- Email: [email protected]

Physica A 240 (1997) 571 585

Nonequilibrium potential approach: Local and global stability of stationary patterns in an activator-inhibitor system with fast inhibition Germ~in Drazer 1, Horacio S. Wio* Centro At6mico Bariloche (CNEA) and lnstituto Balseiro (UNC), 8400 S. C de Bariloche, Argentina

Received 6 September 1996

Abstract

We study the formation and global stability of stationary patterns in a finite one-dimensional reaction~liffusion model of the activator-inhibitor type. The analysis proceeds through the study of the nonequilibrium potential or Lyapunov functional for this system considering the fast inhibitor case and, in order to obtain analytical results, the adoption of a piecewise linear version of the model. We have studied the changes in relative stability among the different patterns as the ratio between the diffusion coefficients is varied and have discussed the meaning of the different contributions to the nonequilibrium potential.

1. Introduction

The subject o f pattern formation in systems far from equilibrium has become one of the most active fields in the physics of complex systems, both from the theoretical and experimental points o f view [1]. From the theoretical side the overwhelming variety of systems that it is possible to consider might suggest that the number o f possible different descriptions will also be large. However, there are some common principles that make it possible to apply similar mathematical descriptions even to different systems. We will concentrate on one o f those descriptions, the reaction-diffusion (RD) approach [2], that has shown to be extremely versatile for modelling realistically many interesting pattern forming phenomena both in natural and social sciences. Recent analysis for one- and two-component RD systems have shown that boundary conditions (b.c.) play a relevant role in the formation and stability o f patterns [3 5]. * Corresponding author. Tel: +54 944 45193; fax: +54 944 45299; E-mail: [email protected] t Permanent address: Departamento de Fisica, Facultad de lngenieria, UNBA, Argentina. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PIIS0378-4371(97)00047-2

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For one-component systems it was also shown how those b.c. can control the relative stability of the locally stable inhomogeneous stationary patterns [6-8]. In the latter case, the concept of the nonequilibrium potential pioneered by Graham and coworkers [9] was successfully exploited. However, in such one-component models the form of the nonequilibrium potential is well known [10] at variance to what happen in systems with several components. There is a particularly interesting problem in active media and RD systems that has attracted the interest of both experimentalists and theoreticians [11,12], namely the effect of the fluctuations or noise, because they can induce transitions between the different locally stable attractors. The decay of homogeneous metastable states has been studied for a long time [13,14], while the decay of metastable states in excitable RD systems was only recently studied [6-8,15]. For the case of one-component systems [ 6-8] the information obtained from the above-indicated nonequilibrium potential gave the possibility to study the decay of an inhomogeneous metastable state by means of a path integral technique introduced in Ref. [16]. It also offers the possibility of discussing the phenomenon of stochastic resonance in spatially extended systems [17,18]. Here we shall focus on a specific system belonging to a family of two-component models having a broad range of applications known as activato~inhibitor models. We want to present an analysis of the global stability of stationary patterns. In order to reach this goal we shall profit by using the concept of Lyapunov functional. This kind of approach, i.e. based in the use of the Lyapunov functional, has been scarcely exploited within the realm of reaction-diffusion systems because it is not a trivial task, insofar as some potential conditions are not fulfilled, to obtain a Lyapunov function in a general problem. However, in the present case we will consider a particular form of the activator-inhibitor model, assuming a fast inhibitor, where such a Lyapunov functional can be found even though the system does not fulfill the above-mentioned potential conditions [19]. In the following sections we present the specific model we will work with and some particular solutions, the case of fast inhibitor and the form of the nonequilibrium potential, an analysis of the stability of the previous solutions, the behaviour of this nonequilibrium potential for these solutions, and its relation to some special features of the pattern. The last section includes a final discussion of our results.

2. The model

The general formulation for the activato~inhibitor model in one spatial dimension reads [ 10]

Otu =Du~Zu+ f ( u , v ) ,

Otv=Dv~v+g(u,v),

(1)

where Du and Dv are the diffusion coefficients of the activator u(x, t) and the inhibitor v(x, t), respectively. Both, u(x,t) and v(x, t) are real fields representing the magnitudes

G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585

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of interest and the nonlinear terms f ( u , v ) and g(u,v) are the source (or dissipative) terms. The nullclines, that is the intersections of those (generally nonlinear) source terms with the (u,v) plane, show characteristic shapes which can be described, typically, by a convex line for if(u, v) and a general cubic-like one (with two extrema and one inflection point) for f ( u , v) [5,20]. Those projections intersect each other at the origin (which counts for a trivial solution) and eventually on both sides around the local maximum of f . Those extra intersections anticipate nontrivial homogeneous solutions and spatial patterns arising from the bistable situation of the system, which have been analysed by several authors [5,15,20,21]. We start with a simplified (piecewise linear) version of the activato~inhibitor model alluded to above, which preserves the essential features, and fix the parameters so as to allow for nontrivial solutions to exist. After scaling the fields in a standard manner, we get a dimensionless version of the model as ?,u(x,t)=D,O~u

u+O[u-a]

(?,v(x, t) = D,,(?2v + [3u - 7v .

v, (2)

We confine the system to the interval - L < x < L and impose Dirichlet boundary conditions in both extrema. The discussion of the more general albedo (or partially reflecting) boundary conditions, as in Ref. [5], will be left for a future work. According to the values of the parameters a, /3 and 7, we can have a monostable or a bistable situation as qualitatively indicated in Fig. 1. In the second case we have two homogeneous stationary (stable) solutions. One corresponds, in the (u, v) plane, to the point (0,0) while the other is given by (uo, vo) with //0

--

/~+v'

U0

--

/~+:'

implying that the condition 7/(/~ + 7) > a must be fulfilled. Without loss of generality we may assume that 0 < a < g1 and u0 < 2a [20]. The inhomogeneous stationary patterns appear due to the nonlinearity of the system, and ought to have activated regions ( u > a ) coexisting with nonactivated regions (u < a). This fact, together with the symmetry of the evolution equations and boundary conditions, implies the existence of symmetric inhomogeneous stationary solutions. We restrict ourselves to the simplest inhomogeneous, symmetric, stationary solutions. That is, a symmetric pattern consisting of a central region where the activator field is above a certain threshold ( u > a ) and two lateral regions where it is below it (u < a). As it was already discussed [20,5], different analytical forms (which are here linear combinations of hyperbolic functions) should be proposed for u and v depending on whether u > a or u < a . These forms, as well as their first derivatives, need to be matched at the spatial location of the transition point, which we called xc. Through that matching procedure and imposing boundary conditions we get the general solution for the stationary case. In order to identify the matching point x,. we have to solve the equation u(x,.) = a, resulting in general in a transcendental equation for x,. In order to avoid the

574

G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585 V

.2 it / tt /

\ 1 I

t

i Ii t

•

\

Fig. I. The nullclines in the piecewise linear approximation.The monostable situation (1) and the bistable case (2) are shown.

u(xFa 0

Fig. 2. u(x) - a versus x. The roots of this function (transcendental equation) are the possible values of the transition point (xc).

complications arising from the possible spatially oscillatory behaviour o f the solutions, we will work in a parameter range where the diffusion coefficient o f the activator Du is lower than some critical value D °so [20,21], beyond which the solutions became spatially oscillatory. In Fig. 2 we sketched a typical result for the transcendental equation for Xc. As we can see, there are some roots with Xc < L / 2 and others with xc > L / 2 . In particular, there are four different solutions for Xc, and associated with each one we have different stationary solutions that we will indicate by Uel, Ue2, Ue3 and ue4, with increasing values o f the transition point xc. To illustrate this, in Figs. 3(a) and (b), we show the stationary solutions Ue2 and Ue3. m linear stability analysis o f these solutions indicates that uel and Ue3 are unstable while u~2 and u~4 are locally stable. The stable states will correspond to attractors (minima) o f the functional while the unstable ones will be saddle points, defining the barrier height between attractors.

575

G. Drazer, H.S. Wio/Physica A 240 (1997) 571 585

U~VI

lu,v

( b}

(a) u0

X\,\/

i

*L x

-L

-L

.L

Fig. 3. Typical stationary solutions of the activator and inhibitor: (a) the u~2 stationary solution: (b) the u,,3 stationary solution.

3. Nonequilibrium potential We write now the equations of our system specifying the time scale associated with each field. This allows us to perform an adiabatic approximation and obtain a particular form of the nonequilibrium potential for this system. Measuring the time variable on the characteristic time scale of the slow variable u (i.e. z,,), Eqs. (2) adopt the form [10] ~tu(x,t) = DuOZxu(X,t) - u ( x , t ) + O [ u ( x , t ) - a] - v ( x , t ) ,

(3)

qO, v ( x , t ) = D~,OZv(x,t) + flu(x,t) - - 7 v ( x , t ) ,

where q = z,~/z~. At this point we assume that the inhibitor is much faster than the activator (i.e. z~:<~%). In the limit q---+0, we can rewrite Eq. (3) as ? , u ( x , t ) -: D~OZu(x,t) - u ( x , t ) + 6)[u(x,t) - a] - v ( x , t ) , 0 = O~?~v(x, t) + flu(x, t) - 7v(x, t).

(4)

In the last pair of equations we can eliminate the inhibitor (which is now slaved to the activator) by solving the second equation using the Green function method {-D~,a~ + 7 } G ( x , x ' ) = 6(x - x ' ) , v(x) : fl.

/

(5)

dx' G ( x , x ' ) u ( x ' ) ,

where the Green function G ( x , x ~) is given by G(x,x'):

1 sNk(L-x')]Sh[k(L+x)] D,k Sh[2kLl 1 Sh[k(L--x)] q h r ~ t l

ShE2kq ~ - L ~ + x ' ) ] ,

x

'

x>

x t

(6)

,

with k = (7/D~:) 1/2. This slaving procedure reduces our system to a nonlocal equation for the activator only, that has the form ~,u(x, t) = D , ~ u ( x ,

t) - u(x, t) + 6)[u(x, t) - a] - [ 3 [ ' d x ' G ( x , x ' ) u ( x ' ) . J

(7)

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G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585

From this equation, and taking into account the symmetry of the Green function G ( x , S ) , we can obtain the Lyapunov functional for this system, which has the form

~g[u] O,u(x, t) -

6u

'

u. ~[u] =

{axU}2 + - f - (u - a ) O [ u - a]

dx

af ax'G(x,x')u(x')u(x)}

+~-

(8)

The spatial nonlocal term in the nonequilibrium potential takes into account the repulsion between activated zones. When two activated zones come near each other, the exponential tails of the inhibitor concentration overlap, increasing its concentration between both activated zones and creating an effective repulsion between them, the Green function playing the role of an exponential screening between the activated zones. We can now exploit this Lyapunov functional in order to discuss the stability of the stationary solutions found earlier. We propose for the activator that

(9)

U = Ue + (9

where u e is one of the stationary solutions of the system indicated in the previous section. Replacing this into Eq. (8) and expanding up to second order in (9 we get 1

f

,~[U] ~ ff[Ue] + 2 J d x ~ ( x ) ~

f { - D u ~2 q - l - ~ [ U

e --a]}

(9(x) (lO)

"lUel+ il "x" ~2[Ue] = { - O , C3x~+ 1 - 6[Ue - a]} 6[x - x ' ] + f l G ( x , x ' ) .

(11)

(12)

From the previous equations we can see that, obtaining the "curvature" of the potential is equivalent to diagonalizing the operator ~-~2[Ue] and finding its eigenvalues. Such an analysis is completely analogous to the linear stability one. In order to find the eigenfunctions (9k(x) and their associated eigenvalues ogk we propose for (9 in Eq. (9) the form (gk(x, t) = e°'~t (gk(x ) .

(13)

To perform the calculation we need to solve the following equation: (Dk(9k(X) = - - / d x

! ~2[Ue](gk(X') .

(14)

G. Drazer, H.S. Wio/Physica A 240 (1997) 571 585

577

Depending on the sign of cok the stationary solution will be stable or unstable. However, in order to assess the stability of a particular state it is enough to determine the largest ~/9k .

According to Eq. (10), we can rewrite Eq. (14) as

-- (')kOk(x)= {--Ducq2 + l -- Z K-i-'iJ[X-- Xci]} (15 where t,i = Idue/dx]......... and xci are the transition coordinate points referred to at the end of the previous section. We can interpret the last equation as a Schr6dinger equation EkqSk = ~ b ~ ,

E~ = -cok,

(16)

where the "Hamiltonian 9(f" has a kinetic energy pZ/2m = - D , 0 ~ , a potential energy V ( x ) = ( 1 -~-~x,, •716[ x -xci]), plus an additional nonlocal interaction term [3,10,19]. As indicated above, the stability is given by the maximum of all the oJk's (co,,,,x) or, equivalently, by the smaller energy eigenvalue. Hence, this is equivalent to find the ground state of a quantum problem, with this eigenvalue corresponding to the most unstable mode. If the energy of such a bound state is negative, the corresponding stationary state u,, is unstable, while if positive the state is stable. The existence of such bound states is possible due to the attractive 6-contributions to the potential. The sign of the energy of the bound state also depends on the intensity of the 6, which depend inversely on the value of ~i, that is of the derivative of the stationary state at the transition points. Hence, it is reasonable to expect that the stationary solution will be more stable if it has a steeper transition zone between the nonactivated and activated values. From symmetry considerations it follows that the lowest energy will correspond to a state symmetric about the origin. Clearly, this analysis will give the same result as the linear stability analysis indicated in the previous section.

4. Fast diffusion of the inhibitor In this section we want to analyse the stability of the stationary solutions we have just found, as functions of the activator diflhsivity. To do this, we need to obtain expressions for the roots of the characteristic polynomial. Such a polynomial arises within the standard linear stability analysis [2,10,20] of the stationary solutions we have found in Section 2. For such an analysis we need to write, for the activator as well as the inhibitor, forms like the one in Eq. (9) and, after substituting into Eq. (3) or (4), to linearize them for the perturbations ~bu and q~v. The usual assumption that the perturbations have a time dependence of the form exp~t (as indicated in

578

G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585

u(x)-a

L

Fig. 4. u(x) - a versus x. The behavior of the transition points as Du changes. Eq. (9)) yields (imposing continuity and boundary conditions on the spatially dependent part) a polynomial of fourth degree in ~, i.e. the so-called characteristic p o l y n o m i a l . The sign of the real part of the roots ~ determines whether the associated stationary solution is linearly (locally) stable or not. When D, ~Dv, the roots of this characteristic polynomial [19] reduce to ~1 ~ D u I/2 ,

~2 ~ {(fl + Y ) / D v } 1/2 .

(17)

These values indicate that there are two characteristic lengths: a very short one of order ~,r~l/2, and a comparatively large one, of order ~/31/2 . Hence, the variations in the terms including e +~'x will became discontinuities in the limit Du --~ 0 [20]. Strictly in the limit D, = 0, there are only two stationary solutions, as the order of Eq. (2) is reduced by two, as well as the degree of the characteristic polynomial. The behavior of the roots for different values of D, are sketched in Fig. 4. It is also seen that, when Du > D m~x with D m~' some threshold value (determined in Ref. [20]), the stationary solutions cease to exist. In Figs. 5(a), (b) and 6(a), (b) we depicted the same stationary solutions as in Figs. 3(a), (b) but now for the case where D, ~Dv. In Figs. 7 ( a ) - ( d ) we show the dependence of the most unstable eigenvalue (ok for the different stationary patterns (Uel,Ue2,Ue3,Ue4). AS the diffusion coefficient D~ increases, we can see that the solutions become unstable as D, crosses the value D°S( (indicated in the figures). This corresponds to a subcritical bifurcation, and this solution ceases to exist beyond some value D,m a x . In the previous section it was shown that the pattern stability is strongly related to the derivative of the activator's stationary solution at the transition points between the activated and nonactivated regions. In Fig. 8 we depicted the value of this derivative vs. D,. In Fig. 9 we show the dependence of the Lyapunov functional versus /9, for the different patterns. From this, we see that this dependence of the derivative and the Lyapunov functional for the different patterns, Uel through Ue4, is compatible with the results of the stability analysis. Before closing this section it is worth presenting some kind of phases diagram indicating the parameter regions where the different kind of solutions might arise. The

G. Drazer, H.S. WiolPhysica A 240 (1997) 571 585

0.8

579

I Ll,',/ 2

0.6

(a)

0.4

0.2

[

I

i

i

lO

7.5

7.5

10

x

U~V

0.8

A (b) Ue2

0.4

\ I

I

I

10

7.5

75

I

,

10 x

-0.2 Fig. 5. The stationary solutions of activator and inhibitor Ue2, t,e2: (a) L = 10, fi D,, = 0.25; (b) L = 10, fi = 0.4, ? = 0.3; Dv 1, Du = 0.000625.

0.4, 7 = 0 . 3 ; D , = I,

first one, shown in Fig. 10, corresponds to the D ~ - D , , plane for a fixed value of a (the threshold parameter). Such a diagram is obtained analysing the characteristic polynomial as well as the associated diagram for u ( x ) - a (as in Figs. 2 and 4). As is apparent, we find three regions: region (1) with nonoscillatory solutions, region (2) with spatially oscillatory unstable solutions, and finally region (3) without inhomogeneous solutions. The lines separating these regions have been obtained by analytical or numerical means. For instance, the line separating regions (2) and (3) was numerically obtained resulting in a straight line (at least for the indicated range of parameters). The other line, separating regions (1) and (2), can be obtained analytically and is also

G. Drazer, H.S. Wio/Physica A 240 (1997) 571 585

580

a straight line (this happens when L, the system size, is much larger than xc, making the approximation L ~ c~ reasonable). Another possibility is to analyse the a - D~, plane, for a fixed (and small) value of the activator diffusivity (D, ~ 0). In Fig. 11 we show the results. In this case we find only two solutions for the transition coordinate

0.8

u,v

(o)

_ _ ~

o.6

Je3

0.2

-10

-7.5

-5

-2.5

I

I

r

2.5

5

75

10 x

(b)

' LI~V

0.8

1/

Ue3

1

I

I

2.5

5

7.5

¢40×

060.4.

(

0.2E -7.5

I -5

I

-2.5 -0.2

Fig. 6. The stationary solution of activator and inhibitor Ue3,tPe2: (a) L = 10, fl D, =0.25; (b) L = 10, fl=0.4, 7=0.3; Dv 1, D, =0.000625.

0.4, 7=0.3; D~,

I,

(XlmaX (o)

3 2.5 2

"°'"'..

! L

1.5 1

-"

"'"'° "".,..,°....o..,..o..°

°°°.,

0.5 -0.5

I

I

I

I

t

I

0.1

0.2

0.3

0.4

0.5

0.6

I

•

1/2

Du

-1 Fig. 7. The dependence of the most unstable eigenvalue for the different stationary patterns as the diffusion coefficient D. increases.

G. Drazer, H.S. Wio/Physica A 240 (1997) 571 585

581

(b)

(-,Om a x

0.4 0.3 0.2 0.1

Dosc, ..

................... 0.3.

-0.1

o.~

o.s

.-

Din

-0.2 -0.3 -0.4

C0max

(C )

0.4 0.3 D2 .,.°.o. ...... •

01

...........'""" , ......

T ......

0.1

D OSC

[

1

1

1

i

0.2

03

0.4

0.5

0.6

1[

D1/2 U

-0.1

COmax

{d )

0.4 0.2 -0.2

D oSc ..

I

I

I

i

~

OI]

0" 2

0"3

0 I4

0"5 ,.'°

....

D~ 2

.,"

-0.4 -0.6

-0.8

!.-I .....

,..-"

..."

..."

. ,,,,'" °,,.,.,'°""

-1

Fig. 7. Continued.

xc: x + > L/2 and x~- < L/2. Again we can define three regions: (1) where u0 > 2a; (2) a region without (inhomogeneous) solutions, separated from region (3) (where two solutions are possible) by a line corresponding to the condition x c+ -- x~.- = L/2, and on the far right side of the figure, the curve corresponding to x~7 : 0 and x cJr = L ,

5. Final comments In this work we have analysed a simplified piecewise model of the activator-inhibitor type within the framework of the nonequilibrium potential [9]. This nonequilibrium potential approach has been scarcely exploited within the realm of reaction-diffusion

G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585

582

X=X C

2

~e2 :

1.75

".Ue4

Uel •

".i i:::!ii i!!!

1.5 1.25

•

1

0.75

;IL-..

......:::!~::~,

0.5 0.25 I

I

I

Fig. 8. The derivative of the stationary patterns at the transition point versus r)l/2 F [ue]1

0.3

Ue3

0.2 0.1

]

0.

0.2

~

0.4

0.5

0.6 Dul/2

Fig. 9. The value of the nonequilibrium potential at the stationary pa~ems as the diffusion coefficient D, increases.

Dv 2 1.75 1.5 1.25

3

1

•

0000000000~ ,•

0.75 0.5

•

2

1

025 0

~,~'~

~I

0.25

i

I

I

I

I

I

I

05

0.75

1

1.25

1.5

1.75

I

2 Du

Fig. 10. Phase diagram on the Du-Dv plane for a fixed value of a. In region (I) we have nonoscillatory inhomogeneous stable solutions; in region (2) we find spatially oscillatory unstable solutions; while (3) is a region without inhomogeneous solutions.

G. Drazer, H.S. Wio/Physica ,4 240 (1997) 571 585

583

Dv 100

8ol 60 40 20 0

I

[

I

I

0.1

0,2

0.3

0.4

0.5

a

Fig. 11. Phase diagram in the a-D~, plane, for a fixed D, (D, ~ 0): region (1) corresponds to the case x0 > 2a; (2) is a region without (inhomogeneous) solutions; while (3) is a region with two possible solutions. The line on the right of the figure corresponds to x~7 = 0 and x+ = L. systems as the required potential conditions are not fulfilled in general. Here we have assumed the fast inhibitor limit, which is a case where the Lyapunov functional can be found even though the system does not fulfill the above indicated potential conditions [19]. It is worth remarking here that recently, some authors have used the same form of the Lyapunov functional in the limit of fast inhibitor, in order to describe the dynamics o f front motion in a chemical system [22]. In the general case the original two equation system can also be reduced to an effective equation for the activator. However, besides the nonlocality in space a nonlocality in time also arises. Such nonlocality in time is purely due to the fact that the characteristic time for the inhibitor is comparable to the activator one. In this case, asymptotic time oscillatory solutions arise [20,15]. The lack of an analytical knowledge of this solution prevents the possibility of finding a local approximation for the Lyapunov functional. We have found that the usual linear stability analysis is obviously equivalent to finding the curvature o f the nonequilibrium potential around the stationary solutions. Also, the potential allows us to interpret the nonlocal term as an effective repulsion between activated zones, as is known from numerical and experimental evidence [1]. Even more, we found the values o f the potential on the different attractors yielding the globally stationary stable state as well as the barrier height between those attractors. These are necessary information when one wants to estimate the mean-life-time o f a metastable state. Such a barrier height can be obtained for a given value of D, from the difference between the Lyapunov functions of the different stationary stable and unstable solutions. For instance, from Fig. 9, in the region 0.1 < D, < 0.3, the homogeneous stationary state is metastable (its Lyapunov functional is zero) while the most stable state is u,,2. The barrier separating these two local attractors is given by the difference between the Lyapunov functional of the unstable state u,,i (which is a saddle) and the one corresponding to the metastable homogeneous state, i.e. A,YL.0 = , ~ [ u ~ l ] - ,~[u,,0] = .~[u,,i ]. Also the barrier separating the other metastable state from the globally stable one, will involve the other saddle Ue3, and is given by A.Y3,4 = - ~ [ U e 3 ] - ~[Ue4]. In the present

584

G. Drazer, H.S. Wio/Physica A 240 (1997) 571-585

case we have used adimensional variables, hence, in order to estimate the possible magnitude of such barriers in a realistic case, we must introduce realistic values for all the parameters (diffusivities, coupling constants, etc.), and get the corresponding realistic values of the Lyapunov functional. The estimate of the decay rate of an inhomogeneous metastable state for the case of one component systems [6-8] was based on a path integral technique introduced in Ref. [16]. However, the application of such techniques here requires a better knowledge of the "topology" of the potential and a detailed study of the validity of the approximations in Ref. [16] for the present case. The present approach can be extended to higher-dimensional systems in a straightforward way (provided the indicated condition, i.e. fast behaviour of the inhibitor, is fulfilled). In such a case, we expect that it will be possible to evaluate the Lyapunov functional for a region with "simple" attractors (for instance, localized disks or stripes, see Ref. [21]), and to discuss their relative stability. However, the case of the so-called breathin9 modes, associated with limit circles, will require a deeper study, beyond the present one, exploiting the ideas from Ref. [9]. Moreover, much care should be exerted when studying infinite systems, due to the possible unboundness of the Lyapunov functional. Finally, we have found a tight connection between the stability of a stationary inhomogeneous pattern and the derivative of this pattern at the transition point. This fact can be understood in terms of the existence of bound states in the associate quantum well. Such a result means that a stationary solution will be more stable when the transition regions are steeper. However, this result is not exclusive of the present piecewise linearized version of the model, but can be found in a more general form of the nonlinear source [23]. Another interesting point to be studied is related to the critical-like behaviour when a stable and an unstable pattern coalesce and disappear as some control parameter is varied [24]. Also, we can exploit these results to discuss stochastic resonance in manycomponent extended systems [17,18]. This as well as other aspects will be the subject of further work.

Acknowledgements The authors thank R. Graham, D.H. Zanette, R. Deza and G. Izfis for fruitful discussions, E. Tapia for his help with the numerical work and V. Griinfeld for a critical reading of the manuscript. Partial support from CONICET, Argentina, through the grant PID 3366/92, and from Fundacirn Antorchas, are also acknowledged.

References [1] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851; E. Meron, Phys. Rep. 218 (1992) 1. [2] G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems (Wiley, New York, 1976); P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics,

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