chaotic behavior in coupled gierer–meinhardt equations

E-mail address: coopercd@bu!alostate.edu, c}cooper}@hot-mail.com (C. Cooper).

Computers & Graphics 25 (2001) 159}170

Chaos and Graphics

Chaotic behavior in coupled Gierer}Meinhardt equations

Crystal Cooper

Department of Physics, Buwalo State College, 1300, Elmwood Avenue, Buwalo, NY 14222-1095, USA

Abstract

In this paper, a computer program was modi"ed to simulate the dynamics of four modi"ed Gierer}Meinhardtreaction}di!usion equations. A time series analysis was performed on the results. It was found that the modi"edGierer}Meinhardt equations demonstrated chaotic behavior under certain conditions. The dynamics included "xedpoints, limit cycles, transient chaos, intermittent chaos, and strange attractors. The creation and destruction of fractaltorii was found. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Pattern formation; Self-organizing systems; Reaction-di!usion systems; Chemical waves; Gierer}Meinhardt equations

1. Introduction

Alan Turing, a noted mathematician and a pioneer inthe design of computers, made an important break-through in the "eld of pattern formation in biology in1952. He postulated that spatial concentration patternscan be formed from an initially homogeneous system iftwo or more chemicals with di!erent di!usion rates reactwith each other [1]. He proposed a system with a self-organizing chemical concentration gradient that pro-vided cells with the `knowledgea of their position. As onemodel, he chose two or more coupled non-linear equa-tions which he linearized to the following:

dX/dt"aXr#b>

r#t (X

r`1!2X

r#X

r~1), (1)

d>/dt"cXr#d>

r#l(>

r`1!2>

r#>

r~1), (2)

where a, b, c, and d represent rate constants, k and lare di!usivities, X and > represent displacements for asteady-state concentration for the chemicals, and r rep-resents the iterations. These are reaction}di!usion equa-tions of the following form:

LC/Lt!fsource

"D+2C. (3)

The "rst term on the left shows how the concentrationC changes at a "xed location, and the next term displaysthe chemical reactions that change the concentration.These are all equal to the term on the right. The term onthe right involves the Laplacian, which is representedin "nite di!erence form in Eqs. (1) and (2) above. InCartesian coordinates in three dimensions the Laplaciangives

+2"L2/Lx2#L2/Ly2#L2/Lz2. (4)

The Laplacian has an `averaginga e!ect. This means thatthe concentration is spread or di!used so that e!ectively+2C is a transport term. The combination of the right-and left-hand sides of Eq. (3) gives the Lagrangian, whichrepresents the time dependence seen by an observermoving with a volume element of the #uid [2]. Thiscombination also gives rise to a con#ict. The di!usion orLaplacian term on the right, tends to smooth out concen-tration di!erences in contradistinction to the time depen-dent term on the left, which produces concentrationdi!erences. It is this con#ict between the two sides thatleads to pattern formation.

Turing formulated these reaction}di!usion equationsto theorize the following process in pattern formation.Under proper conditions, i.e. with chemicals that havedi!erent di!usion rates, waves of chemical concentra-tions will form in chemical systems that are initiallyhomogeneous. A wave with one of the rate constants at

0097-8493/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 1 1 5 - 1

the highest positive value will dominate the system tosuch an extent that its wavelength will become the chem-ical wavelength of the emerging pattern. In other words,the solution to the equations with the proper parametersform periodic distributions that are stable in time, knownas standing waves. It is Turing who proposed calling theparameters X and > morphogens. According to his the-ory, development then is a two step process: a localizedhigh concentration of the morphogen is formed, and thenthe morphogen determines cell di!erentiation and devel-opment. The morphogen is therefore functioning asa pre-pattern builder. The chemical systems thereforehave two basic requirements before the patterns form.The "rst requirement is that of autocatalysis, or strongpositive feedback. Since Turing viewed pattern forma-tions as small deviations from an initially homogeneoussystem, his autocatalysis requirement merely stated thatwhen the small deviations with strong positive self-feedback increase the deviations, then patterns areestablished. The second requirement is that of negativefeedback or inhibition, since the deviations could not goon to in"nity.

The Gierer}Meinhardt model is a Turing type systemwith two morphogens or chemicals called activators andinhibitors. In one dimension it is as follows:

LA/Lt"p0p#cpA2/H!uA#D

aL2A/Lx2, (5)

LH/Lt"c@p@A2!vH#DhL2H/Lx2. (6)

Unlike Turing's original equations, these involve non-linear terms. P and p@ are the source concentrations ofA and H, respectively. A is the activator concentration,which they proposed as a large, slowly di!using moleculethat activated activator and inhibitor synthesis. H is theinhibitor concentration, which they proposed as a small,fast-di!using molecule that inhibits activator and inhibi-tor degradation. The term p

0is a constant and is a basic

production activator. The symbol u represents the de-struction or removal of activator A, and v represents thedestruction or removal of inhibitor H. C and c@ areconstants involving activator and inhibitor production,respectively. D

aand D

hare the di!usion constants of the

activator and inhibitor.Reaction}di!usion cells are standard models for trans-

formation and transport studies for not just cells ortissues, but for physiological systems, ecological models,and all forms of chemical, biochemical, and biologicalreactants [3]. Reaction}di!usion models are dissipative.Dissipative systems, as a consequence of the LiouvilleTheorem, have motion, where the phase-space volumecontracts asymptotically onto a lower dimension. Cha-otic behavior in dissipative systems in the form of strangeattractors are well known; some famous examples are theRoK ssler and Henon maps [4].

Most of the studies of non-linear analysis of reac-tion}di!usion equations involve chemical reactions. In

1982, Schreiber and Marek published a paper where theystudied the development of strange attractors in coupledreaction}di!usion cells [5]. The model they used wasa numerical simulation of a well-known kinetic schemecalled the Brusselator, a trimolecular system, which isoften used in chemical analysis. By varying a parameter,they were able to "nd periodic and aperiodic (chaotic)behavior. Their time-series analysis involved the stan-dard tests of Lyapunov exponents, power spectra,and Poincare surfaces of section. Lyapunov exponentsmeasure the exponential divergence of trajectories.A positive one is a sign of chaos. A power spectrum, alsocalled the spectral density, is found by plotting the squareof the Fourier transform against the frequency. Chaoticspectra demonstrate no preferred frequency. A Poincaresurface of section is a cross-section of a phase-space plot.In the summer of that same year, Marek and Schreiberpublished another paper, where they speci"cally exam-ined the transition to chaos via a two torus system incoupled reaction}di!usion cells [6]. Again, they useda Brusselator model. They discussed how the Brusselatorexhibited all types of chaotic behavior, including Feigen-baum period doubling bifurcations, intermittent chaos,and metastable chaos. Intermittent chaos is characterizedby periodic or quasiperiodic behavior interspersed withchaotic `burstsa. Metastable chaos is transient chaoticbehavior that settles down to a steady state, such asa limit cycle. See Yorke and Yorke for a study of meta-stable chaos in the Lorenz system [7].

In 1987, Barkley et al. published a paper on anotherchaotic transition via a torus [8], where they useda chemical reaction called Belousov}Zhabotinskii astheir model. The Belousov}Zhabotinskii reaction in-volved the cerium-catalyzed bromination and oxidationof malonic acid by a sulfuric acid of bromate. Barkleyet al. observed how transitions occur between a smoothtorus and a wrinkled torus via a secondary Hopf bifurca-tion, and how the wrinkled tori exhibited twisting andfolding characteristic of chaotic behavior. The connec-tion between distorted tori and chaos was also studied bySteinmetz and Larter [9], who made their observationsby visually examining phase space. They listed fourstages of a torus in the transition from quasi periodicityto chaos: an undistorted torus, a wrinkled torus, a fractaltorus, and a broken torus. An undistorted torus was theusual elliptical shape. A wrinkled torus was distorted, butwas quasiperiodic. A fractal torus appeared similar to thewrinkled one, not only in the phase space but in thepower spectrum, but was chaotic and not quasi periodic.A broken torus was not continuous.

A study of a reaction}di!usion model using theGierer}Meinhardt equations was done by Tsonis et al.[10]. Two of the equations were examined for chaoticbehavior, with a numerical simulation involving a linearone-dimensional array of cells. Two models were used:regular, and forced. The regular model started with

160 C. Cooper / Computers & Graphics 25 (2001) 159}170

Fig. 1. Bifurcation diagram of u as a function of v with y3"0.

20 cells and used a phase or state space of inhibitor vs.activator concentrations. These concentrations are foundwith the usual methods as demonstrated in earlier chap-ters. Again, the standard techniques of power series andPoincare sections were used to test for chaos. The unfor-ced model demonstrated periodicity. The forced model,which was simulated nearly the same way save for anarray of 100 cells and the addition of a forcing term to theGierer}Meinhardt equations, showed chaotic behaviorthat was dependent upon the parameters of the forcingterm. The transition to chaos via a Hopf bifurcation wasobserved. A Hopf bifurcation is one of the most observedbifurcations in non-linear dynamics. It is de"ned as atransition from a "xed point to a limit cycle via a changein a control parameter [11].

2. The research model

This theoretical research uses four Gierer}Meinhardtequations with a Conway}Cooper modi"cation, ar-ranged in a circle of 18 units, with each unit possessingfour reaction}di!usion concentrations. The modi"cationwas a link added that represented the di!erence betweenthe "rst and second equations. The link was added to thethird equation, and had the e!ect of modifying the orig-inal Gierer}Meinhardt equation (5) to that of the form:

Lc/Lt"p0p#c

0pc2/d!uc#ky3(a!b#-oor)

#dcL2c/LX2, (7)

where k times y3 is a constant that involves the size ofthe radius and changes the strength of the link betweenthe equations, and the constant yoor controls the size ofthe di!erence.

The two sets of the equations therefore used were

La/Lt"p0p#c

0pa2/b!ua#d

aL2a/Lx2, (8)

Lb/Lt"c@p@a2!vb#dbL2b/Lx2, (9)

Lc/Lt"p0p#c

0pc2/d!uc#ky3(a!b#-oor)

#dcL2c/Lx2, (10)

Ld/Lt"c@p@c2!vd#ddL2d/Lx2. (11)

The sampling depended on the amplitude of the oscil-lations of the concentrations. The di!usion constantsu and v were varied to alter the results. Restrictionslimiting the linking term in the third equation to positivenumbers were used.

The following software was used as an aid in the timeseries analysis: Chaos Data Analyzer, by Julien C. Sprott;Santis, by R. Vandenhouten, M. Rasche, H. Tegtmeier,and G. Goebbels; DaDiSP/SE, by Dsp DevelopmentCorporation; and Microcal Origin.

3. Operation

The parameters u and v were chosen. The programwas run using the initial gradients with y3 set to 0, anda result was recorded. The program was run again withthe gradient command, but this time y3 was givena value, usually 0.003 (a weak link) or 0.009 (a stronglink). Results were again recorded.

The transition to chaos in the present research modelproceeded as follows: "xed points, transient or meta-stable quasiperiodic or limit cycle behavior; and theneither intermittent chaos, or a fractal torus, broken torusor chaos. The bifurcation is clearly of Hopf type. Thechaos was in the form of strange attractors.

4. Results

Fig. 1 shows the results obtained by varying u, v, andy3. F

pmeans "xed point, t

crepresents transient chaos,

q means quasiperiodic, and c means chaos.The parameters u and v proved to be essential elements

in the determination of chaos. Generally, if v is greaterthan u, "xed points are formed. Exceptions are found athigh levels of u (u'0.05), where values close to a limitcycle yielded fractal or broken tori. If v equaled u, thenmetastable or transient periodic or quasiperiodic cyclesare formed. The larger the values of u and v, the greaterthe radius of the limit cycles, until the transients disap-pear and chaos immediately forms. This chaos is similarto that found in chemical equations, where the motion isaperiodic in that the amplitudes of the oscillations areerratic. This corresponds to chaotic motion on a strangeattractor [12]. If v is less than u, some form of chaosalways resulted, from fractal tori to strange attractors.As u and v increased, so did the amount of chaos. Forexample, the Lyapunov exponents of the trajectoriesgrew larger. The Lyapunov exponent at u"0.2 v"0.15

C. Cooper / Computers & Graphics 25 (2001) 159}170 161

Fig. 2. A typical phase space for u much greater than v.

Fig. 3. A time spectrum reveals intermittent chaos.

Fig. 4. The power spectrum shows no preferred frequency.

Fig. 5. The Poincare section shows two chaotic attractors in theshape of an ¸.

had a value of 0.01. Also, at u"v and near a limit cycle,the exponents are smaller than for those farther away.

The general e!ect of y3 is to dampen the amplitude ofthe oscillations, usually by approximately 10% for 0.003,and approximately 30% for 0.009. However, it also dis-torted tori, yielding the creation of metastable chaos, andfractal or broken tori, particularly when v and u are equalto each other and greater than 0.01. It also had a tend-ency to increase the amount of quasiperiodicity.

5. Two examples

The examples below are taken with respect to reactioncells c1 and d1. This can be done because generally, allthe cells sharing the same parameters exhibited similarbifurcation behavior. The patterns in the unreconstruc-ted phase space for reaction cells with the same para-meters frequently di!ered, however, especially whena torus became distorted. Unless otherwise noted, thePoincare sections are taken with respect to the averagevalue of x or 0. The embedding dimension n and timedelay q are set at 4. Since the model has a minimum of 72dimensions, there are at least 72 Lyapunov exponents.The largest one is used to indicate chaos:

u"0.07 v"0.01.

Figs. 2 show typical behavior for u much greater thenv. The time spectrum in Fig. 3 shows intermittent chaos.

The Lyapunov exponent is 0.0006. The power spec-trum in Fig. 4 yields no preferred frequency. The Poin-care section (Fig. 5) shows two chaotic attractors in theshape of an ¸, with indistinguishable points "lling theouter edges. The CD is 0.569$0.033.

A change is introduced when y3 is increased to 0.003.The Lyapunov exponent is the same, but the phase spacehas changed (Fig. 6). The time spectrum, in Fig. 7, is


Fig. 6. An increase in y3 has distorted the phase space.

Fig. 7. An increase in y3 damps the amplitudes.

Fig. 8. The Poincare section has retained its shape despitea change in y3.

Fig. 9. The phase-space changes as y3 is increased.

Fig. 10. The time spectrum of c1 is damped even more as y3 isincreased.

damped. The points on the Poincare section, shown inFig. 8, have moved closer together. The CD changed to0.868$0.063.

When the value for y3 rises to 0.009, again, the phase-space changes to a more uniform shape that is clearlydamped (Fig. 9). The amplitudes for the time spectrumare damped even further (Fig. 10). The Poincare sectionhas retained its ¸ shape (Fig. 11). The Lyapunov expo-nent is decreased to 0.0004, signalling that the trajectoriesdiverge more slowly. The CD is 1$0.057.

u"v"0.07.

This example shows how a torus is deformed. The "rst"gure (Fig. 12) is a chaotic limit cycle at y3"0, witha power spectrum as shown (Fig. 13). Fig. 14 shows thetime spectra. The largest Lyapunov exponent is 0.0007.The CD is 1.368$0.089.

Fig. 15 shows how the torus changes as y3 increases to0.003. The area is decreased as the orbits move closer


Fig. 11. The poincare section of u"0.07 and v"0.01, withy3"0.009.

Fig. 12. A limit cycle for u"v"0.07, with y3"0.

Fig. 13. The power spectrum for u"v"0.07, with y3"0.

Fig. 14. The time spectrum of c1 for u"v"0.07, with y3"0.

Fig. 15. As y3 is increased to 0.003, the trajectories are no longercon"ned to a single-limit cycle.

Fig. 16. The power spectrum for u"v"0.07, with y3"0.003.

together. The high peak in the power spectrum has shif-ted to the middle (Fig. 16). The amplitudes are damped inthe time spectra (Fig. 17). The Lyapunov exponent is0.016. The CD is 1.585$0.173.


Fig. 17. The time spectrum for u"v"0.003, with y3"0.003.

Fig. 18. The phase space for u"v"0.07, y3"0.009 showsa strange attractor.

Fig. 19. The power spectrum for u"v"0.07, with y3"0.009.

Fig. 20. The time spectrum of c1, for u"v"0.07, y3"0.009.

Fig. 21. The Poincare section for u"v"0.07, with y3"0.

Fig. 22. The Poincare section for u"v"0.07, with y3"0.003.

Fig. 18 demonstrates how y3 at 0.009 gives rise tomany additional orbits in phase space. The powerspectrum has fewer frequencies (Fig. 19), and the timespectra are again damped (Fig. 20). The Lyapunov


Fig. 23. The Poincare section for u"v"0.07, with y3"0.009. Fig. 24. The trajectories approach the "xed point slowly.

Fig. 25. The trajectories approach the "xed point from di!erentdirections.

Fig. 26. A typical "xed point.

exponent has decreased to 0.002. The CD is loweredto 0.322$0.029.

The next three "gures show how the Poincare sectionschange for each successive y3. Here, the sections aretaken with respect to zero, to show the torus moreclearly. The chaotic attractors are at "rst separated alongthe axes, as seen in Fig. 21. For y3"0.003, the attractorsmerge and form a fractal torus as seen in Fig. 22. Finally,for y3"0.009, the torus breaks (Fig. 23).

6. More interesting points

This paper concludes with pictures of especially inter-esting patterns found in the course of the research.

6.1. Some typical xxed points

The "xed points encountered in the simulation areattractive, which is normal in dissipative systems, and isproven by the Liouville Theorem. The "rst picture,Fig. 24, is called a spiral node, where the trajectoriesspiral about a node on a surface. For higher values ofu"v, a limit cycle forms as this node is destroyed, whichwhy this process is a Hopf bifurcation. The next picture,Fig. 25, also shows up quite frequently in non-lineardynamics, usually for high values of u and v that are veryclose to a limit cycle. The trajectories approach the pointattractor along di!erent paths.

The next Fig. 26, shows a phase space, where theconcentration values are "rst very small; then they in-crease rapidly, and "nally fall to a permanent value. The"gure after this one, Fig. 27, shows trajectories that swirlaround in phase space before orbiting in tighter andtighter circles until the "xed point is reached. The trajec-tories move from the right to the left.


Fig. 27. The trajectories spiral into a "xed point on the left.

Fig. 28. The trajectories settle quickly to a "xed point.

Fig. 29. The attractors move slowly to the left before remainingin an elliptical area.

Fig. 30. The trajectories widen, #atten, then jump further right.

Fig. 31. The trajectories widen, #atten, then move inward.

Finally, typical behavior is shown for values of v muchgreater than u (Fig. 28). The di!usion is limited so muchthat the trajectories quickly reach a constant value of d,change slowly along c, and "nally remain constant.

6.1.1. Quasiperiodic

u"0.08 v"0.09 y3"0.009 LE"!0.0008.

The pattern formation in Fig. 29 proceeds by develop-ing smaller limit cycles before it jumps to the "rst ellipseon the right. It then moves over twice more before stop-ping. The second one, Fig. 30, creates narrow circles thatwiden and #atten, then jump to the right. The last "gure,Fig. 31, has cycles that move inward instead of outward.These patterns indicate the presence of more than oneattractor.


Fig. 32. Shown are two attractors that are limit cycles.

Fig. 33. Elliptical limit cycles.

Fig. 34. The attractors show a knit-like pattern.

Fig. 35. The trajectories #atten to the right and then widen toa limit cycle.

6.1.2. Transient chaosu"v"0.05 y3"0.003 Radius"0.009

LE"0.0098.

In Fig. 32, the trajectories move back and forth alongtwo limit cycles. Fig. 33 has trajectories that move up andback until an ellipse is formed; then other ellipses beginto form, moving inward:

u"0.06 v"0.07 y3"0.009 LE"0.0006.

Fig. 34 shows a strange, knit-like pattern. After transi-ents, decreasing limit cycles are formed. By the thou-sandth iteration, they begin con"ning themselves to thatshape, spreading rightward over time. The second frame,Fig. 35, shows the trajectories forming thin circles thatmove right and widen in time. Fig. 36 behaves in the same

fashion:

u"v"0.07 y3"0.009 LE"0.002.

After transients, the trajectories form the smaller limitcycle shown near the bottom (Fig. 37). Then they beginbouncing up and down, "lling in the until the bubble-likepattern is formed. The picture indicates the presence of atleast four attractors.

u"0.07 v"0.08 y3"0 LE"0.0098.

Fig. 38, shows trajectories that proceed around di!er-ent paths to a limit cycle and not a "xed point. This isshowing the formation of the limit cycle to be found atu"v"0.08.

u"v"0.2 y3"0.


Fig. 36. The attractors form widening limit cycles.

Fig. 37. The trajectories continually form the limit cycle, moveup, then approach the limit cycle again.

Fig. 38. Trajectories that converge to a limit cycle.

Fig. 39. A limit cycle that starts from the right and moves to theleft.

Fig. 40. A limit cycle that starts from the left and goes to theright.

In Fig. 39, the trajectories form limit cycles beginningat the right and moving to the left before stopping.Fig. 40 shows the opposite.

In the "rst frame below, Fig. 41, the trajectories againmove to the right before stopping and forming a limitcycle. The second frame (Fig. 42) plots reaction cells b1and a1 where, after a large transient around 1E12, thetrajectories alternate between low and high peaks.

7. Conclusion

This paper re-emphasizes the well-known idea thatchaotic dynamics may play a strong role in patternformation and other areas, where reaction}di!usion cellsare used. In biology, we know that parameter-dependenttransitions may result in clinically observed defects; these


Fig. 42. Trajectories that move up and down.

Fig. 41. A limit cycle that begins on the left and #attens as itmoves right.

have been called dynamical diseases [13]. This paper alsoshows that forced oscillations are not necessary in orderto observe chaotic behavior in Gierer}Meinhardtequations.

This paper also proves that the parameters u and v aresigni"cant in the transition from steady state to chaoticmotion. To recall, parameters u and v control the re-moval or destruction of the concentrations. Parameteru concerns the removal or destruction of reaction}di!u-sion cells c and a, while v concerns reaction}di!usioncells d and b. Dependent on how they are chosen, "xedpoints, limit cycles, and some forms of chaos are observedin the simulation. The chaos observed, which was alwaysin the form of strange attractors, included transient,metastable, intermittent, and aperiodic chaos. The para-meter y3 changes a pattern's shape and/or area in phasespace, by damping the amplitudes and/or changing thefrequencies of the oscillations. In addition, it has a tend-ency to alter the capacity or fractal dimension for theselected set of parameters.

Acknowledgements

The author wishes to thank Dr. Bruce Flanders andDr. Nikki Hatzilambrou for their critiques as the re-search was being developed.

References

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[4] Lichtenberg AJ, Lieberman MA. Regular and stochasticmotion. Berlin: Springer, 1983.

[5] Marek M, Schreiber I. Strange attractors in coupled reac-tion}di!usion cells. Physica 1982;5D:258}72.

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[7] Yorke J, Yorke E. Metastable chaos: the transition tosustained chaotic oscillations in a model of Lorenz.Journal of Statistical Physics 1979;21(3):263}77.

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[10] Tsonis A, Elsner J, Tsonis P. On the dynamics of a forcedreaction}di!usion model for biological pattern formation.Proceedings of the National Academy of Science USA1988;86:4938}42.

[11] Tomita K. Periodically forced nonlinear oscillators. In:Holden AV, editor. Chaos. Princeton, NJ: Princeton Uni-versity Press, 1986.

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[13] Rapp PE. Oscillations and chaos in cellularmetabolism and physiological systems. In: Holden AV,editor. Chaos, Princeton, NJ: Princeton University Press,1986.


chaotic behavior in coupled gierer–meinhardt equations

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