Pattern formation for a reaction-diffusion system: Catalytic-dependent diffusion coefficients

Pattern formation for a reaction-diffusion system: Catalytic-dependent diffusion coefficients García Cortí , J . L. Fernández, 1. Pacheco, J .M . Resumen Se considera un modelo de reacción-difusión para dos reactantes en presencia de un tercero, que actúa de cata lizador. La esca la temporal para el catalizador se compara con la de los reactantes y los coeficientes de difusión dependen sol ame nte de la conce ntra ción en el estado de equilibrio del cata li zador. Se realizan experimentos para diferentes cinéticas. Abstract A reaction-diffusion mode l is considered for two rea cta nts in the presence of a third chemical, acting as a cata lysator. The time sca le for the cata l ysator is fast compa red with that of the reactants and thus the diffusion coefficie nt s depend only on the steady-state concentr ation of the cata lysator. Expe riments are carr ied over for different kineti cs . AMS Classifications: 35825 , 35832, 35K57 Key Words: Patt ern formation, reaction-diffusion, cata lysis. 1 INTRODUCTION The pr esent paper is a developm ent of [2] for the case of a two-dimensional system . AIso, it represe nt s an evolution of [1] in cl uding severa l kinetics. Many processes in vo l ve interaction between two species -either chemica l, biologi ca l, eve ll social onesin the presence of a third one act ing as enhancer or 309 catalysator. This last one does not react with the other two; therefore the model systcm in dilllensionless form can be written as:[3J oX I ,fl(X I, X2) + div(D¡gradX¡) ( 1 ) al aX 2 ,12(X I, X2} + div(D2g radX 2) (2) al D Xl ,!:3(X 3) + div(D 3 grad}(d (3) Dt = plus initial ano bOllndary conditions . In most cases, dlle to symmetry conditions (e .g. a we lls tirr ed cylinorical reactor. .. ) the spatial extension of the problem is one-dimensional, so the diffusive terms reduce to ~(D.aXi) ox 10X Moreover, the catalysator llsna ll y has a decaying dynamic behaviour and diffuses at a co nstant rat e, while the oiffusion coeficients D¡ and D2 depend on the e nhancer collcentratioll. Thus the model reads: oX¡ . a áx¡ (4) al ,fl(X ¡ "\2) + Ox(D ¡ (X 3) OX ) oX 2 _ o oX 2 (5) ol ,12 (.\ ¡ , X 2) + ox (D 2( x 3) ox ) oX 3 . 02X3 (6) al -d'\3 + k- - ax 2 The spatio-tempora l oomain is [O , 1J x [0 ,(0 ) and the boundary conditions are zero-flux ones for X¡ and X2 at both ends of the i nterva l, while X3 is subject to the zero-flux conditioll at one end, say x = O, while it is kept constant at the other, i.e. X3 (1,l) = c. As a r ul e, X3 has a fast time scale, so for practical purposes a steady state for X3 can be considered. Putting 0~3 = O and solving the r esulting boundary problem for an ordinary differential equatio n it is found (see figure 1 below) that (7) 310 and in this way the model reduces to: ax¡ at aX 2 at a ax¡ = "J¡(X¡,X 2)+ ax(D¡(X 3 (x)) ax) a aX2 = "h(X¡,X 2 )+ ax(D 2 (X 3 (X)) ax) (8) (9) In practice D2 is taken as sorne scalar multiple mD¡ of D¡, and here it will be imposed that D¡(X 3) be an affine function (see figure 2 below) of X3, i.e.: D¡(X3) = oX 3 + (3 D2 = mD¡ (o < 0,(3 > O,m > O) Care must be taken in order that o and (3 define a straight line segment such that D¡ > O. In any case, the diffusion coefficients are written as functions of x alone: Ch(xlf) D¡ = D¡(X3(X)) = oX 3 (x) + (3 = oc Id + (3 = D¡(x) Ch( V f) .., Figure 1 Figure 2 The case of a single autocatalytic reactant has been dealt with hy the authors elsewhere [2] , and other conditions, as time dependen ce of the diffusion coefficients can be also considered. See e.g.[4]' though they are not studied in this papero 2 LINEAR SINGULAR POINT ANALYSIS Let !i( X ¡, X 2) satisfy conditions for the diffusion-free system to have a singular 311 point in the first orthant, say (X lO ,X 20 ), and let be the jacobian matrix of the J; at the singular point. In order to discover possible pattern formation [5], solutions to the model are looked for in the form : Xl = X lO + eAtYl(X) X2 = X20 + eAtY2(X) Putting this into the equations and writing Di(X 3) in the aboye defined form, the following system of ordinary differential equations is obtained: (aX3 + (J)Y {' + aX~Y{ + (all -A)Yl + al2Y2 = O m( aX 3 + (J)y;' + maX~Yl + a2lYl + (an -A)Y2 = O This system can be reduced to a single equation in a new variable Yl + AY2 through a judicious choice of Ao The first equation plus A times the second one yield: m . ( aX 3 + (J)(Y{' + AYl') + aXHx)(Y{ + AYl)+ +((a - A) + fW..A )(Y + aI2+(a22-'\)~ Y) = O 11 m 1 al1 -A+~ 2 (10) h o h h o o A al2 + (a22 - A) ~ Th o o d o o h so t e ng t e Olce IS = A o IS IS a qua ratlc equatlOn w ose all -A + a;:. roots are Al,A2o By writing Zj = Yl + AjY 2, two equations are obtained: . a A- (aX3 + (J)Zj' + aX~(x)Zj + ((all - A) + ~ )Zj = O (j = 1 ,2) (11) m The so lutiori s to this equation have the general form: where Zj(x) = cljex (H¡-H2)) + C2je x (H¡+H 2 )) H -- exc JISh(x/f) 1 - 2(¡1 Ch ( /f)+ excCh (x/f) -4 ¡1p)Ch2( /f)-4 ex p) cCh( /f)Ch(x/f)+ex 2c2 ~Sh2(x/f) 2(¡1Ch( ~)+excCh(x/f) 312 with Pj = (all _ A) + a2¡A, m . The zero-flux condition at x = O implies after sorne alg ebra that c¡j = C2j, so the following system is obtained : Z¡(x) = Y¡ + A¡Y 2 = c¡¡(ex(H¡-H2¡) + eX(Hl+H2¡)) Z2(X) = Y¡ + A2Y2 = c12(e x(HI-H22) + eX(Hl+Hn)) and solving for Y¡ and Y2 yield s: Y¡ = 2A:I~~1 (c¡¡A 2 Ch(x H2 ¡) -c12AICh(xH22)) Y2 = 2A:I~~1 (C12Ch(xH22) - cllCh(xH 2 ¡)) Remembering that e..\t yj(x) = Xj -X jO at (0,0), Y¡(O) = X¡(O, O) -X lO and Y2(0) = X2(0, O) -X 20 hold, so the constants C¡¡ and C12 are: Y¡ (O) + Al Y2(0) YI (O) + A2Y2(0) c¡¡ = 2 C12 = 2 3 MORPHOGENESIS A more detailed ana lysis of the Zj equation, ( (3) 11 '( ) ' (( ) a 2l Aj) QX 3 + Zj + QX 3 X Zj + a¡¡ -A + -- Zj = O m is needed to account for pattern formation. The characterist ic equation can be written as 2 QX~ (a¡¡ _ A) + a21A, r + r+ m - O QX3 + (3 QX3 + (3 -(12) QX' Because of the form of X3 and QX 3 + (3, it is always true that X 3 _ < O. A Q 3+ little algeb ra shows that 1 QX' H¡ = -2QX3~(3 > O Therefore, if there are complex roots, their real parts are positive. For complex roots to appear, 313 ( aX~ )2 _ 4 (an - A) +~ < O aX 3 + f3 aX 3 + ¡3 (13) must hold, and this condition is readily seen to be equivalent to H2j being imaginary, i.e. H 2j = if(2j for sorne real f(2j . Putting aH this together , the final expressions for Y¡ and Y2 are found: Y¡ = A~z::~1 ((Y¡(O) + A¡ Y2(0))A2cos(xf(2¡) -(Y¡(O) + A2Y2(0))A¡cos(xf(22)) Y2 = A~z::~1 ((Y¡(O) + A2Y2 (0»)cos(xf(n) - (Y¡(O) + A¡Y 2(0))cOS(xf(2¡)) showing the osciHatory nature of Yj. Condition (13) can be expressed as Fj(x,A) < O, where (14 ) Figures 3 and 4 show graphs for Z¡ = F¡(x,A) and Z2 = F2 (x,A) respectively. Figure 3 Figure 4 In both figures the parameter fi:. = 1. For any fixed x, the various A indicate V"k different temporal evolutionary behaviours. 314 4 NUMERICAL EXPERIMENTS Exp erillH'lIts arp ca rriNI on for two versions of system (4)(5)(6) with diffe rent rPilctio ll !. e nll s. Thp first 0111' c orre s ponds to the s o-called Schnackenberg kinetics: JI = P - Xl + X? );2 (15 ) h q-X¡X 2 (16) Pa.rilme !.Pr values , nI' (\' = -1, ¡3 = :20 , I = 1000, e = 15, m = 10 , p = 0.1. q = 0.9 alld fi = I for figures 5a a Ild 5b, while fi = :2 for figures 6a and 6b. Vf Vf 1'h e sec oIld OIle is t he act i vator-i nhi bitor s ystem wher e JI v2 ·~l = p-qX I+ X2 (1+BXi) h xf - Xl (1 i ) (18) Param ete r value s are el = -1, ¡3 = :20. I = 1000, e = 15, m = 12, P = 0.1, q = 1, (d (d B = 0. 05. with V f = 0.1 for figures ia an d ib, while V f = 2 for figures 8a and t3 b. Figure 5a Figure 5b 315 Figure 6a Figure 6b Figure 7a Figure 7b Figure 8a Figure 8b 316 These experiments show that the square root !"i of the quotient between the decay rate of the cata l ysator and its diffusivity coefficie nt acts as a bifurcation parameter, great ly cha nging the final shape of the distributions for Xl and X2. 5 REFERENCES [1] Ga rcía Cortí , J.L ., Pacheco, J.M. (1994) : Morphogenesis in a reactiondiffusion system with catalytic - dependent diffusion coefficients,International Confer'ence on Nonlinear Dynamics and Pattern Formation in the Natu1'a1 Environme nt , Noordwijkerhout, the Netherlands. [2 ] García-Co rtí , J .L., Pacheco, J .M . (1994) : Sobre una ecuación de reaccióndifusión,Revista de la Acad. Canaria de Ciencias (in the pressj. [3] Murray, J. (1988) : Mathematical Biology, Springer Verlag, Berlin. [4] Padilla, 1. , Fernández, 1., Pacheco, J.M. (1993) : Estabilidad de sistemas de ecuaciones de reacción-difusión con difusividades variables,Actas II Congreso de Métodos Numéricos en Ingeniería, Vol. 2, 892-899, La Coruña. [5] Turing, A .M . (1952) : The chemical basis of morphogenesis, Phil. Trans. R. Soco Lond., B2 37, 37-72, Londres. Authors: D epto. de Matemáticas Universidad de Las Palmas de G.C. Campus de Tafira Baja E35017 Las Palm as, Spain 317