Collision Patterns on Mollusc Shells

On mollusc shells one can find famous patterns. Some of them show a great resemblance to the soliton patterns in one-dimensional systems. Other look like Sierpinsky triangles or exhibit very irregular patterns. Meinhardt has shown that those patterns can be well described by reaction–diffusion systems [1]. However, such a description neglects the discrete character of the cell system at the growth front of the mollusc shell.


A INTRODUCTION
One example is the sea shell pattern of the Amoria dampiera (Fig. 1), which seems to be very sim- ple.Stripes perpendicular to the direction of growth look like waves.One gets the impression that all the cells at the growing edge (the lip of the shell) are oscillating in synchrony.The ques- tion that arises is how are they synchronized ?Meinhardt modelled this pattern using his fa- mous activator-inhibitor model of two coupled partial differential equations of the reaction diffu- sion type [1].Here a denotes the activator and b the inhibitor concentration.The constants ba and bb are the permanent activator and inhibitor productions, while s is the so-called "source density" of the activator pro- duction.For a given set of parameters this sys- tem of differential equations will exhibit stripes perpendicular to the direction of growth.But As usual, one can rewrite the Laplacian term by the well known spatial discretisation V2c c(x 1,7-) 2c(x, 7-) + c(x + 1,7-).
So Fick's law can be formulated by the itera- tive equation, where is the discrete time now and c the concentration.Let us call this discrete formulation of Fick's second law the Laplacian diffusion: FIGURE Pattern of the sea shell of Amoria damperia (with the kind permission of Meinhardt [1]).See Color Plate I. models of this type of spatially non-restricted ki- netic reaction-diffusion equations totally neglect the cellular character of the biological system.
During the "13.Winterseminar on Zeinisjoch (March 1996)", G. Baier presented a system com- prising a large number of linearly arranged diffu- sively coupled chemical oscillators.It is very large-scaled to handle such a system, but seems to permit the description of pattern formation in the biological system of the sea shells as well.
However, our basic question is, "Can we con- struct a cellular automaton model, discrete in space and time, which is able to show the pat- terns being observed in the animals?" B AVERAGING AND DIFFUSION To answer this question, we first have to ask "What does diffusion mean in a cellular system?"Moreover, we have to ask whether random events are necessary in order to understand diffu- sion.Let us give a very first and rough answer to both questions.In the linear case, one does not need randomness to describe isotropic diffusion processes, due to the fact that the linear space is isotropic by definition.So diffusion becomes any kind of spatial averaging.For example, let us take Fick's second law OC 02C 07--De ON 2 c(x, + 1) c(x, t) + Dc[c(x 1, t) 2c(x, t) + c(x + l,t)], xEZ;tEN'.Figure 2 shows an example of how this itera- tive equation works.
On the other hand, there is the well-known binomial smoothing procedure of a discrete func- tion Z(i) with N': where n is the number of smoothing generation.
Interpreting this number n as discrete time t, this smoothing procedure formally represents a temporal averaging.One can easily rewrite this smoothing procedure in terms of the diffusion equation, replacing Z(i, n) by c(x,t): c(x, + 1) (c(x 1, t) + 2c(x, t) + c(x + 1, t)) c(x, t) + 1 / 4 (c(x 1, t) 2c(x, t) + c(x + 1, t)) c(x, t) + Dc(c(x-1, t) 2c(x, t) + c(x + 1, t)) with the diffusion coefficient Dc-1 / 4 .Let us call this special procedure the binomial diffusion (Fig. 3).We can now state that, in the case of a linear cellular automaton, diffusion means any kind of a spatial averaging of the states of the cells over time.In order to deal with natural numbers only, we define integral states z(i, t) by means of the Gaussian-brackets notation [u]:  (i, t) 1 / 4 (z(i 1, t) + 2z(i, t) + z(i + 1, t)), where [u] is the largest natural number that is less or equal to u: [u] < u and [u] + > u.
Again, this expression represents a kind of dif- fusion, for which reason we call it a Gaussian diffusion.In contrast to the binomial diffusion, however, it does not spread out the quantities all over the space.If we set a very sharp initial con- centration profile, it does not run but stays in a restricted domain, although it shrinks over time.
(see Fig. 4) One can easily express this iterative equation in the form of a transformation rule of a cellular auto- maton with the abbreviation: E g_lz(i-1, t)+ goz(i,t) +g+iz(i+ 1, t); with the weights g_-- g+l 1, go 2 as shown in Table I z(i,t/l) 0 0 0 0 E 7 8 9 10 11 12 z(i,t+l) Introducing an additive term b in the equation z(i, + 1) [(i, t)] leads to the expression: This iterative equation describes a spatially re- stricted diffusion as well, but the parameter b can now be interpreted chemically.For b > 0 one may think of a production and for b < 0 a destruction, which is added to the diffusion.For example if b 0.25 diffusion will be stopped after some time  by the production (Fig. 5).One obtains a station- ary spatial distribution of states: z(i,t+ 1)= z(i,t) for all cells i.Let us call G the sum of weights of states of the neighbouring cells: +1 G-Zgk.
Increasing b, the stationary state is reached if b becomes larger or equal to 1/G" 1/G < b < 1.
Setting b-1 it generally means a shift of the z(i, + 1)-row of Table I to the left by four digits: Introducing a lower limit L for opening the state growth of a cell: t+l)-f0 if z(i, t) 0 and N <_ L, [2(i, t) + b] otherwise, the states of those cells start to grow which have an appropriate neighbourhood (see Fig. 5).As a result, Gaussian diffusion is not spatially re- stricted anywhere.
This value of b causes a destruction which is added to the normal Gaussian diffusion.Now the state 5'(i,t) of a cell at time is represented by two different quantities: the integral concentration z(i, t) and the phases p(i, t)" C THE VECTOR AUTOMATON MODEL 2'(i, t) (z(i,t)) p(i,t) Both look-up tables (Tables II and III) realize classical cellular automata.One can combine these automata with each other, constructing a new type of automata-cellular vector automata Let us assume that a biological cell possesses two different phases of activity: an active phase with p-0 and a passive phase p-1.The beha- viour of the biological cells in these two phases should be characterized by a production (p-0; b > 0) in the active state, and a destruction (p-1; b_< 0) in the passive state.So the phases may be used to construct a switch between the different rules.But it should be an internal switch, the position of which should again depend on the states of the neighbouring cells.
For example, if p(i, t)-0 then" p(i,t/l) 0 0 0 0 0 0 0 7 8 9 10 11 12 p(i,t/l) 0 which are now the components of the state vector Y.The first rule means that the production rule (Table IV) works up to < 7, whereas the destruction rule (Table V) governs the region for E; > 8.But if the cell is in the phase p(i, t)-and the concentration sum E of the neighbours exceeds 3" > 4, the destruction rule is responsible for the development of the vector state.Only if _< 3 does the production rule start to work again.This means that the cellular vector automaton displays a hysteresis loop.

D DISSIPATION
The system of coupled differential equations, used by Meinhardt [1] is essentially dissipative, as can be shown by the non-vanishing trace of the Jacobian.In cellular automata there is no explicit expression for this proof of the dissipative char- acter of the transformation rule.But looking at the maps one can decide whether the system is conservative or dissipative.For example, if there is a one-to-one map the system is conservative [6].But in our case the map F 5'(i, t) --5'(i, + 1) and if p(i, t)- p(i,t/l) 0 0 0 0 7 8 9 10 11 12 p(i,t+l) is not unique.The graph of this map is com- posed of two tent-like staircases, which are shifted against each other.So we have a many- to-one map which is essentially dissipative.Since the general transformation rule of the vector state 5'(i, t) realizes a dissipative map, the cellular vector automaton becomes a powerful tool for llaodelling the patterns of a natural sys- tem such as mollusc shells.
In particular, if we use a finite number of 599 linearly arranged cells of the automaton and cyc- lic boundary conditions, the periodic state of the automaton is structured as riffled waves perpendicular to the direction of time (Fig. 7), which is also the direction of growth of the mollusc shell lip. Figure 8 shows that there are indeed mollusc shells which exhibit precisely such riffled patterns.
Rather than starting with only one cell, but instead with many cells./" in the states 5'(j, 0)-() distributed randomly, one obtains very plane waves, perpendicular to the direction of growth (Fig. 9).There are also sea shells which display precisely such patterns; (see Fig. 1).
The other very simple shell pattern Meinhardt discussed [1] involves stripes parallel to the direction of growth (Fig. 10).Using the vector if-(1, 1, 1) and a very strong production rule z(i,t+l)-[2+3] with the transformation matrix:   p(i,t)=O z(i,t+l) 0 3 3 4 4 4 5 p(i,t)= z(i,t+l) 0 0 0 0 0 0 p(i,t)=O p(i,t+l) 0 0 0 0 0 0 p(i,t)=l p(i,t+l)0 0 and a few (about 20) arbitrary cells, after some time one obtains stripes parallel to the direction of growth (Fig. 11).These stripes represent spa- tially stable situations of the automaton.They are created by cells which run into a non-vanish- ing stable "fixed point".Finally this means that one obtains a vector state distribution which is stable in space and time.But if one looks carefully at the origin of the stripes, one can see that they are created by the collision of solitary waves.If solitary waves col- lide, a new, localized "collision state" emerges.Such a state may be stable, as in the case of stripes parallel to the direction of growth.However, other solitary waves might exist the ex- cited collision state of which is unstable and vanishes or it decomposes after a while.In the first case one should obtain "chemical waves", while in the second case "solitons" should be observed.
FIGURE 10 The pattern of the cone shell Hirasei (44.2 For example, there is a very simple automaton ram) from Taiwan shows stripes parallel to the direction of rule which results a chemical wave: growth (with the kind permission of Jerry G. Walls   p(i,t)= z(i,t+l) 0 0 0 0 0 p(i,t)=O p(i,t+l)0 o The excited state which is created by the colli- sion vanishes after a few time steps.The zero state of the cells will form the only stable fixed point of the dynamics of the collision system (Fig. 12).This simple rule is nothing other  VII, causing stripes parallel to the direction of growth.
FIGURE 12 The spatio-temporal development of a "chemical wave" in the one-dimensional space of the automaton according to Table VIII.
than a reformulation of the famous Wiener- Rosenblueth automaton for a one-dimensional cellular automaton.It therefore comes as no sur- prise that some mollusc shells exhibit patterns which resemble the annihilation of colliding soli- tary waves (Fig. 13).
A closer look at such mollusc patterns reveals that this collision behaviour of chemical waves is often accompanied by soliton-like behaviour.If solitons collide, they pass each other by a phase shift.The following transformation matrix en- ables the creation of such solitons (Fig. 14): p(i,t)=O z(i,t/l) 0 2 p(i,t)= z(i,t+l) 0 0 0 0 p(i, t) O p(i, / 0 0 0 p(i,t)= p(i,t+l) 0 0 5 6 7 p(i,t)=O z(i,t+l) 0 0 0 p(i, t) O z(i, / 0 0 p(i, t) O z(i, / p(i,t)=O z(i,t+l) These two different kinds of solitary waves are not the only ones, however.There is a variety of solitary waves the collision states of which differ considerably.In the case of the chemical and the soliton waves respectively, one can classify the solitary waves purely in terms of their collision behaviour.Classifying waves in this way is very fundamental [11].For example, there are solitary waves which cross each other without any phase shift (Fig. 15).They can generated by the trans- fomation matrix: p(i,t)=0 z(i,t/l) 0 3 0 0 p(i,t)=l z(i,t/l) 0 0 3 0 p(i,t)=O p(i,t+l) 0 0 0 0 p(i,t)= p(i,t+l)0 0 5 6 7 p(i, t) z(i, + 0 0 0 p(i, t) z(i, + 0 0 0 p(i, t) O z(i, + 0 0 p(i,j) O z(i,t / 0 0 0 In this respect they behave like classical waves.In this case the collision state is very unstable.Its lifetime tends to zero.As has been demonstrated, there are many struc- tures for the collision states of solitary waves.The question arises as to whether one can create chaotic collision patterns as well.To answer this question, one should investigate the automata under consideration in greater detail.The proce- dure of averaging does not enable the creation of waves, even in the case of integral numbers, for example: z(i, + 1) [ 1 / 2 ].However, taking a productive diffusion z(i, + 1) [ 1 / 2 E + 1] and in addition if E(i, t) 0 z(i, + 1) 0, one may obtain an automaton which will create a chemi- cal wave with a saw-tooth profile.

E STRUCTURES OF COLLISION
FIGURE 14 The automaton in Table IX, which creates solitons in the one-dimensional space.FIGURE 15 Solitary waves crossing each other without any phase shift similar to "classical" waves according to Table X p(i,t)-0 z(i, t4-1) 3 3 4 4 4 182 183 184 185 186 p(i,t)--0 z(i, t4-1) 61 62 62 62 0 dependent on the state z0 z(i, t) of the cell under consideration.Meinhardt [1] used such linear terms raa and rbb in his system of differential equations mentioned in the Introduction.
Playing around with the transformation rules one may ask what happens if one uses an inverse sequence like IO-E; ifO<E< 10, Y(i,t+l)-0 ifP-OorE>_ 10.TABLE XVII N 0 p(i,t)=0 z(i,t+l) 0 9 p(i,t) z(i,t / l) 0 p(i, t) =0 p(i, / 0 p(i, t) p(i, / 0 6 7 z(i,t+l) 4 3 z(i,t/l) 0 p(i,t) =0 p(i,t) p(i, t) O p(i, + l) p(i, t) p(i, + 2 3 4 5 8 7 6 5 This means that for larger states there is a strong diffusion but almost no production, whereas for smaller states there is a strong production but diffusion can be neglected.This transformation rule simulates the competition between an inhibition for higher concentrations and a strong autocatalysis for low concentrations (see Fig. 20).
The patterns, produced by this automaton are very sophisticated even if one starts with only FIGURE 19 "Turbulent" pattern in the temporal development of the one-dimensional automaton according to Table XVI (see [8] ).We set out to elaborate an automaton produ- cing solitary waves which will create all the fan- tastic spatio-temporal phenomena if they collide.We now have all the ingredients for its con- struction.If the concentration states of the cells are small, they should diffuse classically, with the diffusion coefficient D_<1/2.However, if the concentration states of the cells are large en- ough, an autocatalytic production should govern the behaviour of the system.Through the colli- sion of the low-state solitary waves, states be- come neigbouring thus throwing the system into the autocatalytic regime.The following transfor- mation rule will create such a behaviour (Fig. 21): TABLE XVIII E 0 p(i,t)--0 z(i,t+l) 0 p(i,t) z(i,t+l) 0 p(i, t) O p(i, + 0 p(i, t) p(i, + l) 0 p(i,t) --0 p(i,t) E z(i,t+ 1) z(i,t+ 1) p(i, t) O p(i, + l) p(i, t) If the low-state solitary waves collide, they create a turbulent system in the inner part of the spreading "reflected" waves.Moreover, there are transfor- mation rules (Table XIX), constructed in a similar manner, which create turbulent areas spreading with half the velocity of the original solitary waves (Fig. 22): p(i, t) z(i, + 0 p(i, t) O p(i, + 0 p(i,t)--1 p(i,t+l) E 6 7 8 9 10 p(i,t)=O z(i,t/l) 3 3 3 4 4 p(i,t)--1 z(i,t+l) 0 p(i,t)=0 p(i,t+l) 0 p(i,t)=l p(i,t/l) 11 12 13 14 p(i,t)=0 z(i,t/l) p(i, t) p(i, + Even the "strings of pearls" (Fig. 23) can be generated in a similar way by the automaton (Table XX): FIGURE 22 Spreading of the "turbulent" interval with half the velocity of the original colliding solitary waves according to Table XIX.

F CONCLUSION
We have shown a very few examples in order to demonstrate the power of our vector automaton model for simulating a high variety of collision patterns of solitary waves.Our aim was neither to present a complete classification of the possible patterns which can be achieved by this type of automata, nor to simulate all natural sea shell patterns.
We pursued the idea that the natural patterns among biological populations of neighbouring in- dividual cells can be understood by means of discrete mathematical tools such as cellular vec- tor automata.The classical ideas of diffusion and production can easily be induced using these automata models.
As we have shown, a new idea was generated in the process.The dynamics of localized excita- tions created by the collision of solitary waves can be described in terms similar to those of   Between these systems and the spatially and temporally stable systems there exist pulsating systems.If these systems are always pushing out new pairs of solitary waves, they are commonly known as reverberators.But again, this class of automata embraces more behaviour patterns than only the classical ones.
This article provides an initial insight into this fascinating area.Much more work has to be done in the future.We have not investigated the reflection behaviour of the different solitary waves, which will bring us to a much better understanding of natural sea-shell patterns.But this is certainly not the only question we have left aside for the future.ordinary dynamic systems.There are localized stable stationary states which give rise to the for- mation of stripes in the 2D-space-time parallel to the direction of growth, as can be observed on the sea shells (Figs. 10, 11).
There also exist localized oscillating patterns (Fig. 24) and localized chaotic patterns (Fig. 25) similar to the pearl string patterns.
A new class of patterns has been found: delo- calized states.This means an excitation spreads over the automaton leaving all cells in an excited state.Examples include not only the well known front wave, but many other phenomena as well, such as the spreading of turbulent or spatially intermittent patterns.
Furthermore, there are colliding systems which are spatially and temporally instable.Such sys-

FIGURE 2
FIGURE 2 Graphs of the iterative formulation of Fick's second law in one-dimensional space.

FIGURE 3
FIGURE3 Graphs of the binomial diffusion of a sharp concentration profile at the beginning.

FIGURE 4
FIGURE 4 Graph of the spatially restricted Gaussian diffusion.

FIGURE 5
FIGURE 5 Development of the productive Gaussian diffusion with the limitation of the state growth if z(i, t)= 0 and E _< L.

[ 2 - 5 ]
. The advantage of such cellular vector automata is that one can select different rules within one expression by selecting different va- lues of b.

FIGURE 7
FIGURE 7  Spatio-temporal development of the automaton Table VI after long period of time.Ritfled waves perpendicular to the direction of growth develop under cyclic boundary conditions.
,t)=o p(i,+) o o o o o p(i,t)=l p(i,t+l)1

FIGURE 8
FIGURE 8 Pattern ot" riffled stripes (perpendicular to the direction of growth) on the cone shell principles (with the kind permission of Jerry G. Walls [7a]).See Color Plate II.

FIGURE 9 "
FIGURE 9 "Phase waves" in the automaton in TableVIwith cyclic boundary conditions.They arise if one starts with hundreds of randomly distributed cells in the state Z(./, 0) ().

FIGURE 11
FIGURE 11 Development of the automaton in TableVII, causing stripes parallel to the direction of growth.

FIGURE 13
FIGURE13  The cone shell Conus textile from the Great Barrier Reef (with the kind permission of Jerry G. Walls [7c]).See Color Plate IV.
this leads to a transformation rule z=[DE]=[1/2]; b=O of the cellular automaton"In the examples mentioned above, the diffusion coefficient D was either D 1 / 4 or D 1 / 2 .But what happens if smaller diffusion coefficients are chosen?

FIGURE 18
FIGURE 18 Chaotic "string of pearls" generated by the automaton in TableXIV.

FIGURE 20
FIGURE 20 Pattern of the "inverse" automaton rule in Table XVII.This pattern resembles the pattern of the spatio-temporal intermittency discussed by Miguel [9].

FIGURE 21 "
FIGURE 21 "Turbulent" pattern created by the collision of solitary waves according to Table XVIII.

FIGURE 23 "
FIGURE 23 "String of pearls" generated by the collision of solitary waves according to Table XX.

FIGURE 24 A
FIGURE 24 A localized oscillating pattern of the cone-shell Glaucus, Solomon Is (with the kind permission of Jerry G. Walls [7d]).See Color Plate V.
terns are classically described by solitons moving in excitable systems.

TABLE IV E
This row represents a diffusion process FIGURE 6 Spatio-temporal development of the automaton Table VI starting with one cell.The time axis runs from top to bottom.aswell, 1.

TABLE XIV 0 2
FIGURE 17 Part of the spatio-temporal development of the automaton in TableXIIIwith a periodic "string of pearls" pattern.