Turing Patterns and Wavefronts for Reaction-Diffusion Systems in an Infinite Channel

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<ul><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>SIAM J. APPL. MATH. c 2010 Society for Industrial and Applied MathematicsVol. 70, No. 8, pp. 28222843</p><p>TURING PATTERNS AND WAVEFRONTS FORREACTION-DIFFUSION SYSTEMS IN AN INFINITE CHANNEL</p><p>CHAO-NIEN CHEN , SHIN-ICHIRO EI , AND YA-PING LIN</p><p>Dedicated to the memory of Chung-Wei Ha</p><p>Abstract. This paper deals with reaction-diusion systems on an innitely long strip in R2.Through a pitchfork bifurcation, spatially heterogeneous patterns exist in a neighborhood of Turinginstability. Motivated by the works of Kondo and Asai, we study wavefront solution heteroclinic toTuring patterns. It will be seen that the dynamics of a wavefront can be approximated by a fourthorder equation of buckling type.</p><p>Key words. Turing pattern, wavefront, reaction-diusion system</p><p>AMS subject classifications. 35J50, 35K55, 35K57, 37L65</p><p>DOI. 10.1137/090747348</p><p>1. Introduction. In 1995, an article by Kondo and Asai [17] showed that somechemical waves have been observed in the skin of angelsh. By using reaction-diusionsystems in simulations, their results seem to t into the formation of patterns verywell. A typical pattern structure is the rearrangement of the stripe patterns. Forexample, when the width of body varies in locations, the number of stripes thatappear on the sh skin become dierent. Moreover, patterns with dierent numbersof stripes seem to be joined with a heteroclinic-like solution and some defects appearin between. According to the observation on the growth of skin, the locations ofthe defects change. After the work [17], although there have been many simulationsrelated to the investigation of such phenomena (e.g., [1, 20, 30, 31]), there have beenno theoretical results yet.</p><p>Fig. 1.1. Turing patterns observed on angelsh [17].</p><p>Received by the editors January 21, 2009; accepted for publication (in revised form) June 6, 2010;published electronically September 16, 2010. This research was supported in part by the NationalScience Council, Taiwan, Republic of China, and Grants-in-Aid for Scientic Research from JSPS.</p><p>http://www.siam.org/journals/siap/70-8/74734.htmlDepartment of Mathematics, National Changhua University of Education, Changhua, 500,</p><p>Taiwan (macnchen@cc.ncue.edu.tw, ylin@cc.ncue.edu.tw).Faculty of Mathematics, Kyushu University, Motooka Nishi-ku, Fukuoka 819-0395, Japan</p><p>(ichiro@math.kyushu-u.ac.jp).</p><p>2822</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>TURING PATTERNS AND WAVEFRONTS 2823</p><p>To seek a theoretical framework to the above phenomena (see Figure 1.1), webegin with studying a mathematical problem on a cylindrical domain and constructinga solution heteroclinic to two stripe patterns; that is, consider a reaction-diusionsystem</p><p>(1.1) Wt = DW + F (W ), t &gt; 0, x = (x, y) ,</p><p>under homogeneous Neumann boundary conditions, where := (,) (0, l) R2, W = t(w1, . . . , wN ) RN , F is a smooth function on RN and D =diag(d1, . . . , dN ), a diagonal matrix with positive entries dj . We look for a solutionW (t,x) of (1.1) satisfying</p><p>(1.2) W (t,, y) = W(y).</p><p>A heteroclinic solution of this type is a standing or traveling front of (1.1). Of particu-lar interest is the situation where W+(y) and W(y) are two nonconstant stationarysolutions of</p><p>(1.3) Wt = DWyy + F (W ), t &gt; 0, y (0, l),</p><p>with the boundary conditions Wy = 0 at y = 0, l.</p><p>The aim of this paper is to start with the investigation of the problems (1.1)(1.2) in the neighborhood of Turing instability [33]. Through a pitchfork bifurcation,we obtain two nonconstant stationary solutions for (1.3), say W(y), which will bereferred to as Turing patterns. In addition, we study standing wave solutions of (1.1)joining with W(y) as x . It will be seen that the dynamics of such a wavefrontare essentially governed by</p><p>(1.4)R</p><p>T= 1 </p><p>4R</p><p>4+R(M2 M1R2), T &gt; 0, R,</p><p>where all the coecients 1, M1, and M2 are positive numbers.In the past, a number of authors [2, 21, 22, 34] studied wavefront solutions in</p><p>cylindrical domains. If a solution of (1.1) has the form W (t, x, y) = U(z, y) withz = x ct, then U satises</p><p>(1.5) D(Uzz + Uyy) + cUz + F (U) = 0, U(, y) = W(y).</p><p>In case (1.1) is a scalar equation. Vega [34] considered the solutions of (1.5) satisfying</p><p>(1.6) W(y) &lt; U(z, y) &lt; W+(y) for all z R.</p><p>Under certain stability conditions onW+ andW, Vega proved existence and unique-ness results [34] for the solutions of (1.5). As a consequence of the maximum principle,such a wave is monotone in the z-direction. Interesting examples including the KPPequation, bistable reaction-diusion equation, and combustion model have been in-vestigated in [2, 22, 34].</p><p>The ow generated by a scalar reaction-diusion equation is order-preserving. Ina convex domain, it is known [5, 19] that a stable solution satisfying homogeneousNeumann boundary condition must be constant. This type of result has been extendedby Jimbo and Morita [14] and Lopes [18] to the case of minimizers in a gradient system.On the other hand, many interesting patterns [6, 8, 9, 10, 11, 15, 16, 24, 25, 26, 27,</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2824 CHAO-NIEN CHEN, SHIN-ICHIRO EI, AND YA-PING LIN</p><p>28, 29, 32, 37] have been found in reaction-diusion systems of activator-inhibitortype. Particular examples including FitzHughNagumo type equations [9, 23] and theGiererMainhardt system [11] have received a great deal of attention. Based on a skew-gradient structure proposed in [35, 36], a number of stability criteria [6, 35, 36] forthe stationary solutions of FitzHughNagumo type equations have been established.Although in the absence of order-preserving, calculus of variations provides a tool inanalyzing Turing patterns of skew-gradient systems. However, further work is neededto use this approach to study wavefront joining with two Turing patterns. In contrastwith the results [2, 23] obtained for the scalar reaction-diusion equation, numericalsimulation (see Figures 5.3 and 5.4) indicates that in a FitzHughNagumo type modela standing wavefront W (x, y) joining with W+(y) and W(y) may not be monotonein the x-direction.</p><p>This paper is organized as follows: In section 2, a bifurcation analysis under thecondition of Turing instability will be derived. Through a pitchfork bifurcation, weobtain two nonconstant stationary solutions W(y) of (1.3). Furthermore, on thecylindrical domain , by taking as the x-independent solutions of (1.1), the stabil-ity of W(y) will be studied in section 3. For a standing wave of (1.1) joining withW+(y) and W(y), an approximate solution will be constructed in section 4. Forthe FitzHughNagumo type model, it is convinced by numerical study that the ap-proximate solution ts in quite well. In certain situations, according to numericalsimulation, defect can be found out between Turing patterns with a dierent numberof stripes. Moreover, the variational method provides an additional tool for studyingwavefront in the FitzHughNagumo type model. Detailed analysis will be given insection 5. Lastly, in section 6 we end up with some discussions on future works.</p><p>2. Bifurcation analysis of Turing patterns. In this section the existenceof nonconstant stationary solutions of (1.3) in a neighborhood of Turing instabilitywill be investigated. Let 0 be a linearly stable equilibrium of Wt = F (W ); that is,F (0) = 0 and all the eigenvalues of the linearized matrix B := F (0) are of negativereal parts. The function F treated here is in a general setting. To nd out non-constant stationary solutions, we use bifurcation analysis. We introduce a parameter and assume a bifurcation occurs at = 0. Then the problem near the bifurcationpoint = 0 is</p><p>(2.1) Wt = DWyy + F (W ) + G(W ), t &gt; 0, 0 &lt; y &lt; l.</p><p>Here the function G satises G(0) = 0. Let L be the linearized operator with respectto the zero solution of</p><p>(2.2) DWyy + F (W ) = 0, y (0, l).Substituting the Fourier series</p><p>n=0 cosCnyan (an RN ) for the function W into</p><p>the eigenvalue problem LW = W gives</p><p>{C2nD +B}an = an,where Cn :=</p><p>nl . This leads to the consideration of matrix () := D+B parame-</p><p>terized by 0. Let j() (j = 1, . . . , N) be the eigenvalues of (). We assume thatthere exist positive numbers 0 and 0 such that Re(j()) &lt; 0 for j = 2, . . . , Nand 1() 0 for any 0, and 1() = 0 if and only if = 0 (Figure 2.1). It isalso assumed that 1() is a simple eigenvalue of () with eigenvector () RNfor all 0.</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>TURING PATTERNS AND WAVEFRONTS 2825</p><p>Fig. 2.1. Eigenvalue 1() and others.</p><p>Let n(y) := cosCnyan. If the width l of satises(l</p><p>)2= 0 and (0) = a1,</p><p>then L1 = 0. In this situation, if unstable solutions in the neighborhood of Turinginstability exist, they must be 1-mod solutions and close to a multiple of 1(y).</p><p>To obtain the existence of nonconstant stationary solutions of (2.1), we studythe ow on an invariant manifold: Let 0 := (0) and </p><p>0 be the vector satisfying</p><p>t(0)0 = 0 and 0,0 = 1. Dene</p><p>M1 := { F (0)b0 0,0 +</p><p>1</p><p>8 F (0)30,0 +</p><p>1</p><p>2 F (0)b2 0,0 </p><p>}and</p><p>M2 := G(0)0,0 ,where bn := 14 (C2nD+B)1F (0)20 (n = 0, 2, 3, . . .). Applying the center manifoldtheory (e.g., [4, 13]) yields the following properties for the solution W (t, y) of (2.1)on the invariant manifold.</p><p>Proposition 2.1. If is suciently small, there exists a function = (r; )(y)with = O(|| + r2), (0; ) = 0, and (r; )(y) = r22 +O(|| + r3) such that</p><p>W (t, y) = r(t)1(y) + (r(t); )(y)</p><p>with r(t) being a solution of</p><p>(2.3) r = H(r; ) := M1r3 +M2r +O(||2 + r4),where 2 := b0 + cosC2yb2.</p><p>The proof of Proposition 2.1 is standard. We omit it.Remark 1. The matrix (C21D+B) is not invertible due to the assumptions of</p><p>() and C21 = 0.Since 1(0) = 0, the eigenvalue 1() of the matrix () has the following ex-</p><p>pansion:</p><p>(2.4) 1(0 + ) = 12 +O(3).If 1 &gt; 0, the stationary solutions of (2.1) in a small neighborhood of Turing instabilityare stable (Figure 2.1). The constant 1 can be determined through the followingderivation.</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2826 CHAO-NIEN CHEN, SHIN-ICHIRO EI, AND YA-PING LIN</p><p>Substituting (0 + ) = 0 + 1 + and (2.4) into the eigenvalue problem()() = 1()(), we see from the coecients of the terms with order of </p><p>1 that</p><p>(2.5) (0D +B)1 D0 = 0.</p><p>Taking the inner product with 0 in (2.5) yields D0,0 = 0.Next, look at the coecients for the terms of order 2:</p><p>(0D +B)2 D1 = 10,</p><p>from which we know</p><p>(2.6) D1,0 = 1,</p><p>by taking the inner product with 0. To determine 1, we calculate 1 from (2.5)rst. Since the matrix (0D+B) has 0 eigenvalue with the eigenvector 0, it followsthat</p><p>1 = e1 + a0,</p><p>where e1 is a uniquely determined vector obtained by solving (0D+B)e1D0 = 0and using e1,0 = 0. We remark that the matrix (0D +B) is invertible in thesubspace {v RN ; v,0 = 0}. Then we see from (2.6) that</p><p>1 = D1,0 = D(e1 + a0),0 = De1,0 + a D0,0 = De1,0 ,</p><p>by making use of D0,0 = 0.In summary, we have the following lemma.Lemma 2.2. If (0 + ) = 0+ 1+ and 1(0+ ) = 12+O(3), then</p><p> D0,0 = 0 and 1 = De1,0 .If both M1 and M2 are positive, a supercritical pitchfork bifurcation diagram</p><p>occurs. This case holds for the FitzHughNagumo equations, as will be shown insection 5. If M1 is negative, the bifurcation diagram is subcritical, which can occur in(5.1) if is negative and the nonlinearity is dierent.</p><p>In what follows, we consider the case where both M1 and M2 are positive. Then</p><p>if &gt; 0 and suciently small, r := </p><p>M2M1</p><p>+O(||) are the stable equilibria of theODE r = H(r; ) as stated in Proposition 2.1. Furthermore,</p><p>W(y) := r1(y) + (r; )(y) = </p><p>M2</p><p>M1cosC1y0 +O(||)</p><p>are the stable stationary solutions of (2.1).</p><p>3. Stability of Turing patterns as x-independent solutions in . In thissection, the stability of W+(y) and W(y) on the cylinder will be investigated.Since W+(y) and W(y) are x-independent solutions of</p><p>(3.1) Wt = DW + F (W ) + G(W ), t &gt; 0, x = (x, y) ,</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>TURING PATTERNS AND WAVEFRONTS 2827</p><p>they will be referred as planar stationary solutions.Let () be the eigenvector satisfying t()() = 1()() and</p><p> (),() = 1. Recall that 0 = (0). It will be seen that the stability ofW(y) depends on the sign of 0 dened by</p><p>0 :=1</p><p>8 F (0)0 (v1 b2),0 +</p><p>1</p><p>8 F (0)30,0 ,</p><p>where vn := ((C2n + 0)D +B)1F (0)20 (n = 1, 2, . . .).Theorem 3.1. If 0 &lt; 0 (&gt; 0), then W</p><p>(y) are stable (respectively, unstable)planar stationary solutions of (3.1).</p><p>Proof. Since the solution under consideration is of order O(), setting :=</p><p>and W = Z in (3.1), we arrive at</p><p>(3.2) Zt = DZ + F (Z) + 2G(Z), t &gt; 0, x = (x, y) ,</p><p>where F = F (Z; ) := F (Z)/ and G = G(Z; ) := G(Z)/. Notice that</p><p>F (Z; ) = F (0)Z +1</p><p>2F (0)Z2 + 2</p><p>1</p><p>6F (0)Z3 + .</p><p>A similar expansion holds for G(Z; ).Let Z(y) := 1W</p><p>(y). Consider the linearized operator L+ dened by</p><p>(3.3) L+Z = DZ + F (Z+(y))Z + 2G(Z+(y))Z.</p><p>We remark that Z+(y) = R+ cosC1y0+R2+2+O(</p><p>2), where 2 := b0+cosC2yb2</p><p>and R+ :=1r+ =</p><p>M2M1</p><p>+O(), as noted in section 2. From (3.3), we obtain</p><p>L+Z = DZ +BZ + R+ cosC1yF(0)0 Z</p><p>+ 2{R2+F</p><p>(0)2 Z + 12R2+ cos</p><p>2 C1yF(0)20 Z +G(0)Z</p><p>}+O(3)Z</p><p>=: L2Z +O(3)Z.</p><p>Taking Fourier transformation of this equation with respect to the x-variable, weobtain</p><p>L+Z = L+Z = 2DZ + LyZ +O(3)Z,where is the Fourier transformation variable and</p><p>LyU := DUyy +BU + R+ cosC1yF(0)0 U</p><p>+ 2{R2+F</p><p>(0)2 U + 12R2+ cos</p><p>2 C1yF(0)20 U +G(0)U</p><p>}.</p><p>Now, the small eigenvalues of L+ will be under investigation. Since L+U = ()U +</p><p>DUyy + O()U , we denote L+ by L(, 2). Note that there are only two situations</p><p>that L(, 2) could have eigenvalues close to 0, namely, 2 0 or 2 0.Let us rst look at the case in the neighborhood of 2 = 0; that is, 2 = </p><p>1. Let L0()U := ()U + DUyy and 0() be the eigenvalue of L0(). SinceC21 = 0, it follows that L0()(cosC1y( + 0)) = cosC1y( + 0)( + 0) =</p><p>Dow</p><p>nloa</p><p>ded </p><p>12/3</p><p>1/12</p><p> to 1</p><p>28.1</p><p>48.2</p><p>52.3</p><p>5. R</p><p>edist</p><p>ribut</p><p>ion </p><p>subje</p><p>ct to </p><p>SIAM </p><p>licen</p><p>se or </p><p>copy</p><p>right;</p><p> see h</p><p>ttp://w</p><p>ww.si</p><p>am.or</p><p>g/jou</p><p>rnals/</p><p>ojsa.p</p><p>hp</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2828 CHAO-NIEN CHEN, SHIN-ICHIRO EI, AND YA-PING LIN</p><p>1(+ 0) cosC1y(+ 0), from which we know 0() = 1(0+ ), where 1(0 + )is the rst eigenvalue of (0 + ) as stated in sectio...</p></li></ul>