Unified Theory of Temporal and Spatial Instabilities Institute of Physics AND H. DEGN Institute of Biochemistry Odense University Denmark Receiced 29th July 1974 Chemical reaction-diffusion systems are discussed with the view of clarifying the relation between spatial and temporal concentration variations. The predicted absence of oscillating or propagating structures in closed two component systems which are not oscillatory when stirred points to the necessity of studying systems with more than two components. 1. INTRODUCTION The occurrence of concentration oscillations in homogeneous chemical systems was a matter of controversy for several decades until the sixties when there came a boom in experimental and theoretical work on the subject. The existence of homogeneous chemical oscillations then became firmly established.This had hardly happened before the logical extension of temporal periodicity namely spatial periodicity took over the role of the flying saucers of chemistry. The possibility of the emergence of a spatial concentration pattern in an initially homogeneous chemical system seems to have first occurred to Turing.' He was led to the idea when searching for an explanation of the stage in morphogenesis when a small spherical array of identical cells assigns different parts of its surface to develop into different organs. Turing's work published in 1952 only became famous after 1967 when the Brussels school began the publication of a series of papers on spatial instability in homogeneous chemical systems.The report by Zhabotinskii on spatial phenomena in the Belousov reaction at the Prague Symposium on Oscillating Reactions in 1968 inspired experimental work on spatial patterns in many labora- tories. Despite the parallel courses of the theoretical and experimental approaches it is not always evident that the theoreticians and the experimentalists talk about the same thing. The pretentious title of this paper reflects the fact that temporal and spatial instab- ilities are closely related phenomena as they both require reaction schemes with feed- back. The aim of the paper is to clarify this relationship and to give operational interpretations of theoretical results regarding spatial instabilities and spatial period- icity. The concept of concentration oscillations in a homogeneous phase is so straight-forward that it needs no special introduction.Spatial periodicity on the other hand is an equivccal expression covering different phenomena which may be easily confused. Before we proceed to theoretical analyses we shall therefore classify these phenomena in operational terms. 233 TEMPORAL AND SPATIAL INSTABILITIES Imagine a solution in an ordinary vessel made of non-reactive material. We stir the solution and it does not exhibit time periodicity as long as it is stirred. When we stop the stirring conceivably we may observe that concentration gradients form spontaneously in the solution. These gradients may tend to form a time-invariant structure or they may be periodic in time either as propagating or as oscillating standing concentration waves.A reaction solution which forms spatial structures after it was homogenized by stirring and did not exhibit concentration oscillations whilc stirred is said to be spatially unstable. Current theory predicts spatial instability in reaction systems which comprise diffusion of chemical components and certain types of chemical reactions with feed-back loops. In the discussion below systems which produce time-invariant structures are said to have aperiodic spatial instability and systems giving rise to coilcentration waves are said to have oscillatory spatial instability. Let us now return to an imaginary experiment where a solution in an ordinary vessel with non-reacting walls exhibits concentration oscillations when stirred.When we stop the stirring conceivably we may observe the emergence of concentration structures which are oscillating with the same period as the previous homogeneous oscillation. In this case spatial instability is not involved. Time-dependent structur:can be explained by small local heterogeneities or externally induced gradients causing different parts of the oscillating solution to come out of phase. Such patterns in oscillating systems will tend to be destroyed by diffusion of the reaction components of the chemical system. This paper is concerned with structures and waves which result from spatial instability and thus grow and are maintained with the assistance of diffusion. 2. TWO-COMPONENT SYSTEMS We consider a system in which two substances A and B are present with concen- trations A(r t) and B(r t) as function of space P and time t.The relevant reactions are assumed to be fully described by two rate equations 8AS -= fA(A B)+DAV2A at dB -= fn(A B)+DBV2B at wheref andf describe the chemical reaction rates DA and DBare diffusion cozfTi-cients and V is the gradient operator. We shall first neglect the spatial dependence of A and B by assuming stirring or large diffusion coefficients. As shown in several works a linear theory provides important information on the dynamics near singular points (As,B,) in the A-B phase plane where For sniall deviations from the singular point a(t)= A(t)-A b(r) = B(t)-R one obtains da -= mAAa+mABb dt I.BALSLEV AND H. DEGN 235 171 are partial derivatives where the coefficients a f (3) in the singular point. Solations to eqn (2) may be written a = a. exp((,u-iio)t) b = bo exp((p-ico)t) (4) where p is the growth rate and co is the angular frequency. The complex growth rate 3. = ji-ico is then determined as the roots of the secular equation (?nAA -l)(mBB -?L) -mABn1BA = 0. (5) Let us denote these roots by A1 = pl+io and l2= p2-io. For LO + 0 we have pl = p2 and we shall in this case denote the singular point as oscillatory unstable if ,ul > 0 and oscillatory stabZe if p1 < 0. For CD = 0 the singular point is aperiodically stabZe if both ,ul and p2 are negative. Two positive values correspond to aperiodical instability and different signs lead to a saddle type instability.In the later discussion of spatial effects this notation is slightly more suitable than the conventional terms focus node and saddle. Asit turns out that the case mABmBA 2 0 is uninteresting in connection with spatial effects included later we shall assume mABmBA < 0. In a qualitative discussion it is convenient to scale the time unit and the unit for the concentrations a and b. In such rationalized units the complex growth rate is obtained from solutions to (.-A)(p-n)+l = 0 (6) where The solutions of eqn (6) are conveniently represented in the ap parameter space with domains for different types of dynamics according to the values of and &. The domains are shown in the ap diagram in fig.1. This graphical representation is useful in the analysis of specific chemical reaction schemes. For fixed rate constants the scheme defines a single point (a,p). For example in the reaction scheme of Lotka (1910) the points obtained for different values of rate constants fall on the negative p-axis and the Volterra-Lotka scheme has always (E p) = (0,O). In the reaction scheme of Prigogine and Lefever it is possible to obtain points in the region between the hyperbola branches and the axes. This latter scheme and reaction systems involving forward inhibition or an enzymatic step were analyzed previously by transforming the ap diagram into a parameter space directly related to the chemical rate constants. Including the spatial dependence of A and B of eqn (1) and assuming a spatially periodic solutioii we find in the linear theory a(r t) = a exp(,ut -iot + ik.r) b(r,t) = boexp(pt -iot + ikx) where k is thc wave vector defining a propagation direction and a wave length of 2x/(kl.Introducing a scaled unit for length we obtain the secular equation (u-DAk2-A(k))(p-D,k2-i(k))+1 = 0 (9) where k is the magnitude of k and I@) = p(k)-ico(k) is the scaled complex growth rate for wave vector k. 236 TEMPORAL AND SPATIAL INSTABILITIES FIG. l.-Domains in the ap-plane for different types of dynamics near singular points in two. component systems. 0s and AS correspond to oscillatory and aperiodic stability respectively- The character of instabilities are abbreviated to 01 A1 and SI corresponding to oscillatory aperiodic and saddle type instability respectively.Comparing eqn (6) and (9) we note that the dynamics of periodic patterns are obtained by subtracting DAk2and DBk2from a and /I,respectively corresponding to a displacement in fig. 1 towards negative parameter values along a line with slope D,/DA. These considerations lead to a classification of spatio-temporal behaviour near singu- lar points shown as domains 1-111 in fig. 2. On the left hand side of these domains there is global stability and on the right hand side the long wave limit is aperiodically unstable. In the latter case the non-linear terms are generally unable to prevent a FIG.Z.-Instability domains used in the discussion of spatio-temporal development beyond the linear approximation.The slope of the straight line is DB/DA. Consequently no lines with this slope through domain I intersects the domain for saddle type instability to the left of the hyperbola. In the figure DslD~m-9. I. BALSLEV AND H. DEGN global drift away from the singular point. In domains I and I11 the long wave limit* is oscillatory unstable while domains I1 and 111 have aperiodically growing solutions in a finite wave vector interval. Domain I1 and 111 vanish unless the self- activating component (here the A-component) is diffusing slower than the other component. The necessity of having D,/DA > 1 for obtaining aperiodical growth for finite wave vectors has been clearly stated by Edelstein and by Segel and Jackson.' The discussion of the further development of unstable systems beyond the linear approximation is limited to systems which do not drift away to other singular points and to closed systems having zero gradient of all concentrations perpendicular to the to the surface of the system.Such boundary conditions allow all space-independent solutions to the rate equations. Under these conditions the nature of the final state is closely related to the position in the parameter space of the point (a,/3) (see fig. 2). In Domain I-systems the homogeneous solution (k = 0) has the largest growth rate and is allowed by the boundary conditions. The final state of such systems is a homogeneous limit cycle. In Domain I1 the long wave limit is stable and there is saddle type instability in a limited wave vector range k < k < k,.If the boundary conditions allow the pres- ence of Fourier components in this range the system develops into states with large deviatims of the steady state. So far the ultimate destiny of such systems has not been analysed in general. Numerical integrations 6* on specific reaction schemes seem to indicate that the above situation generally leads to stable structures. How-ever at present it is not established if a system with specific spatial and chemical parameters allows a multiplicity of different stable patterns and to what extent these are accessible from a a slightly perturbed homogeneous steady state. From the linear theory it is obvious that below a critical size L,xn/k of the largest linear dimension there is spatial stability k2 being the largest wave vector with positive growth rate.In the case of niuch larger systems it would be interesting to search for general trends concerning the favoured spatial wave length of the final pattern. Oscillatory instability in the long wave limit and aperiodic growth of patterns is simultaneously present in Domain I11 (see fig. 2). Numerical integration in one dimension on such systems indicates that homogeneous temporal oscillations are strongly favoured if the concentration gradients are zero at the boundaries." In fact the existence of any form of stable patterns in closed Domain 111 systems is questionable. We can summarize that in closed two-component systems spontaneous oscillations tend to be homogeneous and spatial instabilities will develop into time-independent structures.For the experimentalist the most pertinent result of the above analysis is first that the formation of stable spatial structures may occur from an initially homogeneous solution. Secondly the interpretation of observed propagating structures in systems which are not oscillating when stirred cannot be based on two-component models. Thirdly in order to obtain spatial instability the self-activating component must diffuse considerably slower than the other one. In the search for systems capable of forming time-independent patterns one should look for reaction systems where an autocatalytic component is considerably more bulky than its precursor. This condition may not be fulfilled in the Belousov reaction where the autocatalytic species almost certainly is a fragment such as BrO of the bromate ion." Since bromate is present in excess it should not be considered as a variable component.The functional precursor of the autocatalytic species in the * The long wave limit corresponds to homogeneous solutions. TEMPORAL AND SPATIAL INSTABILITIES Belousov reaction is cerous ion. We have no data on the diffusion coefficients of these substances. However cerous ion is probably the bulkier one because of its tendency to form complexes. In the Bray reaction the autocatalytic substance is probably a fragment of the iodate ion and its functional precursor is molecular iodine. The large mass of molecular iodine and possibly its tendency to form complexes suggest a large diffusion coefficient compared to that of an 10 species.Thus neither of the systems mentioned above seem to have ratio of diffusion Coefficients required for spatial instability in a two-component model. 3. THREE- AND FOUR-COMPONENT SYSTEMS The absence from two-component systems of oscillatory spatial instabilities is believed to be a consequence of their simplicity rather than a general trend in reaction- diffusion systems. This suggests that models with more than two components should be investigated. We have studied a special class of three- and four-component systems nameiy those containing a chain of first order reactions with a feed-back on the first reaction from the last intermediate in the chain. Such systems can be symbolized as folIows s-I -+A-+B-+C-+ -1 I -+A+B-+C-+D-+ where the feed-back is activation or inhibition.The structure of the matrix of partial derivatives (cf. eqn (3)) for a chain of three components is given by where AfP = IF],+ Dpk2 t 12) and niA,mBand rn are positive (no self-activation). This system can be shown to be globally stable ifm c 0 (backward inhibition). For nz0 > 0 the system may have oscillatory or aperiodic instabilities. However as shown in the Appendix the oscillatory solutions have increasing growth rate for increasing wave lengths in the vicinity of marginal stability (dp(k2)/d(k2) < 0 for A(k) = +iu) whereas aperiodic spatial instability may occur. Thus this three-component system does not offer any spatial phenomena unknown in two-component systems.Preliminary studies lo of four-component systems by the methods shown in the Appendix indicate that backward inhibition may lead to oscillatory spatial instability. This requires a stable long wave limit (all four roots of the secular equation have negative real parts for k = 0) and the existence of wave vectors for which two rwts with p(k) = 0 and ~(k) + 0 have dp(k)/d(k2) > 0. From eqn (12) we have Thus oscillatory spatial instability requires that at least one of the derivatives dpld :VIP is positive for a marginally stable oscillatory solution. This is the case for large diffusion coefficients of the A-component and a large rate constant of the reaction which is subject to inhibition.I. BALSLEV AND €1. DEGX The above preliminary results for models with more than two components indicate that such models should be explored in more detail. The results should be related to chemical reaction schemes involving backward inhibition such as the model for the Belousov reaction. If models predicting oscillatory spatial instabilities turn out to be chemically realistic then a quite new field is opened not only in connection with instabilities. It is known in general linear wave mechanics that globally stable steady states in systems which are close to oscillatory spatial instability allow wave propagation with little spatial attenuation. APPENDIX Considering the matrix in eqn (11) of a three component system with backward inhibition we find that the complex growth rate A = p-iw is the solution to (MA+A)(MB+?J(Mc+A)-m,m,MA i-inOmAmB= 0.(14) With reference to eqn (13) we concentrate on the signs of ap/aMp. Investigating first the behaviour near marginally stable aperiodic solutions (E. 2~ 0) we require for a finite wave vector ko MAMBMC -(MA-mA)mOmB= 0. In the vicinity of ko Evaluating dp/dMp and inserting in eqn (12) and (15) we find and ap/akfA = (M&f,m~3/(DAk2N) ap/dh!f = -MAhfJN (17) where ap/aM = -M~M~/N N = MAMB+MAMc- MBhfC172A/DAk1e (18) The different signs of the derivatives ap/aMpallow the existence of aperiodic spatial instab- ility. For oscillatory solutions we find in the case of marginal stability In the evaluation of @/dMp we consider the left hand side of eqn (14) as a function MA MB,Mc A) and use the fact that Inserting in this expression the above value of A = +iowe find that all derivatives &/dMp are negative.Consequently oscillatory spatial instabiIity is absent in this three-component system. ' A. M. Turing Phil. Trans. B 1952 237 37. A. M. Zhabotinskii in B. Chance E. K. Pye A. K. Ghosh and R. Hess (eds.) Biologied and Riochemical Oscillators (Academic Press New York 1973) p. 89. P. Ortoleva and J. Ross J. Chem. Phys. 1973 58 5673. A. Lotka J. Plzys. Chem. 1910 14 271. TEMPORAL AND SPATIAL INSTABILITIES V. Volterra Lepw sur la Theorie Mathematique de In Lutte pour la Vie (Gautier-Villars Paris 193 1). I. Prigogine and R. Lefever J. Chem. Phys. 1968,48 1695. I. Balslev and H. Degn J. Theor. Biol.,1974. B. B. Edelstein J. Theor. Bid 1970 26 227. L. A. Segel and J. L. Jackson J. Theor. Biol. 1972 37 545. lo I. Balslev unpublished work. R. J. Field E. Koros and R. M. Noyes J. Amer. Chem. Soc. 1972 25 8649. l2 H. Degn J. Chem. Ed. 1972,49,302.