible to induce a Turing instability by globaldynamics alternation because the alternation destabilizes the equilibrium points. Wefurther argue that the mechanism for the generation of new equilibrium points is notrelated to relaxation dynamics [6] because it may occur even when each of the twodynamics by itself relaxes to the same equilibrium point.The concept of “slow” and “fast” dynamics implies a comparison between temporal

scales: an external one imposed by the timing of the alternation, text, and a internal


one related to the dynamics, tint, so that a parameter r= text=tint separates slow and fastdynamics. A “slow” alternation leads to no interesting behavior; indeed, it is not evenobservable if both dynamics by themselves lead to the same equilibrium. A suLciently“fast” alternation allows us to replace the switching control parameter by its average,and the system may exhibit stationary patterns, as we will show. Furthermore, wewill show that the most interesting behavior occurs at the “boundary” r≈ 1 betweenslow (r ¿ 1) and fast (r ¡ 1) switching: in this case, if the alternation is periodic, aresonance phenomenon leads to oscillatory behavior.We will quantify the time scales text and tint in the following section. However, 9rst

we consider what alternation schemes for models such as in Eq. (1) lead to a Turinginstability.

3.1. Di4usion alternation

Consider the case of alternating di�usion coeLcients:

Dz(t) = D1z�(t) + D2z[1− �(t)] ;where again z stands for u and v. At a given time all sites are driven either bythe di�usion constant set {D1u; D1v} or by the set {D2u; D2v}. For a homogeneousequilibrium state to be stable with respect to each dynamics when r ¿ 1, Eqs. (9)–(11)must be satis9ed for each dynamics. On the other hand, when alternating fast we candescribe the system by a single dynamics with e4ective di�usion coeLcients

QDu = 12(D1u + D2u); QDv = 1

2(D1v + D2v) :

For a Turing instability to develop, a necessary condition according to Eq. (12) is thatQDugv + QDvfu¿ 0. However, this violates condition (11) for each dynamics, Diugv +Divfu ¡ 0. Therefore, alternation of di�usion coeLcients does not lead to pattern for-mation.

3.2. Reaction alternation

Next, we consider switching of the reaction terms,

f(u; v; t) = f1(u; v)�(t) + f2(u; v)[1− �(t)] ;g(u; v; t) = g1(u; v)�(t) + g2(u; v)[1− �(t)] :

As in the previous example, at any time all sites are driven by either {f1(u; v); g1(u; v)}or by {f2(u; v); g2(u; v)}. When the alternation is slow, the stability of the homogeneousstate for each dynamics requires that Eqs. (9)–(11) be satis9ed for each dynamics. Inthe fast dynamics limit r ¡ 1 we de9ne the e4ective reaction terms

Qf(u; v) = 12 [f1(u; v) + f2(u; v)]; Qg(u; v) = 1

2 [g1(u; v) + g2(u; v)] :

Again, the e�ective dynamics would present a Turing pattern if Du Qgv+Dv Qfu¿ 0. How-ever, condition (11) for each dynamics eliminates this possibility. Therefore, alternationof reaction terms does not lead to pattern formation.


3.3. Reaction–di4usion alternation

The only remaining choice is the alternation of both reaction and di�usion terms,that is,

f(u; v; t) = f1(u; v)�(t) + f2(u; v)[1− �(t)] ;g(u; v; t) = g1(u; v)�(t) + g2(u; v)[1− �(t)] ;Du(t) = D1u�(t) + D2u[1− �(t)] ;Dv(t) = D1v�(t) + D2v[1− �(t)] :

Therefore, the entire system will switch in time between the dynamics de9ned by thesets {f1(u; v); g1(u; v); D1u; D1v} and {f2(u; v); g2(u; v); D2u; D2v}.

When slow dynamics applies, r ¿ 1, the equilibrium point is stable with respect tothe two dynamics if Eqs. (9)–(11) hold for each dynamics. On the other hand whenalternating fast, r ¡ 1, the e4ective dynamics must satisfy the conditions for Turingpattern formation, that is,

Qfu + Qgv ¡ 0 ; (14)

Qfu Qgv − Qfv Qgu ¿ 0 ; (15)

QDu Qgv + QDv Qfu¿ 2[ QDu QDv( Qfu Qgv − Qfv Qgu)]1=2 : (16)

Note that here conditions (9)–(11) do not negate conditions (14)–(16). Consequently,this last mechanism may induce a Turing instability.Let us now focus on some particular examples of this type of alternation scheme.

We stress that these are purely mathematical illustrations that may be neither physicallysigni9cant nor even physically realizable.One example that does not lead to pattern formation (listed here since it might be

an obvious trial choice) because Eq. (10) cannot be satis9ed by each of the dynamicsis a dynamical interchange of the form

f1(u; v) = g2(u; v); f2(u; v) = g1(u; v); D1u = D2v; D2u = D1v :

Another that also does not lead to patterns is a simple scaling of the form

fi(u; v) = aif(u; v); gi(u; v) = big(u; v); Diu = aiDu; Div = biDv

with positive constants ai and bi. It is then easy to show that the conditions (9)–(11)for each dynamics precludes the necessary condition QDu Qgv + QDv Qfu¿ 0 for pattern for-mation upon rapid alternation.A case where the alternation mechanism may induce not only spatial but also spa-

tiotemporal structures is the following:

fi(u; v) = aif(u; v); gi(u; v) = g(u; v); Diu = biDu; Div = ciDv ;


where ai, bi, and ci are positive constants. For no patterns to form in the slow switchingregime we must require that

aifu + gv ¡ 0 ; (17)

ai(fugv − fvgu)¿ 0 ; (18)

biDugv + aiciDvfu ¡ 0 : (19)

In the limit r ¡ 1 a Turing instability develops if12afu + gv ¡ 0 ; (20)

a(fugv − fvgu)¿ 0 ; (21)

2bDugv + acDvfu¿ 0 ; (22)

[2bDugv + acDvfu]2¿ 8abcDuDv(fugv − fvgu) ; (23)

where we have de9ned a ≡ (a1 + a2) and similarly for b and c. There is an in2nitenumber of sets S ≡ {a1; a2; b1; b2; c1; c2; fu; fv; gu; gv; Du; Dv} that satisfy conditions(17)–(23) and that will therefore induce Turing patterns by global dynamics alternation.Moreover, note that di�erent nonlinear systems may share the same set of values S.A subsequent simpli9cation is possible since b1 = b2 (or c1 = c2) may also lead to aTuring instability.

4. Estimation of the time scales

As noted in the previous section, the control parameter r = text=tint measures a ratioof temporal scales. The external time is determined by the switching and is the averagetime that the system spends in each dynamics. In the case of pure periodic switchingwe set text = T=2, where T is the period of the alternation process. The other time, tint,is related to the short time instability that drives the system evolution at early stages.A rough estimate of this time is tint ∼ O(1=max{fu; fv; gu; gv}). A precise estimationof tint can be obtained as follows. Suppose the alternation of dynamics starts with thedynamics 1 (the result is the same if we start with the other dynamics). From Eq. (3),

Zq(nT ) =[(

exp(J1qT2

)exp(J2qT2

))]nZq(0) = [G1�→2

q (T )]nZq(0) : (24)

Then, the real parts of the eigenvalues �q(T ) of G1�→2q (T ) determine the stability of the

alternation process. The eigenvalues are of course independent of the initial condition.The characteristic polynomial of the operator Gi �→j

q (T ) is given by

�q(T )2 − �q(T )tr[Gi �→jq (T )] + det[Gi �→j

q (T )] = 0 :

If Re[�q(T )] ? 1 the alternation process will produce destabilization/stabilization ofthe initial instability. Therefore, the condition Re[�q(T )] = 1 determines the criticalstability curve. When switching produces a Turing instability, there exists a critical


value of the period T̃ below which Re[�q(T )]¿ 1. The internal time tint can be de9nedas half the value of the critical period T̃ such that

9Re[�q(T̃ )]9q

∣∣∣∣q=q∗

= 0; Re[�q∗(T̃ )] = 1 : (25)

For T ¡ T̃ one obtains a Turing instability of wavevector q∗ that maximizes �q(T ),

9�q(T )9q

∣∣∣∣q=q∗

= 0;92�q(T )9q2

∣∣∣∣q=q∗

¡ 0 : (26)

In the limit T → 0, the most unstable mode coincides with the one obtained by solvingEq. (13) using the e4ective average dynamics.

5. A particular example

Let us now illustrate the mechanism with a particular system. The simplest versionof the so called activator–substrate model reads [9]:

9u9t = a(u

2v− u) +∇2u;9v9t = (1− u2v) + D∇2v ; (27)

where convenient dimensionless units have been chosen. This model has been used todescribe pigmentation patterns in seashells [10,11] as well as the ontogeny of ribbingon ammonoid shells [12]. The variable v can be interpreted as the concentration ofsubstrate being consumed by an activator of concentration u. The positive constant adenotes a cross-reaction coeLcient. It is easy to check that there is a unique 9xed point(us = 1; vs = 1) independently of the values of {a; D}, but whose stability depends onthe values of these constants.According to the alternation scheme (17)–(23), the activator–substrate model presents

a switching-induced Turing instability if

ai ¡ 1 ; (28)

aiDi ¡ 1 ; (29)

(a1 + a2)(D1 + D2)¿ 2 ; (30)

(a1 + a2)(D1 + D2)7 (3∓ 2√2) ; (31)

where we have de9ned Di = Dvi = ciDv. Condition (31) means that one of the twoinequalities must be chosen. Notice that if the “greater than” condition is chosen forEq. (31), the inequality (30) is automatically satis9ed. We stress that if the systemevolves under either {a1; D1} or {a2; D2} satisfying Eqs. (28)–(31), no pattern developsand the system simply relaxes to the homogeneous state (u=1; v=1). However, globalalternation between the sets of constants {ai; Di} leads to a Turing instability whenT ¡ T̃ .


Fig. 1. Loci of points where Re[�q(T )]=1 for the activator–substrate model (solid line). A Turing instabilitydevelops if the period of alternation T is smaller than T̃ � 0:34. The most unstable mode as a function ofT is also plotted (dotted line).

A possible family of solutions of the system of inequalities (28)–(31) is

a1¡ 1 ; (32)

a2¡a1

(8√2− 117

); (33)

D1¡1a1; (34)

8√2 + 12

a1 + a2− D1¡D2¡

1a2: (35)

Moreover, two particular sets of constants {ai; Di} ful9lling conditions (32)–(35) are{ 34 ; 1} and { 3

100 ; 30}. We use these values in Fig. 1, where we depict the loci whereRe[�q(T )] = 1 (solid line) and the most unstable Fourier mode as a function of theperiod in the alternation process, q∗(T ) (dotted line). That is, inside the solid linethe alternation process leads to a Turing instability and the resulting pattern presentsa wavelength ∗(T ) = 2�=q∗(T ). Also, note that the tip of the loci determines theinternal time according to Eq. (25). The value obtained for the internal time is tint 0:17(T̃ 0:34). There is almost no variation of q∗ as T increases, decreasing monotonicallyfrom q∗(0) = 0:399 to q∗(T̃ ) = 0:393.To support these 9ndings, we have performed numerical simulations of the activator–

substrate model on a 64-site one-dimensional lattice using a second order Runge–Kuttascheme with periodic boundary conditions. The initial conditions for the concentrationsof the species u and v in these simulations were the stationary concentrations of thespecies obtained after a lengthy simulation with r → 0. This allows us to avoid theinitial transients for the cases r 1 and r ¡ 1. Other relevant parameters are Rt=10−3

and Rx=1. Note that under these conditions we expect L= ∗(T ) 4 pattern wavelengthssince q∗ 0:4.


Fig. 2. Gray-scale density plot of the concentrations of the chemical species u (left) and v (right) for r� 10.The species relax to the homogeneous state u = 1 and v = 1.

Fig. 3. Density plot of the species u (left) and v (right) for r� 0:1: a stationary pattern develops.

We focus on a periodic alternation scheme for di�erent values of the parameter r.As shown in Fig. 2, when r 10 the system relaxes to a homogeneous state where theconcentrations of the activator and the substrate are respectively us = 1 and vs = 1, asexpected. As r decreases a Turing instability develops, and when r approaches zero theinstability leads to a stationary pattern. In Fig. 3, where r 0:1, the stationary roll-likespatiotemporal concentrations of u and v are shown. The 9gure shows approximatelyfour full wavelength patterns, as expected.The most dramatic behavior is obtained when r 1. A resonance phenomenon be-

tween the two characteristic times yields oscillatory patterns. Fig. 4 shows the


Fig. 4. Density plot of the species u (left) and v (right) for r� 1. Note the oscillatory behavior of v.

Fig. 5. Detailed behavior of the oscillatory pattern shown in Fig. 4 for the species v during a period ofoscillation and throughout a wavelength.

oscillatory pattern obtained for the concentration v for r 1. The oscillations are de-tailed in Fig. 5, where v(x; t) is shown for one period T of the external forcing andthroughout a wavelength ∗ of the pattern.


6. Conclusions

We have shown how the non-equilibrium process of global alternation may lead topattern formation in reaction–di�usion systems. In contrast with other alternation mech-anisms for pattern formation, here relaxational processes do not play a role. Instead,pattern formation is triggered by short time instabilities ampli9ed by the switching pro-cess and by collective e�ects. We have shown that not all global alternation schemeslead to a Turing instability. Global reaction and di�usion alternation is a requirementfor the proposed mechanism. We have also discussed a number of simple reactionand di�usion switching schemes and 9nd an uncomplicated alternation scenario thatproduces patterns. We have illustrated the mechanism with the well known activator–substrate model. By performing numerical simulations on the model we have shownthat three di�erent scenarios can be obtained as a function of a control parameter r thatquanti9es the ratio between the characteristic times involved in the dynamics. Whenr ¿ 1 (slow alternation) the system relaxes to a homogeneous state. If r ¡ 1 (fastswitching) an e�ective average dynamics drives the system, producing a Turing insta-bility that leads to a stationary pattern. Finally, when r 1 a resonance phenomenonproduces oscillatory patterns. It is interesting to speculate on the possible connectionbetween the mechanism presented here and the Gashing ratchet mechanism in whichswitching between two sets of reaction rates in a chemical circular reaction with threestates also produces patterns [13].

Acknowledgements

This work was supported by the Engineering Research Program of the OLce ofBasic Energy Sciences at the U. S. Department of Energy under Grant No. DE-FG03-86ER13606. Support was also provided by MCYT-Spain Grant BFM 2001-0291 andby MECD-Spain Grant EX2001-02880680.

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[10] H. Meinhardt, J. Embryol. Exp. Morph. 83 (1984) 289;H. Meinhardt, M. Kingler, J. Theoret. Biol. 126 (1987) 63.

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patterns in reaction–diffusion systems generated by global alternation of dynamics

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