Mechanism Generating Spatial Patternsin Reaction-Diffusion Systems

Kanako SUZUKI�

Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received June 8, 2011; final version accepted October 17, 2011

In 1952, A. M. Turing proposed the notion of ‘‘diffusion-driven instability’’ in his attempt of modeling biologicalpattern formation. Following his ingenious idea, many reaction-diffusion systems have been proposed later on. Onthe other hand, Turing patterns can be explained by some cellular automata. Cellular automata are theoreticalmodels which consist of a regular grid of cells, and they exhibit the complex behavior from quite simple rules. Inthis paper, we describe the mathematical properties of reaction-diffusion systems modeling pattern formation, inparticular, Turing patterns. Moreover, we explain ideas which connect differential equations with cellularautomata.

KEYWORDS: pattern formation, activator-inhibitor system, cellular automata

1. Introduction

1.1 Pattern formation

In biology, the development of patterns and forms is caused by chemical reactions inside cells and communicationsamong cells. It is one of the most interesting subjects to understand the mechanism of morphogenesis, which is the partof embryology. A. M. Turing [15] wanted to understand how it is possible that a symmetric embryo evolves anasymmetric form, and proposed the idea of diffusion driven instability in his seminal paper in 1952. It is explained asfollows. The reaction between two chemicals with different diffusion rates may cause the destabilization of a spatiallyhomogeneous state, thus leading to the formation of nontrivial spatial structures. His idea was demonstrated by using alinear reaction-diffusion system, however, many other systems based on his idea have been proposed later on. Spatialpatterns generated by that idea have been often called Turing patterns, and they have the following fundamentalproperties:

. They are the stationary spatial structures generated by reaction-diffusion systems.

. Small inhomogeneous perturbations to a stable steady state in kinetic system grow in the presence of diffusion.

. They have intrinsic wavelength determined by parameters of the reaction-diffusion process, which does notdepend on boundary conditions, external forces.

These insights provide, for example, a possible mechanism for producing patterns in animal skin. Original observationof the dynamic properties of Turing patterns in nature was made by Kondo and Asai [8] in 1995. They explained themechanism of the formation of horizontal stripes in the tropical fish Pomacanthus imperator (Figure 1.1) by using areaction-diffusion system based on Turing’s idea. Since stripe patterns on Pomacanthus imperator are not fixed in theirskin, they maintain the spaces between the lines by the continuous rearrangement of patterns. This phenomenon ispredicted by numerical simulations. On the other hand, concerning stripe patterns on zebra, Turing instability isexhibited in the early stages of embryonic development–around 21–35 days of gestation. The visible stripes on zebraenlarge proportionally during its body growth. This is not a Turing patterning mechanism because it contradicts thethird property of Turing patterns listed above.

In order to obtain Turing patterns under laboratory conditions, it is required that diffusion constants of the twospecies have a sufficiently large ratio. However, small molecules or ions all diffuse more or less at the same rate.Therefore, no clear experimental evidence of stationary Turing structures had been reported until the work of DeKepper and coworkers [1] in 1990. They used the chlorite-iodide-malonic acid starch (CIMA) reaction in an openunstirred gel reactor to visualize Turing’s pattern. Starch (S), which is used as a color indicator for the concentration ofiodide (I�) ion, forms a complex with iodide according to the scheme

2010 Mathematics Subject Classification: 35K57, 92C15, 68Q80.� Corresponding author. E-mail: kasuzu@m.tohoku.ac.jp

Interdisciplinary Information Sciences Vol. 17, No. 3 (2011) 131–153#Graduate School of Information Sciences, Tohoku UniversityISSN 1340-9050 print/1347-6157 onlineDOI 10.4036/iis.2011.131

Sþ I� þ I2 � SI�3 :

The SI�3 complex is considered to be practically immobile in the gel. On the other hand, chlorite (ClO�2 ) is

uncomplexed iodide. It can be easily seen that the diffusion of immobile molecules is substantially slower than that ofchemicals which are not bound.

The CIMA reaction is described by a five-variable model involving the concentrations of I�, CIO�2 , MA, CIO2 and

I2. Lengyel and Epstein [9] simplified the model to a two-variable model:

@u

@t¼ D1

@2u

@x2� u�

4uv

1þ u2þ a

@v

@t¼ � D2

@2v

@x2þ b u�

uv

1þ u2

� �� �:

8>>><>>>:

Here, D1, D2, �, a, b are positive constants. The unknown functions uðx; tÞ and vðx; tÞ represent the concentrations ofiodide (I�) and chlorite (ClO�

2 ) ions, respectively. � is a rescaling parameter which primarily depends upon theconcentration of starch. The higher starch concentration means larger �. Since the diffusion constant for v is �D2, it ismuch larger then the diffusion constant for u if the concentration of starch is sufficiently high.

As an example which generates a different kind of pattern, the Belousov–Zhabotinsky reaction (the BZ reaction) isfamous as a spatio-temporal chemical oscillator. A typical recipe of the BZ reaction is the mixture of Ce4þ, BrO�

3 ,CH2(COOH)2 and H2SO4. In a thin unstirred layer of reaction solution, self-organized target patterns or spiral wavesare developed. The Oregonator is known as a mathematical model which helps us to understand how the BZ reactionproduces oscillations and waves (see e.g., [2]):

"dx

dt¼ qy� xyþ xð1� xÞ;

�dy

dt¼ �qy� xyþ 2 fz;

dz

dt¼ x� z:

8>>>>>><>>>>>>:

Here, ", �, q, f are positive constants, and the functions x ¼ xðtÞ, y ¼ yðtÞ and z ¼ ðtÞ represent the concentrations ofbromous acid (HBrO2), bromine ion (Br�), cerium ion (Ce4þ), respectively.

In the Oregonator modeling of the BZ reaction, it is assumed that there are a huge number of molecules, and thenumber of molecules is approximated by the density. But, this idea cannot be applied to the case where the number ofmolecules is limited. Concerning Turing’s pattern, it is analyzed by reaction-diffusion equations on uniform media,where it is assumed that the size of each cell is extremely small. However, it is natural to consider discrete models toexplain the mechanism of Turing patterns when we discuss it in a microscopic scale. Cellular automata have beenknown as discrete models for pattern formation. A cellular automaton consists of a regular grid of cells. For each cell, aset of cells called ‘‘neighborhood’’ is defined relative to the specified cell. If a fixed rule is given and an initial condition

Fig. 1.1. Pomacanthus imperator from Wikipedia http://en.wikipedia.org/wiki/Emperor angelfish, June 9, 2011.

132 SUZUKI

is selected by assigning a state for each cell, a new state of each cell is determined by the current state of the cell andthe states of the cells in its neighborhood. Turing patterns and BZ reactions can be simulated by some kinds of cellularautomata. If the both, a reaction-diffusion system and a cellular automaton, produce similar spatial patterns, then it isinteresting to understand a common mechanism in both models.

In Section 2, we describe the mathematical properties of reaction-diffusion equations which lead to Turing patterns.In Section 3, we study the Lengyel–Epstein model to illustrate that the diffusion driven instability occur and somespatial patterns can emerge. In the last section, we consider the relation between reaction-diffusion systems and discretemodels.

2. Turing’s Idea: Diffusion Driven Instability

The diffusion phenomenon is expressed by the following differential equation:

@u

@t¼ D�u; x 2 �; t > 0:

Here, � � Rn is an open set in the n-dimensional Euclidean space R

n. The function u ¼ uðx; tÞ represents theconcentration or the density of particles or a species at position x and at time t. The symbol � is the Laplace operator inR

n, which is defined by

� ¼@2

@x21þ@2

@x22if n ¼ 2:

The constant D > 0 denotes the diffusion rate. The diffusion describes the spread of particles from regions of higherconcentration to regions of lower concentration, and enhance spatial homogeneity. Therefore, it is impossible to get thenonhomogeneous spatial patterns by a diffusion process, only.

In 1952, A. M. Turing discovered that the reaction between two chemicals with different diffusion rates may causethe destabilization of the spatially homogeneous states, and lead to the formation of nontrivial spatial structures. Thissurprising idea is called the diffusion driven instability. In order to explain this phenomenon, let us consider thefollowing two component system:

@u

@t¼ D1�uþ f ðu; vÞ; x 2 �, t > 0,

@v

@t¼ D2�vþ gðu; vÞ; x 2 �, t > 0.

8>><>>: ð2:1Þ

The first terms on the right-hand sides of (2:1) model diffusion, and D1 > 0 and D2 > 0 denote the diffusioncoefficients of u and v, respectively. The second terms on the right-hand sides of (2:1) describe the reaction between u

and v, which will be always nonlinear. The system (2:1) is called reaction-diffusion equations. The ordinary differentialequations corresponding to (2:1) are called as the kinetic system and are of the form:

du

dt¼ f ðu; vÞ; t > 0,

dv

dt¼ gðu; vÞ; t > 0.

8>><>>: ð2:2Þ

The following conditions are necessary to generate spatial patterns by the diffusion driven instability:1. u and v have different diffusion rates, for example, D1 � D2;2. if ð �uu; �vvÞ is a stable steady state of (2:2), then it is unstable as a stationary solution of (2:1).

The diffusion driven instability is sometimes called the Turing instability, and the spatial patterns caused by the Turinginstability are called Turing’s pattern. In the following, we formulate this problem mathematically and we discussnecessary conditions for the Turing instability.

2.1 Linearized stability of steady states to ODE system

We consider the following initial-value problem:

du

dt¼ f ðu; vÞ;

dv

dt¼ gðu; vÞ;

uð0Þ ¼ a; vð0Þ ¼ b:

8>>>><>>>>:

ð2:3Þ

Let ð �uu; �vvÞ be a steady state of (2:3), which is defined by the equations f ð �uu; �vvÞ ¼ 0 and gð �uu; �vvÞ ¼ 0. Therefore, anintersection of the curves f ðu; vÞ ¼ 0 and gðu; vÞ ¼ 0 in the uv-plane corresponds to a steady state.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 133

Definition 2.1. Let ð �uu; �vvÞ be a steady state of (2:3). It is said to be stable if for any " > 0, there exists � > 0 such that asolution ðuðtÞ; vðtÞÞ of (2:3) with initial data ða; bÞ satisfying jða; bÞ � ð �uu; �vvÞj < � satisfies jðuðtÞ; vðtÞÞ � ð �uu; �vvÞj < " for allt > 0. The steady state ð �uu; �vvÞ is said to be unstable if it is not stable.

Moreover, if ð �uu; �vvÞ is stable and satisfies ðuðtÞ; vðtÞÞ ! ð �uu; �vvÞ as t ! þ1, then it is called asymptotically stable.

In order to check the stability of ð �uu; �vvÞ, we use the linearization of (2:3) at ð �uu; �vvÞ. Letting

uðtÞ ¼ �uuþ "�ðtÞ; vðtÞ ¼ �vvþ " ðtÞ; ð2:4Þ

and substituting (2:4) into (2:3), we have

"d�

dt¼ f ð �uuþ "�; �vvþ " Þ;

"d

dt¼ gð �uuþ "�; �vvþ " Þ:

The Taylor expansion of f ð �uuþ "�; �vvþ " Þ at ð �uu; �vvÞ is

f ð �uuþ "�; �vvþ " Þ ¼ f ð �uu; �vvÞ þ@ f

@uð �uu; �vvÞ"�þ

@ f

@vð �uu; �vvÞ"

þ1

2

@2f

@u2ð �uu; �vvÞ"2�2 þ

@2f

@u@vð �uu; �vvÞ"�" þ

@2f

@v2ð �uu; �vvÞ"2 2

� �þ � � �

¼ "@f

@uð �uu; �vvÞ"�þ

@f

@vð �uu; �vvÞ"

� �þ Oð"2Þ;

where f ð �uu; �vvÞ ¼ 0 because ð �uu; �vvÞ is an equilibrium of (2:3). Similarly, gð �uuþ "�; �vvþ " Þ is expanded as follows:

gð �uuþ "�; �vvþ " Þ ¼ "@g

@uð �uu; �vvÞ"�þ

@g

@vð �uu; �vvÞ"

� �þ Oð"2Þ:

Therefore, the equations in (2:3) become

d�

dt¼@ f

@uð �uu; �vvÞ�þ

@ f

@vð �uu; �vvÞ þ Oð"Þ; ð2:5Þ

d

dt¼@g

@uð �uu; �vvÞ�þ

@g

@vð �uu; �vvÞ þ Oð"Þ: ð2:6Þ

Here, we assume that " > 0 is sufficiently small and both �ðtÞ and ðtÞ are bounded, for example they satisfy�2ðtÞ þ 2ðtÞ � 1. Then, terms Oð"Þ on the right-hand sides of (2:5) and (2:6) can be neglected. Hence, the followinglinearized system gives a good approximation to (2:3) around ð �uu; �vvÞ:

d�

dt¼@ f

@uð �uu; �vvÞ�þ

@ f

@vð �uu; �vvÞ ;

d

dt¼@g

@uð �uu; �vvÞ�þ

@g

@vð �uu; �vvÞ :

8>><>>: ð2:7Þ

Let L be the Jacobian matrix on the right-hand side of (2:7), that is,

L ¼

@ f

@uð �uu; �vvÞ

@ f

@vð �uu; �vvÞ

@g

@uð �uu; �vvÞ

@g

@vð �uu; �vvÞ

0BB@

1CCA:

In the following theorem, we explain how to use eigenvalues of L to analyze the stability of ð �uu; �vvÞ.

Theorem 2.2.(i) If the real parts of both eigenvalues of L are negative, then ð �uu; �vvÞ is asymptotically stable.(ii) If L has an eigenvalue with positive real part, then ð �uu; �vvÞ is unstable.(iii) If L has purely imaginary eigenvalues, then the stability of ð �uu; �vvÞ cannot be determined by the eigenvalues only.

For simplicity, we denote @ f@u ð �uu; �vvÞ,

@ f@v ð �uu; �vvÞ,

@g@u ð �uu; �vvÞ and

@g@v ð �uu; �vvÞ by fu, fv, gu and gv, respectively. The eigenvalues of

L are solutions � to the characteristic equation

detðL� �IÞ ¼ 0;

where the left-hand side is the determinant of the matrixL� �I and I is the identity matrix. Therefore, � is a solution of

�2 � ð fu þ gvÞ� þ ð fugv � fvguÞ ¼ 0: ð2:8Þ

The both of two roots of (2:8) have negative real parts if and only if

134 SUZUKI

fu þ gv < 0 and fugv � fvgu > 0:

On the other hand, L has an eigenvalue with positive real part if and only if it satisfies

either fu þ gv > 0 or fugv � fvgu < 0:

2.2 Turing instability

We consider the following initial-boundary value problem:

@u

@t¼ D1

@2u

@x2þ f ðu; vÞ; 0 < x < L, t > 0,

@v

@t¼ D2

@2v

@x2þ gðu; vÞ; 0 < x < L, t > 0,

@u

@x¼@v

@x¼ 0; x 2 f0;Lg, t > 0,

uðx; 0Þ ¼ u0ðxÞ; vðx; 0Þ ¼ v0ðxÞ; 0 < x < L.

8>>>>>>>>><>>>>>>>>>:

ð2:9Þ

Here, we focus on the problem on one-dimensional region ½0;L� for L > 0. The functions u ¼ uðx; tÞ and v ¼ vðx; tÞrepresent the concentrations of particles at position x and at time t. D1 and D2 are positive, which denote the diffusioncoefficients. The Neumann boundary condition is imposed in (2:9), which means that u and v cannot go out from norcome into the region through boundaries. The ordinary differential equations corresponding to (2:9) are (2:3). Indeed, asolution of (2:9) with initial data ðu0ðxÞ; v0ðxÞÞ ¼ ða; bÞ is also a solution of (2:3) thanks to the boundary condition in(2:9).

Assume ð �uu; �vvÞ is an asymptotically stable steady state of (2:3). Hence, we impose the following conditions on thenonlinear terms at the point ð �uu; �vvÞ:

fu þ gv < 0 and fugv � fvgu > 0: ð2:10Þ

Here, we use the same notations as in the previous subsection. For the Turing instability, the stable equilibrium ð �uu; �vvÞhas to be destabilized in the presence of unequal diffusions. To illustrate this phenomenon, we use the linearization of(2:9) at ð �uu; �vvÞ again and we see that ð �uu; �vvÞ becomes unstable thanks to diffusions. Let

uðx; tÞ ¼ �uuþ "zðx; tÞ; vðx; tÞ ¼ �vvþ "wðx; tÞ:

Substituting these new variables into (2:9), after some calculations as shown in the previous subsection, we obtain thelinearized system of (2:9) at ð �uu; �vvÞ:

@z

dt¼ D1

@2z

@x2þ@ f

@uð �uu; �vvÞzþ

@ f

@vð �uu; �vvÞw; 0 < x < L, t > 0,

dw

dt¼ D2

@2w

@x2þ@g

@uð �uu; �vvÞzþ

@g

@vð �uu; �vvÞw; 0 < x < L, t > 0,

@z

@x¼@w

@x¼ 0; x 2 f0;Lg, t > 0.

8>>>>>><>>>>>>:

ð2:11Þ

It follows from the theory of Fourier series that solutions of (2:11) can be expressed as

zðx; tÞ ¼X1n¼0

znðtÞ cos�n

Lx; wðx; tÞ ¼

X1n¼0

wnðtÞ cos�n

Lx: ð2:12Þ

This idea, which was introduced by J. Fourier, says that the spatial distribution can be expressed in terms of the sum oftrigonometric functions, and the coefficient of each spatial frequency develops in time independently around the steadystate. Substituting (2:12) into (2:11) and comparing the coefficients of cos �n

Lx on both sides of the resulting system, we

obtain, for n ¼ 0; 1; 2; . . . , that

d

dt

zn

wn

� �¼

�D1

�n

L

� �2

þ fu fv

gu �D2

�n

L

� �2

þ gv

0BBB@

1CCCA zn

wn

� �: ð2:13Þ

Since the case of n ¼ 0 corresponds to (2:3), ð �uu; �vvÞ is stable for n ¼ 0 because of the assumption. If we have, for alln � 1, that

� D1

�n

L

� �2

þ fu � D2

�n

L

� �2

þ gv < 0 ð2:14Þ

and

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 135

�D1

�n

L

� �2

þ fu

!�D2

�n

L

� �2

þ gv

!� fvgu > 0; ð2:15Þ

then ðznðx; tÞ;wnðx; tÞÞ ! ð0; 0Þ as t ! þ1 for all n � 1. This implies that ð �uu; �vvÞ is an asymptotically stable stationarysolution of (2:9). Thus, the diffusion driven instability never occurs. Note that inequality (2:14) is always satisfiedbecause D1 > 0 and D2 > 0. Therefore, in order that ð �uu; �vvÞ is unstable, it is sufficient that there exists at least one n � 1

such that

�D1

�n

L

� �2

þ fu

!�D2

�n

L

� �2

þ gv

!� fvgu < 0:

After some calculations, we obtain

�n

L

� �4

D1D2 <�n

L

� �2

ð fuD2 þ gvD1Þ � ð fugv � fvguÞ: ð2:16Þ

Since it is clear that the left-hand side of (2:16) is positive, the right-hand side of (2:16) should be positive as well. Itfollows from (2:10) that fugv � fvgu > 0. Therefore, the first term on the right-hand side of (2:16) has to be positive.Noting that fu þ gv < 0 is also satisfied by (2:10), we see that the quantity fuD2 þ gvD1 is positive if and only if one ofthe fu and gv is positive and the other negative. Now, let us consider the case fu > 0 and gv < 0. Then, D1 and D2

should satisfy

D2 > �gv

fuD1 ð2:17Þ

to get the both

fu þ gv < 0 and fuD2 þ gvD1 > 0:

This implies that (2:17) is a necessary condition for the diffusion driven instability. If we suppose fu < 0 and gv > 0,then fuD2 þ gvD1 is positive provided that D1 > �ð fu=gvÞD2.

Next we seek a sufficient condition for the diffusion driven instability under the conditions fu > 0 and gv < 0. Letð�n=LÞ2 ¼ � for a fixed n � 1. We see that inequality (2:16) becomes

D1D2�2 � ðD2 fu þ D1gvÞ� þ ð fugv � fvguÞ < 0: ð2:18Þ

Put � ¼ �D1 and � ¼ �D2. Then, this inequality may be regarded as

��� ð� fu þ �gvÞ þ ð fugv � fvguÞ < 0: ð2:19Þ

Let � ¼ ��. Then, the left-hand side of (2:19) is equal to

�ð�; �Þ ¼ ��2 � ð� fu þ gvÞ�þ ð fugv � fvguÞ:

The quadratic equation �ð�; �Þ ¼ 0 in � has positive roots if and only if

gv þ � fu > 0 and ðgv þ � fuÞ2 � 4�ð fugv � fvguÞ � 0: ð2:20Þ

Note that (i) ðgv þ � fuÞ2 � 4�ð fugv � fvguÞ < 0 if � ¼ �gv= fu > 0, and (ii) ðgv þ � fuÞ2 � 4�ð fugv � fvguÞ ! þ1 as� ! þ1. Moreover, we have

(iii)d

d�

ðgv þ � fuÞ2

4�¼

ð� fu þ gvÞð� fu � gvÞ4�2

> 0;

provided that � fu þ gv > 0. Combining (i)–(iii), we obtain that there exists a unique �c > �gv= fu > 0 such that�ð�; �cÞ ¼ 0 has a positive double zero, and �ð�; �Þ ¼ 0 has two positive zeros 0 < �1 < �2 if � > �c. In fact, �c is a root of

f 2u �2 þ 2ð2 fvgu � fugvÞ� þ g2v ¼ 0:

It is noted that �c > 1 due to the first condition of (2:20). Consequently, we see that if � > �c and �1 < � < �2, then�ð�; �Þ < 0. In other words, if

D2

D1

> �c and�1L

2

ð�nÞ2< D1 <

�2L2

ð�nÞ2;

then (2:16) is satisfied and the stationary solution ð �uu; �vvÞ is unstable.

2.3 Activator-inhibitor system

We consider the mechanism of the pattern formation in (2:9) when fu > 0 and gv < 0. From the assumption (2:10),we have fugv � fvgu > 0. Therefore, fv and gu should satisfy fvgu < 0. This means that the signs of fv and gu should beopposite. Hence, there are only two possibilities:

136 SUZUKI

fu fv

gu gv

� �¼ either

þ �þ �

� �or

þ þ� �

� �:

The former on the right-hand side above is called an activator-inhibitor system, and the latter is a resource-consumersystem. In the following, we explain the mechanism of the pattern formation in activator-inhibitor systems.

Let u and v be the concentrations of an activator and an inhibitor, respectively. Activator stimulates the production ofitself ( fu > 0), and promotes the production of the inhibitor (gu > 0). On the other hand, the inhibitor suppresses boththe activator and the inhibitor ( fv < 0 and gv < 0). From the assumption (2:17), the inhibitor spreads by diffusion fasterthan the activator. This system is expected to make the following spatial pattern. From an initially almost uniformdistribution, the activator u starts to grow at a point x. Thanks to the self-enhancement of u, the production of u isstimulated at this point. The inhibitor v diffuses faster than the activator and it blocks further production of the activatorin distant places. Now, u grows more and more at the point where it was enhanced initially, and the region of highconcentration for u becomes narrower and narrower. As a result, a striking pattern of the activator concentrationemerges.

A. Gierer and H. Meinhardt [3] developed Turing’s idea, proposing several reaction-diffusion systems to model thebiological pattern formation. The following is an example:

@u

@t¼ Da

@2u

@x2� uþ

u2

vþ �; 0 < x < 1, t > 0,

@v

@t¼ Dh

@2v

@x2� vþ u2; 0 < x < 1, t > 0,

8>><>>: ð2:21Þ

where, Da, , Dh and � are positive constants. This is an activator-inhibitor system in which u ¼ uðx; tÞ and v ¼ vðx; tÞrepresent the concentrations of an activator and an inhibitor, respectively. We suppose the activator diffuses slowly andthe inhibitor diffuses rapidly, that is Da � Dh. The spatial pattern generated by (2:21) is shown in Figure 2.1. It isassumed that a change in cells or tissue takes place in the region where the activator concentration is high. They usedthis system to simulate the regeneration and transplantation experiments on hydra. Hydra is an animal with body lengthof a few millimeters. If the head is amputated, then a new head regenerates within a few days. When the head isremoved, the head-activating substance represented by u is released from cells, and activation proceeds by mechanismin which the released substances themselves influence further release. The results with localized pattern of the activatorconcentration as in Figure 2.1 are in agreement with experiments.

3. Lengyel–Epstein Model

In this section, we demonstrate how a stable steady state loses its stability and spatially inhomogeneous patterns canemerge. We use the Lengyel–Epstein model, which is a typical and important example which gives rise to the Turinginstability:

@u

@t¼ D1

@2u

@x2� u�

4uv

1þ u2þ a; 0 < x < L, t > 0,

@v

@t¼ � D2

@2v

@x2þ b u�

uv

1þ u2

� �� �; 0 < x < L, t > 0,

@u

@x¼@v

@x¼ 0; x 2 f0;Lg, t > 0,

8>>>>>>><>>>>>>>:

ð3:1Þ

0 0.5 1.0

10.0

0 0.5 1.0

10.0

0 0.5 1.0

10.0

0 0.5 1.0

10.0

Time

Fig. 2.1. The horizontal axis is the interval ½0; 1�, the vertical axis denotes the concentrations of u and v. Activator is represented bya red curve and inhibitor is represented by blue one. This figure shows the spatial pattern of the activator concentration emergesstarting from an almost homogeneous state.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 137

where D1, D2, a, b, � are positive constants. In the following, we use a ¼ 30, b ¼ 2:8, � ¼ 8 for the sake ofconvenience.

The Lengyel–Epstein model is an activator-inhibitor system, where u ¼ uðx; tÞ represents the concentration of anactivator, v ¼ vðx; tÞ represents the concentration of an inhibitor. We study only positive solutions of (3:1). Thecorresponding system of ordinary differential equations of is given by

du

dt¼ �u�

4uv

1þ u2þ a;

dv

dt¼ �b u�

uv

1þ u2

� �:

8>><>>: ð3:2Þ

First, we find steady-state solutions of (3:2). Letting

f ðu; vÞ ¼ �u�4uv

1þ u2þ a; gðu; vÞ ¼ �b u�

uv

1þ u2

� �;

we solve f ðu; vÞ ¼ 0 and gðu; vÞ ¼ 0. We see that (3:2) has only one equilibrium point ð �uu; �vvÞ in the first quadrant ofuv-plane (see Figure 3.1), which is

ð �uu; �vvÞ ¼a

5; 1þ

a2

25

� �¼ ð6; 37Þ:

Next, we check the stability of the equilibrium ð �uu; �vvÞ. Since we use the linearization of system (3:2) at ð �uu; �vvÞ, let

uðtÞ ¼ �uuþ "�ðtÞ; vðtÞ ¼ �vvþ " ðtÞ:

Then, we obtain the linearized system of (3:2) at ð �uu; �vvÞ:

d

dt

�

� �¼

�1�4ð1� �uu2Þ1þ �uu2

�4 �uu

1þ �uu2

�b 1�1� �uu2

1þ �uu2

� ���b

�uu

1þ �uu2

0BBB@

1CCCA �

� �:

We study the signs of the real parts of eigenvalues of the matrix on the right-hand side above. If ð �uu; �vvÞ is asymptoticallystable, then it is satisfied

fu þ gv < 0 and fugv � fvgu > 0:

Therefore, we have

� 1�4ð1� �uu2Þ1þ �uu2

� �b�uu

1þ �uu2< 0; ð3:3Þ

head

foot

tentacles

Fig. 2.2. Hydra (by courtesy of M. Nakayama).

138 SUZUKI

�1�4ð1� �uu2Þ1þ �uu2

� ���b

�uu

1þ �uu2

� �þ

4 �uu

1þ �uu2�b 1�

1� �uu2

1þ �uu2

� �> 0 ð3:4Þ

if ð �uu; �vvÞ is stable. It turns out that both (3:3) and (3:4) are satisfied when �uu ¼ 6, b ¼ 2:8, � ¼ 8. Hence, the uniqueequilibrium point ð �uu; �vvÞ ¼ ð6; 37Þ of (3:2) is asymptotically stable.

In order to obtain Turing patterns, the stable equilibrium ð �uu; �vvÞ of (3:2) has to be destabilized in (3:1) with unequaldiffusions. To see it, we study eigenvalues of a linearized operator of (3:1). We substitute the functions

uðx; tÞ ¼ �uuþ "zðx; tÞ; vðx; tÞ ¼ �vvþ "wðx; tÞ

into (3:1) to obtain the linearized system of (3:1) at ð �uu; �vvÞ:

@

@t

z

w

� �¼

D1

@2

@x2� 1�

4ð1� �uu2Þ1þ �uu2

�4 �uu

1þ �uu2

�b 1�1� �uu2

1þ �uu2

� ��D2

@2

@x2� �b

�uu

1þ �uu2

0BBB@

1CCCA z

w

� �:

We express the functions z and w in terms of trigonometric series (2:12). Substitute these expansions into the linearizedequation above and compare the coefficients of cos �n

Lx on the both sides to obtain, for n ¼ 0; 1; 2; . . . ,

d

dt

zn

wn

� �¼

�D1

�n

L

� �2

� 1�4ð1� �uu2Þ1þ �uu2

�4 �uu

1þ �uu2

�b 1�1� �uu2

1þ �uu2

� ���D2

�n

L

� �2

� �b�uu

1þ �uu2

0BBB@

1CCCA zn

wn

� �: ð3:5Þ

It follows from D1 > 0, D2 > 0 and (3:3) that the following inequality always holds:

�D1

�n

L

� �2

� 1�4ð1� �uu2Þ1þ �uu2

� �D2

�n

L

� �2

� �b�uu

1þ �uu2< 0:

Therefore, ð �uu; �vvÞ becomes an unstable stationary solution of (3:1) if and only if there exists an n � 1 such that

�D1

�n

L

� �2

� 1�4ð1� �uu2Þ1þ �uu2

!��D2

�n

L

� �2

� �b�uu

1þ �uu2

!þ

4 �uu

1þ �uu2�b 1�

1� �uu2

1þ �uu2

� �< 0:

Thus, after some calculations, we obtain the following inequality:

D1 <L2

ð�nÞ24ð �uu2 � 1Þ�uu2 þ 1

� 1�L28b �uu3

ð �uu2 þ 1ÞðD2ð�nÞ2ð �uu2 þ 1Þ þ L2b �uuÞ

� �: ð3:6Þ

For any fixed D2, if the right-hand side of (3:6) is positive for some n � 1, then ð �uu; �vvÞ becomes unstable for D1

sufficiently small. Then, a spatial pattern of mode n is expected to appear. The integer n determines the spatial period !of the solution, which is given by ! ¼ 2�=n. Let us study it more precisely.

Fig. 3.1. Red and blue curves are given by f ðu; vÞ ¼ 0 and gðu; vÞ ¼ 0, respectively. The intersection denotes an equilibrium pointð6; 37Þ of the kinetic system (3:2).

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 139

For each n � 1, we draw a graph of the following equation in the first quadrant of D2D1-plane:

D1 ¼L2

ð�nÞ24ð �uu2 � 1Þ�uu2 þ 1

� 1�L28b �uu3

ð �uu2 þ 1ÞðD2ð�nÞ2ð �uu2 þ 1Þ þ L2b �uuÞ

� �: ð3:7Þ

In Figure 3.2, each curve approaches D1 ¼ L2

ð�nÞ2 ð4ð �uu2�1Þ�uu2þ1

� 1Þ as D2 ! þ1. The abscissa of this asymptote tends to 0as n ! þ1. Each curve intersects the D2-axis at

D2 ¼L25b �uu

ð�nÞ2ð3 �uu2 � 5Þ;

and this intersection moves to the origin as n ! þ1.Letting L ¼ 10, we shall observe the mechanism of the Turing instability, and see how Turing patterns emerge by

numerical simulations. For fixed D2 ¼ 1:5, we use D1 as a parameter to obtain the diffusion driven instability.Figure 3.4 is focused on the intersection of curves n ¼ 3 and n ¼ 4. The value of D1 corresponding to the intersectionsof the curves n ¼ 3 and n ¼ 4 is 1:011822867 � � � . The yellow star in Figure 3.3 denotes a point with D1 ¼ 0:95, whichis above the curve n ¼ 3, while it is below the curve n ¼ 4. This means that ð �uu; �vvÞ does not lose its stability against the

n=1

n=3

n=2

n=4

n=5

Fig. 3.2. The curves described by equation (3:7) with �uu ¼ 6, b ¼ 2:8 and L ¼ 10.

n=3

n=4

n=5

Fig. 3.3. The curves described by (3:7) with �uu ¼ 6, b ¼ 2:8 and L ¼ 10. The vertical dashed line corresponds to D2 ¼ 1:5.

140 SUZUKI

spatial perturbation of mode n ¼ 3. On the other hand, it is destabilized by the spatial perturbation of mode n ¼ 4.Therefore, it is expected that if we take D1 ¼ 0:95 and D2 ¼ 1:5, then small perturbations to ð �uu; �vvÞ grow and a spatialpattern with mode n ¼ 4 emerges. Indeed, it can be demonstrated by numerical simulations, see Figure 3.5.

Now, let D1 ¼ 1:5 and D2 ¼ 1:5. This case is marked by the pink disk in Figure 3.4. We solve (3:1) numerically,choosing a spatial pattern with mode n ¼ 4 as its initial data. Since the condition where D1 ¼ 1:5 and D2 ¼ 1:5 isabove the curve n ¼ 4, the constant solution ð �uu; �vvÞ is not destabilized by this spatial mode. Therefore, we can see inFigure 3.6 that a solution starting from the pattern with mode n ¼ 4 comes back to ð �uu; �vvÞ.

Time

Fig. 3.5. Solutions of (3:1) with D1 ¼ 0:95 and D2 ¼ 1:5 (the yellow star in Figure 3.4) with initial conditions which are taken in asmall neighborhood of ð �uu; �vvÞ ¼ ð6; 37Þ.

Fig. 3.4. The intersection of the curves (3:7) for n ¼ 3 (red) and n ¼ 4 (blue).

Time

Fig. 3.6. At the pink circle in Figure 3.4, we have D1 ¼ 1:5 and D2 ¼ 1:5. A solution of (3:1) with these diffusion coefficients andwith an initial condition which is a spatial pattern with mode n ¼ 4.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 141

4. Discrete Models

Cellular automata are theoretical models which consist of a regular grid of cells. Each cell is in one of a finite numberof states such as {green, red, yellow}. For each cell, a set of cells called ‘‘neighborhood’’ is defined relatively to aspecified cell. If a fixed rule is given and an initial condition is selected by assigning a state for each cell, a new state ofeach cell is determined by the current state of the cell and the states of cells in its neighborhood. In Figure 4.1, where aone-dimensional grid of finitely many cells is considered and each line denotes the states of cells at time t, it isillustrated that two waves propagate and they retain their forms after collision. The rule to determine a state at time t isas follows: for each cell from left to right at each state,

. if a cell is empty and you have balls, you put a ball into the cell,

. if a cell is empty and you have no ball, then nothing should be done for the cell, and you go to the next cell,

. if there is a ball in a cell, you remove it from the cell and carry it,

. if you have come to the end of the state (the right-hand side), you go to the next time step t þ 1.In Figure 4.1, the top line shows the initial state (t ¼ 0), and you start without balls in your hand. The second line andbelow show the subsequent development.

Cellular automata exhibit the complex behavior from quite simple rules. Therefore, they have been widelyinvestigated in physics, chemistry, biology and computer sciences. In the pattern formation, a cellular automatonsimulates the BZ reaction which creates self-organized target patterns or spiral waves in a thin unstirred layer of areaction solution. Moreover, Turing patterns can be explained by some cellular automata. It is known that patterns onsome seashell (Figure 5.1) are simulated by cellular automata following simple rules. It is an interesting problem tounderstand the relation between reaction-diffusion models (continuous models) and cellular automata.

In this section, we show how to derive nonlinear difference–difference equations from nonlinear differentialequations taking the Lotka–Volterra equations as examples.

4.1 Lotka–Volterra equations

The Lotka–Volterra equations are nonlinear differential equations describing the dynamics of the growth of twopopulations, one a predator and the other a prey:

du

dt¼ ð� vÞu;

dv

dt¼ ðu� �Þv;

8>><>>: ð4:1Þ

where � and are positive constants. The function u ¼ uðtÞ is the density of prey and v ¼ vðtÞ is the density of predator.Let

V ¼ Vðu; vÞ ¼ uþ v� � log u� log v:

tim

e

Fig. 4.1. Cellular automaton with two propagating waves.

142 SUZUKI

A direct calculations shows that Vt ¼ 0, which implies that V is a conserved quantity for system (4:1). Therefore,solution orbits for (4:1) correspond level curves of V , see Figure 4.2. In Figure 4.3, the progression of two species overtime is plotted. The Lotka–Volterra equations can be generalized to systems of more than two species. Let Xj be aspecies at time t and ujðtÞ be its population density. Assuming Xj preys on Xj�1 and Xjþ1 is its predator, we obtain, foreach j, the following equations:

duj

dt¼ ujðuj�1 � ujþ1Þ: ð4:2Þ

Here, all parameters were set 1, for simplicity.Suppose there are only four species X0, X1, X2, X3 and the numbers of X0 and X3 are constants, that is, u0ðtÞ ¼ a,

u3ðtÞ ¼ b for some a > 0 and b > 0. Then, the dynamics of X1 and X2 is given by the system

du1

dt¼ u1ða� u2Þ;

du2

dt¼ u2ðu1 � bÞ;

which is exactly the same as (4:1). Therefore, (4:1) is a special case of the generalized system (4:2). Hereafter, we call(4:2) the Lotka–Volterra system, and let �1 < j < þ1.

4.2 Problems caused by nonlinearity

In general, it is difficult to expect that there is an explicit formula of solutions to nonlinear differential equations. Insuch a case, numerical simulations are helpful in understanding the dynamics of solutions. We want to believe thatnumerical simulations give us good approximations to solutions of reaction-diffusion systems. But a nonlinearitysometimes causes strange or unusual behavior of solutions.

The Lorenz oscillator is one of the examples which exhibit chaotic behavior. The Lorenz oscillator is a nonlinearthree-component dynamical system:

du

dt¼ �puþ py;

dv

dt¼ �uwþ ru� v;

dw

dt¼ uv� bw:

8>>>>>><>>>>>>:

ð4:3Þ

Here, p, r, b are positive constants. Figures 4.4 and 4.5 show the behavior of solutions of (4:3) starting from a slightlydifferent initial data. They are calculated for the same values of the parameters p, r, b and the same time interval. It isobserved that the two orbits do not look close to each other.

Fig. 4.2. Orbits of the Lotka–Volterra system (4:1) with ¼ 0:5 and � ¼ 0:5. Green curve: ðuð0Þ; vð0ÞÞ ¼ð0:8; 1:0Þ, blue: ðuð0Þ; vð0ÞÞ ¼ ð1:0; 1:0Þ, cyan:ðuð0Þ; vð0ÞÞ ¼ ð0:5; 1:5Þ.

Fig. 4.3. A solution to system (4:1). Red curve is thepopulation of preys, blue one is the population of predators.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 143

4.3 Finite difference methods

In the one-dimensional case, a finite difference method is a fundamental way to approximate a solution of differentialequations. First, the space-time plane is divided into a uniform grid. Let the mesh size of the x-axis be h and that of thet-axis be k. We consider the following reaction-diffusion equation:

@u

@t¼@2u

@x2þ f ðuÞ for 0 < x < 1, 0 < t < T ð4:4Þ

associated with the non-flux boundary condition and the initial condition:

@u

@xð0; tÞ ¼

@u

@xð1; tÞ ¼ 0 for t > 0; ð4:5Þ

uðx; 0Þ ¼ u0ðxÞ for 0 < x < 1: ð4:6Þ

Here, u ¼ uðx; tÞ represents the concentration of a chemical substance, and f ðuÞ is a nonlinear term, for example,f ðuÞ ¼ u2. The initial data u0ðxÞ is a given function. The approximation of a solution uðx; tÞ of (4:4) at point xj ¼ jh andat time t ¼ nk is denoted by unj for 0 � j � J and 0 � n � N. There are two examples of finite difference schemes:

explicit method:unþ1j � unj

k¼

unjþ1 � 2unj þ unj�1

h2þ ðunj Þ

2; ð4:7Þ

implicit method:unþ1j � unj

k¼

unþ1jþ1 � 2unþ1

j þ unþ1j�1

h2þ ðunþ1

j Þ2: ð4:8Þ

On the boundary, un�1 and unJþ1 for 0 � n � N are replaced by un1 and unJ�1, respectively. In the explicit method (4:7),we obtain the values of unþ1

j , 0 � j � J, immediately if the values at time n are given. On the other hand, the implicitmethod (4:8) requires to solve a system of numerical equations on each time step.

Let uðx; tÞ be an exact solution of (4:4) with initial data �ðxÞ, and unj for 0 � j � J, 0 � n � N be a solution of adiscretized equation of (4:4) with initial data �j. Here, �j means the value of �ðxÞ at x ¼ jh. For the moment, we assumethat f ðuÞ 0 in (4:4) and consider the linear diffusion equation. Let m be an arbitrary positive integer and let us studyhow the amplitude behaves for the specific initial data �j ¼ sinð�mð jhÞÞ with 0 � j � J. For this purpose, we putunj ¼ AnXj, where Xj ¼ sinð�mð jhÞÞ, and substitute these in (4:7). Then, we have

Anþ1 ¼ ð1� 2� þ 2� cosð�mhÞÞAn:

If the sequence fAng1n¼0 remains bounded for any positive integer m, then the scheme is said to be stable. Clearly, theexplicit scheme is stable if and only if � � 1=2. On the other hand, if we substitute unj ¼ AnXj in (4:8), then we obtain

ð1þ 2� � 2� cosð�mhÞÞAnþ1 ¼ An:

Since 1þ 2�ð1� cosð�mhÞÞ � 1 for any positive integer m, we see that fAng1n¼0 is bounded for any m no matter how� > 0 is. In this case, the implicit scheme (4:8) is unconditionally stable.

The error in a solution of a discretized equation is defined as the difference between its approximation and the exactsolution of differential equation. There are two sources of errors. One is called a round-off error, which is the loss ofprecision due to rounding of decimals in computers. The other is called a discretization error, which is the differencebetween the solution of the discretized equation and the exact solution of the original differential equation. In general, it

Fig. 4.4. A trajectory of the Lorenz system withðuð0Þ; vð0Þ;wð0ÞÞ ¼ ð1:0; 5:0; 10:0Þ.

Fig. 4.5. A trajectory of the Lorenz system withðuð0Þ; vð0Þ;wð0ÞÞ ¼ ð1:1; 5:1; 10:1Þ.

144 SUZUKI

is not expected to obtain the formula for exact solutions of nonlinear differential equations. Then, it is difficult toestimate the difference between solutions of discretized equations and original differential equations. Moreover, as it isseen in the Lorenz equations, the discretization error can lead to unexpected behavior of solutions to nonlineardifferential equations. Therefore, we have to pay attention to what numerical simulations explain, and to what we canand cannot understand through them.

Recently, much attention has been focused on numerical schemes that conserve the structure of the solution set. Suchschemes have been developed to study e.g. integrable nonlinear equations, describing the propagation of soliton. Asoliton is a solitary wave which keeps its form after collision. The KdV equation and the nonlinear Schrodingerequation are examples of soliton equations. Moreover, the generalized Lotka–Volterra equations (4:2) also have solitonsolutions. The structure-conserving numerical schemes are required to preserve such a quality of soliton afterdiscretizing differential equations.

In the following, we introduce one of the methods for deriving a finite difference scheme which keeps its conservedquantity. Using the Lotka–Volterra equation (4:2) as an example, we show how to construct a discrete equation whichkeeps solitons. On the basis of this idea, we would like to understand a connection between reaction-diffusion equationsand cellular automata in the last part of this section.

4.4 Bilinearization methods

In this and next subsections, we describe methods which are introduced in [5, 6]. We discretize the Lotka–Volterraequation (4:2) in the following three steps:

1. transform (4:2) into a bilinear differential equation, where the resulting equation is invariant under gaugetransformations;

2. derive a discrete bilinear equation which keeps the gauge invariance;3. obtain a discrete equation from the discrete bilinear equation.

Let xðtÞ be a solution of a differential equation. The gauge transformation is defined by changing the dependent variablexðtÞ to xðtÞgðtÞ for an arbitrary function gðtÞ of t. If the differential equation for x is invariant under gaugetransformation, then it is said to be gauge invariant.

We shall obtain a discrete Lotka–Volterra equation from (4:2) following those three steps above. Here, we write (4:2)again:

duj

dt¼ ujðuj�1 � ujþ1Þ;

where ujðtÞ is the number of species Xj at time t. We impose boundary conditions ujðtÞ ! 1 as j ! 1 on (4:2).Step 1: From nonlinear differential equations to bilinear differential equations. Introducing new variables f fjðtÞg,we assume that uj is expressed as follows:

uj ¼fj�1 fjþ2

fj fjþ1

: ð4:9Þ

t

x

u

h

k

Fig. 4.6. Discretizing a function on a grid.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 145

In addition, on fj’s imposed are boundary conditions fjþ1= fj ! c as j ! 1. Here, cþ (or c�) is a constantindependent of j and t. These boundary conditions for fj are natural. Indeed, since ujðtÞ ! 1 as j ! 1, it followsfrom (4:9) that fj should satisfy fjþ2= fjþ1 ¼ fj= fj�1 as j ! 1.

Substituting (4:9) into (4:2) leads to the following bilinear differential equation

fjþ1

d fj

dt� fj

d fjþ1

dt¼ fj�1 fjþ2 � fj fjþ1: ð4:10Þ

Indeed, dividing both sides of (4:2) by uj, we obtain

d log ujðtÞdt

¼ uj�1 � ujþ1:

Substituting (4:9) into the left-hand side above, we have

d log ujðtÞdt

¼d

dtlog

fj�1 fjþ2

fj fjþ1

�������� ¼ d

dtlog

fjþ2

fjþ1

��������þ d

dtlog

fj�1

fj

��������:

This implies, for all j, thatd

dtlog

fjþ2

fjþ1

��������þ ujþ1 ¼ �

d

dtlog

fj�1

fj

��������þ uj�1

¼d

dtlog

fj�1

fj

���������1

þ uj�1

¼d

dtlog

fj

fj�1

��������þ uj�1: ð4:11Þ

By induction on j we find that the right-hand side is equal to

d

dtlog

fj�m

fj�m�1

��������þ uj�m�1

for all positive integer m. Then, it follows from the boundary condition for fj that the right-hand side of (4:11) tendsto 1 as j ! �1 (assuming that fjþ1= fj is convergent together with its derivative). Hence, we see, for all j,that

d

dtlog

fjþ1

fj

��������þ fj�1 fjþ2

fj fjþ1

¼ 1:

After direct calculations, we get

1

fjþ1

d fjþ1

dt�

1

fj

d fj

dtþ

fj�1 fjþ2

fj fjþ1

¼ 1;

which implies (4:10).The bilinear differential equation (4:10) has the gauge invariance. To see this, transform fl to gðtÞ flðtÞ for l ¼

j� 1; j; jþ 1; jþ 2 and compute

ðg fjþ1Þdðg fjÞdt

� ðg fjÞdðg fjþ1Þ

dt� ðg fj�1Þðg fjþ2Þ � ðg fjÞðg fjþ1Þ

¼ g fjþ1 fjdg

dtþ g

d fj

dt

� �� g fj fjþ1

dg

dtþ g

d fjþ1

dt

� �� g2ð fj�1 fjþ2 � fj fjþ1Þ

¼ g2 fjþ1

d fj

dt� fj

d fjþ1

dt

� �� g2ð fj�1 fjþ2 � fj fj�1Þ:

Therefore, if the fj’s satisfy (4:10), so do the g fj’s.Step 2: From bilinear differential equations to discrete bilinear equations. Our goal is to obtain a discreteequation for (4:2) which has conserved quantities. To do so, we first derive a discrete bilinear equation from (4:10).

If we discretize (4:10) by the explicit scheme, then we have

fjþ1ðtÞfjðt þ kÞ � fjðtÞ

k� fjðtÞ

fjþ1ðt þ kÞ � fjþ1ðtÞk

¼ fj�1ðtÞ fjþ2ðtÞ � fjðtÞ fjþ1ðtÞ; ð4:12Þ

where k is the mesh size of time interval. It is easy to see that (4:12) does not have any gauge invariance. Indeed, if wereplace fjðtÞ by gðtÞ fjðtÞ in (4:12), then

gðtÞgðt þ kÞfjþ1ðtÞ fjðt þ kÞ

k�

fjðtÞ fjþ1ðt þ kÞk

� �¼ g2ðtÞð fj�1ðtÞ fjþ2ðtÞ � fjðtÞ fjþ1ðtÞÞ;

which is not satisfied unless gðt þ kÞ ¼ gðtÞ. Hence, (4:12) is not invariant under gauge transformations.

146 SUZUKI

Since we want (4:12) to have the gauge invariance, we shift the time variable on the right-hand side of (4:12) toobtain

fjþ1ðtÞfjðt þ kÞ � fjðtÞ

k� fjðtÞ

fjþ1ðt þ kÞ � fjþ1ðtÞk

¼ fj�1ðtÞ fjþ2ðt þ kÞ � fjðt þ kÞ fjþ1ðtÞ: ð4:13Þ

Let t ¼ nk and denote fjðnkÞ by f nj . Then, equation (4:13) can be rewritten as

ð1þ kÞ f nþ1j f njþ1 ¼ f nj f

nþ1jþ1 þ k f nj�1 f

nþ1jþ2 : ð4:14Þ

This is the desired discrete bilinear equation for (4:10).Step 3: From discrete bilinear equation to discrete Lotka–Volterra equations. Finally, we obtain a discreteLotka–Volterra equation from (4:14). To come back to the variable unj from f nj , we set

unj ¼f nj�1 f

nþ1jþ2

f nj fnþ1jþ1

: ð4:15Þ

Then, we have

unþ1j

unj¼

f nþ1j�1 f nþ2

jþ2 f nj fnþ1jþ1

f nþ1j f nþ2

jþ1 f nj�1 fnþ1jþ2

: ð4:16Þ

Since

unj�1 ¼f nj�2 f

nþ1jþ1

f nj�1 fnþ1j

;

we see that identity (4:16) becomes

unþ1j

unj¼

f nþ1j�1 f nþ2

jþ2 f nj unj�1

f nþ2jþ1 f nþ1

jþ2 f nj�2

: ð4:17Þ

From (4:14), the following two equations hold:

ð1þ kÞ f nþ2jþ1 f nþ1

jþ2 ¼ f nþ1jþ1 f nþ2

jþ2 þ k f nþ1j f nþ2

jþ3 ; ð4:18Þ

ð1þ kÞ f nþ1j�1 f nj ¼ f nj�1 f

nþ1j þ k f nj�2 f

nþ1jþ1 : ð4:19Þ

Substituting (4:18) into the right-hand side of (4:17), we obtain

unþ1j

unj¼

ð1þ kÞ f nþ1j�1 f nþ2

jþ2 f nj unj�1

f nþ1jþ1 f nþ2

jþ2 f nj�2

�1þ k

f nþ1j f nþ2

jþ3

f nþ1jþ1 f nþ2

jþ2

�

¼ð1þ kÞ f nþ1

j�1 f nj unj�1

f nþ1jþ1 f nj�2ð1þ kunþ1

jþ1 Þ: ð4:20Þ

Furthermore, it follows from (4:19) that

ð1þ kÞ f nþ1j�1 f nj

f nj�2 fnþ1jþ1

¼f nj�1 f

nþ1j

f nj�2 fnþ1jþ1

þ k:

Therefore, substituting the above equation into the right-hand side of (4:20) gives

unþ1j

unj¼

unj�1

1þ kunþ1jþ1

f nj�1 fnþ1j

f nj�2 fnþ1jþ1

þ k

!: ð4:21Þ

Since ð f nj�1 fnþ1j Þ=ð f nj�2 f

nþ1jþ1 Þ ¼ ðunj�1Þ

�1, we see

unþ1j

unj¼

1þ kunj�1

1þ kunþ1jþ1

: ð4:22Þ

Consequently, the following discrete Lotka–Volterra equation is obtained:

unþ1j � unj

k¼ unj u

nj�1 � unþ1

j unþ1jþ1 : ð4:23Þ

In the following, we shall see that equation (4:23) has a soliton solution which is given by formula (4:42) below and thetransformation (4:15).

We have introduced the procedure to obtain a discrete Lotka–Volterra equation which has conserved quantities. Thisreasoning seems to be complicated because of several calculations and it is not easy to solve (4:23). However, it isimportant to have this relation in order to introduce the corresponding cellular automata.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 147

4.5 Ultradiscretization methods

Cellular automata are discrete dynamical systems that consist of a lattice of discrete identical cells. The states of cellsevolve in discrete time steps according to simple rules. Note that dependent variables of this system are discrete. On theother hand, dependent variables of difference equations are still continuous quantities though they are considered in thespace-time plane which is divided into a uniform grid. For example, unj in (4:23) represents the density of the species Xj

at point xj ¼ jh and at time t ¼ nk, where h and k are the mesh size of space and time, respectively. Here, the word‘‘density’’ suggests that the state varies continuously, not discretely.

In the following, we introduce a technique to discretize dependent variables of differential equations, and we relatedifferential equations with cellular automata.

4.5.1 Ultradiscrete diffusion equations

First, we obtain the discrete counterpart of the following diffusion equation in one-dimensional space:

@u

@t¼@2u

@x2; ð4:24Þ

where u ¼ uðx; tÞ represents the density of particles. Let the space and time variables be divided by N, and let h ¼ 1=Nand k ¼ T=N. We denote the value of uðx; tÞ at point xj ¼ jh and time t ¼ nk by unj . Using the explicit method (4:7), wehave

unþ1j ¼ unj þ

k

h2ðunj�1 � 2unj þ unjþ1Þ; ð4:25Þ

where the dependent variable has not been discretized yet. In the following, we put k=h2 ¼ 1=2 for simplicity, whichassures us that (4:27) is obtained.

Assume unj � unjþ1 for all 0 � j � N at time n. We note that the particles diffuse from the place of higher density tothat of lower. If

� � � ¼ unj�2 ¼ unj�1 < unj ¼ unjþ1 ¼ � � � ; ð4:26Þ

then diffusion occurs left from unj but it never occurs from unjþ1 to unj . Since the state between unjþ1 and unj will be inequilibrium, the next time step becomes

unþ1j�2 < unþ1

j�1 ¼ unþ1j < unþ1

jþ1 : ð4:27Þ

Introduce new variable Vnj as follows. If unj ¼ unjþ1, then there is no diffusion effect from right to left. We denote this

state by Vnj ¼ 0. If unj < unjþ1, then the diffusion occurs from right to left. This is the state Vn

j ¼ 1. Hence, the example(4:26) above shows that Vn

j�1 ¼ 1 and Vnj ¼ 0 becomes Vnþ1

j�1 ¼ 0 and Vnþ1j ¼ 1. As a result, the dependent variable in

(4:25) has been discretized. This kind of discretizing process on a dependent variable is called ultradiscretization. Wenow start to derive an ultradiscretization from the diffusion.

Introducing a new variable vnj defined by

vnj ¼1

h

unjþ1

unj; ð4:28Þ

we will obtain a discrete equation for vnj . Since

vnþ1j

vnj¼

unþ1jþ1 u

nj

unþ1j unjþ1

;

and it follows from (4:25) that

unþ1j

unj¼ 1� 2

k

h2þ

k

h2

unj�1 þ unjþ1

unj;

unþ1jþ1

unjþ1

¼ 1� 2k

h2þ

k

h2

unj þ unjþ2

unjþ1

:

Thus, we have

vnþ1j

vnj¼

1� 2k

h2þ

k

h2

unj þ unjþ2

unjþ1

1� 2k

h2þ

k

h2

unj�1 þ unjþ1

unj

: ð4:29Þ

From the definition of vjn, equation (4:29) becomes

148 SUZUKI

vnþ1j

vnj¼

1� 2k

h2þ

k

h2

�1

hvnjþ hvnjþ1

�

1� 2k

h2þ

k

h2

�1

hvnj�1

þ hvnj

�: ð4:30Þ

Furthermore, we transform vnj into

vnj ¼ eVnj =" ð4:31Þ

for " > 0, and assume h and k satisfy

h2 ¼ e�1="; ð4:32Þh3

k� 2h ¼ e�1=": ð4:33Þ

Substituting (4:31) into (4:30), we obtain

eVnþ1j

=" ¼ eVnj ="

1� 2k

h2þ

k

h3ðe�Vn

j =" þ h2eVnjþ1="Þ

1� 2k

h2þ

k

h3ðe�Vn

j�1=" þ h2eV

nj="Þ:

Relation (4:33) leads to the identity

1� 2k

h2¼

k

h3h3

k� 2h

� �¼

k

h3e�1=";

which, together with (4:32), allows us to transform (4:31) as follows

eVnþ1j

=" ¼ eVnj ="

e�1=" þ e�Vnj =" þ eðV

njþ1

�1Þ="

e�1=" þ e�Vnj�1=" þ eðV

nj�1Þ=" : ð4:34Þ

Computing the logarithms of both sides of (4:34), we see

Vnþ1j ¼ Vn

j þ " log½e�1=" þ e�Vn

j =" þ eðVnjþ1

�1Þ="� � " log½e�1=" þ e�Vnj�1=" þ eðV

nj �1Þ="�: ð4:35Þ

Now, we need the following lemma to connect (4:35) with a cellular automaton.

Lemma 4.1. The following identity holds:

lim"!þ0

" logðeA=" þ eB="Þ ¼ maxðA; BÞ; ð4:36Þ

for all A > 0 and B > 0.

Proof. First, assume that A > B. Then, the left-hand side of (4:36) is

lim"!þ0

" log½eA="ð1þ e�ðA�BÞ="Þ� ¼ Aþ lim"!þ0

" logð1þ e�ðA�BÞ="Þ:

Since A� B > 0, the limit on the right-hand side above is equal to 0. This shows (4:36). The case B > A can be provedby the same argument.

Next, assume A ¼ B. It is easy to see that the left-hand side of (4:36) satisfies

lim"!þ0

" log 2eA=" ¼ Aþ lim"!þ0

" log 2 ¼ A:�

Remark 4.2. By virtue of Lemma 4.1, the following general formula holds:

lim"!þ0

" logðeA1=" þ eA2=" þ � � � þ eAn="Þ ¼ maxðA1; A2; . . . AnÞ: ð4:37Þ

Now, let us come back to the estimate of (4:35) by making use of Lemma 4.1. Computing the limits as "! þ0 ofboth sides of (4:35), we obtain

Vnþ1j ¼ Vn

j þmaxð�1;�Vnj ;�ð1� Vn

jþ1ÞÞ �maxð�1;�Vnj�1;�ð1� Vn

j ÞÞ:Note that minðA;BÞ ¼ �maxð�A;�BÞ. Therefore, we get the ultradiscrete diffusion equation

Vnþ1j ¼ Vn

j �minð1; Vnj ; 1� Vn

jþ1Þ þminð1; Vnj�1; 1� Vn

j Þ: ð4:38ÞIf an initial state fV0

j g0� j�N is set to be 0 or 1 for each j, then Vnj is determined to be either 0 or 1 in each time step.

Hence, (4:38) is a cellular automaton modeling a diffusion process.

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 149

4.5.2 Ultradiscrete Lotka–Volterra equations

Now, we show the process of deriving ultradiscrete equations from nonlinear differential equations, taking theLotka–Volterra equations (4:23) as an example.

First, we use a transformation of variables analogous to that in (4:31). Letting Unj be an integer and " > 0, we

transform unj and k into

unj ¼ eUnj ="; k ¼ e�1=":

Substituting these new variables into (4:22), we have

eUnþ1j="

eUnj="

¼1þ keU

nj�1="

1þ keUnþ1jþ1=":

After computing the logarithm of both sides above and some calculations, we see that the Unj ’s satisfy

Unþ1j ¼ Un

j þ " logð1þ keUnj�1="Þ � " logð1þ keU

nþ1jþ1="Þ

¼ Unj þ " logð1þ eðU

nj�1

�1Þ="Þ � " logð1þ eðUnþ1jþ1

�1Þ="Þ: ð4:39Þ

It follows from Lemma 4.1 that the limit as "! þ0 gives the following ultradiscrete Lotka–Volterra equation

Unþ1j ¼ Un

j þmaxð0; Unj�1 � 1Þ �maxð0; Unþ1

jþ1 � 1Þ: ð4:40Þ

Note that the right-hand side of (4:40) has the term at time nþ 1, namely, Unþ1jþ1 . Therefore, the time evolution of

(4:40) cannot be obtained easily even if an initial state is given, which implies that this has not become a cellularautomaton yet. To obtain a cellular automaton, we need more calculations.

It is known that the bilinear differential equation (4:10) has the following explicit solution:

fjðtÞ ¼X

S�f1;2;...;Ngexp

Xm2S

�mð j; tÞ þXl;m2Sl<m

alm

0B@

1CA; ð4:41Þ

where

�mð j; tÞ ¼ �m j� !mt þ cm; !m ¼ 2 sinh �m;

alm ¼ log sinh2�l � �m

2

�sinh2

�l þ �m2

� �:

If the parameters �m and cm are given for 1 � m � N with a natural number N, then !m, �mð j; tÞ, alm are fixed, and fjðtÞare determined by (4:41).

On the right-hand side of (4:41), the sum is taken over all subsets S of f1; 2; . . . ;Ng (including the empty set).The symbol

Pm2S stands for the sum over all elements in S, and

Pl;m2Sl<m

is the sum over all pairs of elements in S

satisfying l < m. Here, if S ¼ ;, thenP

m2S is equal to 0. If S consists of only one integer, thenP

l;m2Sl<m

is defined to

be 0.Since (4:10) has the explicit solution (4:41), we see that the discrete bilinear equation (4:14) has the following

solution:

f nj ¼X

S�f1;2;...;Ngexp

Xm2S

�mð j; nÞ þXl;m2Sl<m

alm

0B@

1CA; ð4:42Þ

where

�mð j; nÞ ¼ �m j� !mnþ cm; !m ¼ log1þ kð1þ e�m Þ1þ kð1þ e��m Þ

;

alm ¼ log sinh2�l � �m

2

�sinh2

�l þ �m2

� �:

The difference between (4:41) and (4:42) lies in the replacement of t in (4:41) by n in (4:42). The definition of !m isalso changed.

Now, we come to the derivation of a cellular automaton from (4:40). To use the explicit solution (4:42) of (4:14), wefirst make an ultradiscrete equation of (4:14). Transform f nj and k into

f nj ¼ eFnj ="; k ¼ e�1=":

Substituting the above into (4:14) and computing the logarithm of both sides of the resulting equation, we obtain

150 SUZUKI

Fnþ1j þ Fn

jþ1 þ " logð1þ e�1="Þ ¼ " logðeðFnj þFnþ1

jþ1Þ=" þ eðF

nj�1

þFnþ1j�2

�1Þ="Þ: ð4:43ÞLetting "! þ0 on both sides of (4:43), we have the following ultradiscretized counterpart of equation (4:14):

Fnþ1j þ Fn

jþ1 ¼ maxðFnj þ Fnþ1

jþ1 ; Fnj�1 þ Fnþ1

j�2 � 1Þ: ð4:44Þ

Now, recall the formula (4:15) which defines the relation between unj and f nj . Using (4:15) together with unj ¼ eUnj =" and

f nj ¼ eFnj =", we obtain a relation between Un

j and Fnj . Indeed, since it follows from (4:15) that

eUnj =" ¼

eFnj�1="eF

nþ1jþ1="

eFnj="eF

nþ1jþ1=";

we have the following identity after computing the logarithm of both sides:

Unj ¼ Fn

j�1 þ Fnþ1jþ1 � Fn

j � Fnþ1jþ1 : ð4:45Þ

It is clear that equation (4:44) has an explicit solution which is given by ultradiscretizing (4:42). Let �m, !m and cm in(4:42) be transformed into

�m ¼Km

"; !m ¼

�m

"; cm ¼

Cm

":

Moreover, let �mð j; nÞ ¼ �mð j; nÞ=" and alm ¼ Alm=". Then, an explicit solution of (4:44) is obtained by taking the limitof (4:42) as "! þ0:

Fnj ¼ max

S�f1;2;...;Ng

Xm2S

�mð j; nÞ þXl;m2Sl<m

Alm

0B@

1CA; ð4:46Þ

where

�mð j; nÞ ¼ Kmj��mnþ Cm;

�m ¼ maxð0; Km � 1Þ �maxð0; �Km � 1Þ;Alm ¼ jKl � Kmj � jKl þ Kmj:

On the right-hand side of (4:46), maxS�f1;2;...;Ng denotes the maximum over all the subsets S of f1; 2; . . . ;Ng. From (4:45)and (4:46), we get an explicit solution of the ultradiscrete Lotka–Volterra equation in the case N ¼ 1. When N ¼ 1, thequantities Fn

j defined by (4:46) satisfies

Fnj ¼ maxð0; K1 j��1nþ C1Þ:

Letting K1 ¼ 3 and C1 ¼ 0 for simplicity, we see �1 ¼ 2. Therefore, we have

Fnj ¼ maxð0; 3 j� 2nÞ:

It follows from (4:45) that Unj satisfies

Unj ¼ maxð0; 3j� 2n� 3Þ þmaxð0; 3j� 2nþ 4Þ

�maxð0; 3 j� 2nÞ �maxð0; 3j� 2nþ 1Þ: ð4:47Þ

Note that the right-hand side of (4:47) attains not only 0 and 1 but also other integer. If we modify (4:47) to take only 0or 1, then it becomes a cellular automaton. We see the transformation

Bnj ¼ F

jj�nþ1 þ F

j�1j�n�1 � F

j�1j�n � F

jj�n

leads to a cellular automaton. When N ¼ 1, K1 ¼ 3 and C1 ¼ 0, the sequence Bnj satisfies

Bnj ¼ maxð0; j� 3nþ 3Þ þmaxð0; j� 3n� 1Þ

�maxð0; j� 3nþ 2Þ �maxð0; j� 3nÞ ð4:48Þ

because of Fnj ¼ maxð0; 3j� 2nÞ. It can be checked that relation (4:48) defines a cellular automaton.

When N ¼ 2, the sequence Bnj defined by (4:48) exhibits the time evolution shown in Figure 4.1 for appropriate

constants. The rules to obtain Figure 4.1, which were explained at the beginning of Section 4, can be written in terms ofthe mathematical equation as follows:

Bnþ1j ¼ min 1� Bn

j ;Xj�1

l¼�1ðBn

l � Bnþ1l Þ

!: ð4:49Þ

On the right-hand side of (4:49), the term 1� Bnj shows that the cell is either empty or not at position j and at time n.

Indeed, if Bnj ¼ 0 (cell is empty), then 1� Bn

j ¼ 1, while if Bnj ¼ 1 (cell is occupied), then 1� Bn

j ¼ 0. Concerning thesecond term in the minimum function on the right-hand side of (4:49), it means

Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 151

Xj�1

l¼�1ðBn

l � Bnþ1l Þ

¼ (the sum of balls on the left side of the cell at position j and time n)

� (the sum of balls on the left side of the cell with position j and time nþ 1)

This quantity is the amount of balls which we have at position j and at time n. Hence, relation (4:49) becomes

(The state of cell at position j and at time nþ 1)

¼ min (empty cell or not at position j and at time n, balls in hand at time n):

It is clear that the right-hands side of (4:49) returns to 0 if a cell is either empty or we do not have any balls, while itreturns to 1 whenever a cell is empty and we have balls, which corresponds to what is explained in Figure 4.1.

Finally, let us remark that we can introduce the ultradiscrete Lotka–Volterra equation starting from (4:49). However,there are many complicated calculations again.

5. Summary

These notes are devoted to the explanation of dynamic properties of reaction-diffusion systems modeling patternformation, in particular, Turing patterns. Moreover, we described ideas which connect differential equations withcellular automata. More mathematical analysis of the Lengyel–Epstein model may be found in [7]. General theoryabout Turing patterns and their analysis are explained in [12]. Monograph [11] elaborates on the BZ reaction. For theultradiscretization and corresponding cellular automaton, we refer the reader to [13, 14] as well as to [5, 6].Mathematical analysis of biological models are given in [10]. Examples and remarks on numerical simulations arefound in [4].

In general, mathematical models by differential equations are simplified as much as possible. For example, the BZreaction consists of ten chemical reaction processes, but its mathematical models by kinetic systems have two or threecomponents. If we need to know the evolution of spatial patterns for long time, reaction-diffusion equations are suitablefor the purpose. On the other hand, a cellular automaton is the simplest model, which is used as models not only ofpattern formation but also of other biological phenomena, crystal growth, turbulence, and so on. It is surprising thatcellular automata show the complicated behaviours according to simple rules, and give us the understanding of themechanism among a large class of phenomena. Thanks to the universal mathematical structure of cellular automata, wemay be able to have a new approach for other subjects of mathematics. Indeed, complex systems as a subject ofmathematics have been developed based on the studies of cellular automata.

Acknowledgments

I want to express my heartfelt thanks to the referees for their deep and precise comments that helped me to improveconsiderably this paper.

Fig. 5.1. This pattern can be expressed by a cellular automaton. From Wikipedia http://en.wikipedia.org/wiki/Cellular automaton,June 9, 2011.

152 SUZUKI

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Kodansha (1997).[12] Nishiura, Y., Hiheikou Dainamikusu no Suri (Japanese), Iwanami Shoten (2009).[13] Takahashi, D., ‘‘Continuous, discrete, ultradiscrete waves,’’ Surikaisekikenkyusho Kokyuroku, 1191: 104–111 (2001).[14] Tokihiro, T., Takahashi, D., Matsukidaira, J., and Satsuma, J., ‘‘From soliton equations to integrable cellular automata through

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Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems 153