Journal of
J. Math. Biology7, 243-263 (1979)
Illalkmal
l
by Springer-Verlag 1979
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis Masayasu Mimura 1 and Yasumasa Nishiura 2 x Department of Applied Mathematics, Konan University, Kobe Department of Computer Sciences, Kyoto Sangyo University, Kyoto, Japan
Summary. A certain interaction-diffusion equation occurring in morphogenesis is considered. This equation is proposed by Gierer and Meinhardt, which is introduced by Child's gradient theory and Turing's idea about diffusion driven instability. It is shown that slightly asymmetric gradients in the tissue produce stable striking patterns depending on its asymmetry, starting from uniform distribution of morphogens. The tool is the perturbed bifurcation theory. Moreover, from a mathematical point of view, the global existence of steady state solutions with respect to some parameters is discussed. Key words: Bifurcation.
Morphogenesis--Spatial
patterns--Non-linear
diffusion--
1. Introduction
Since Turing's famous paper (1952) has been published, many interesting mathematical models, based on Turing's idea, have been proposed in various fields to explain spatial patterns. Most of the models are described by OU _ 02U -oi- = ~ T ~ x ~ + F ( U ) .
(1.D
The variable U represents quantities such as temperature, concentrations and densities. T h e most interesting problems for (1.1) are related to the analysis of the qualitative behavior. The main tool to study the formation of spatial patterns of (1.1) is the theory of bifurcation (see, e.g., Auchmuty and Nicolis (1975)). Gierer and Meinhardt (1972) proposed several models to interpret cell differentiation. One of their models takes the form Oa . O2a cp(x)a ~ - ~ = a,, ~ + p(X)po + h(1 + Ka2) Oh 02h O~ = dn ~ + c'p'(x)a 2 -
pa
(1.2) vh.
0303-6812/79/0007/0243/$04.20
244
M. Mimura and Y. Nishiura
Here a(t, x) and h(t, x) are the concentrations of a short range activator and a long range inhibitor at time t and position x, respectively, p(x) and o'(x) are the source density distributions of activator and inhibitor, which generally depend on position x. The difference between (1.1) and (1.2) is that the interaction terms are generally functions o f x and U. The reason is that Child's gradient theory is introduced into Turing's idea. The x-dependence of p(x) and p'(x) reflect the appearance of polarity in the tissue, p0, c, K,/z, c' and v are assumed to be positive constants, da and dh are the diffusion constants of activator and inhibitor, respectively (for the biological meaning, see Gierer and Meinhardt (1972)). These authors showed numerically that slightly asymmetric source density distributions produce stable striking patterns starting from uniform distributions of activator and inhibitor. These numerical experiments show the importance of the polarity. From an analytical point of view, Granero, Porati and Zanacca (1977) studied the stationary problem of (1.2) under zero flux boundary conditions when K = 0 and p(x) and p'(x) are both constants. They showed the explicit form of small amplitude steady state solutions by making use o f the Poincarr-Lindstedt method. The aim o f this paper is to discuss the effect of non-constant p(x) and p'(x) on spatial structures of steady state solutions of (1.2) and to interpret Gierer and Meinhardt's numerical experiments analytically. Moreover, we show some global structure of the set of the steady state solutions. The main mathematical tools used here are the perturbed bifurcation theory and the Leray-Schauder degree. In Section 2, we describe the model discussed in this paper. In Section 3, we show the uniform boundedness of a solution (a(t, x), h(t, x)) of the initial-boundary value problem for (1.2) under-zero flux boundary conditions and hence obtain global existence theorem. In Section 4, we discuss the existence of small amplitude steady state solutions by using the Lyapunov-Schmidt method and their stability. The solutions are of two different types, classified as ordinary type and snap-like type. In Section 5, we study a certain continuum which contains the solutions stated in Section 4. Finally we conclude with some comments on our results. We shall use the following notations throughout the paper: 1) R~ = (UtU ~ ~'~ and each component is positive}. 2) L2(I) = space of functions which are square summable on I = (0, l). We denote its norm and inner product by II" II and (,), respectively. 3)
4) 5) 6) 7) 8)
Hg(I)=
mrx~~~ in the usual Sobolev space Hk(/), whose closure of _-~c~ "--l-_,=o
norm is denoted by I1" Ilk{Hg(I)} 2 = Hz~(I) x Hz~(1). We use the same notation I1 Ilk for the norm of this space as in 3). ck(I) = space of k-times cotinuously differentiable functions oh L Bc = {VIV~(H~) 2, IlVl12 ~< c). Bc = interior of Be. J = identity operator. o
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
245
2. Description of the Model
The system treated here is the model (1.2) introduced in Section 1 ; we rewrite it as Oa 02a 07 = d~ ~ + fo(a, h, x){f(a, x ) - h} _
(2.1.1) Oh 02h 0-7 = dh ~ + g(a, x ) -- vh,
wherefo, f a n d g are described by fo(a, h, x ) = at~ - p(X)po h '
f(a, X) =
ep(x)a 2 (1 + Ka2)(atL -- p(x)po)
and g(a, x ) = c'p'(x)a 2.
Here we assume that p(x) and p'(x) are slightly inhomogeneous in the following sense: p(x) = p + ep(x)
and
p'(x) = p' + ,p'(x),
where p and p" are both positive constants, e is a small number compared with p and p', and #(x) and #'(x) are appropriately smooth and bounded functions. The parameters tz, c, v, c', po and K are all positive constants and P0 is sufficiently small. We consider the initial-boundary value problem for (2.1.1) in (t, x) ~ (0, +oo) x I, where I = (0, l), subject to the boundary conditions "t
Oa (t, O) = Oa O/ Ux ~x (t, I) =
ol ,
Oh (t, O)
Ux
Oh
Ux (t, 1)
t>O
(2.1.2)
and the initial conditions a(O, x ) = ao(x)'~
h(O,x)
ho(x)J'
x
(2.1.3)
EZ
For simplicity, we assume Max lp(x)] ~< 1 and XEi
Ma_x ]p'(x)l ~< 1. X~I
246
M. Mimura and Y. Nishiura
3. Global Existence
We show the uniform bound of solutions (a(t, x), h(t, x)) of the problem (2.1). Define the rectangle C in R ~ as follows: C = {(a,
h)ls=
~< a ~< A
and
8n ~< h ~< H},
where e=, A, 8n and H are positive constants satisfying 8d~ - (p + 8)00 > 0,
c'(0'_-- 8) {(0--r 1/
8~
>
(3.1)
> -
"
8h~
c(o + .)A s (1 + KA~){AV -- ( 0 + 8)00}'
c'(p' + 8)A =
< H,
(3.2)
(3.3)
(3.4)
V
and
c(p - 0(.=) ~ > H. {1 + x(sa)z}{eat L - (p - 8)p0}
(3.5)
Lemma 3.1. Consider the initial-boundary value problem (2.1). I f (ao(x ), h0(x)) is in C for x ~ 1, then the solution (a(t,x), h(t, x)) is confined to C for any (t, x) ~ [0, + ~ )
xi. Proof. It suffices to show that the vector field (fo(a, h, x) {f(a, x) - h}, g(a, x) - vh) on the boundary of C is always directed inward for any x E i. Because, with the aid of this result, the maximum principle leads to the proof of Lemma 3.1 (see, for instance, Chueh, Conley and Smoller (1977)). It is easy to see
f(e=, x) > h > f(A, x)
for x ~ i,
(3.6)
if 8n g(a, x) > wh for x e L
(3.7)
by making use of (3.2) and (3.4). Thus, noting that fo(a, h, x) is positive for (a, h, x) ~ C x L we see that the vector field ( f o ( f - h), g - vh) on 0C is directed inward. This completes the proof. [] Using this result and the local existence theorem of a solution (see, for instance, Friedman (1964)), we have immediately Theorem 3.1. Consider the initial-boundary value problem (2.1). I f (ao(x ), ho(x )) is in
C for x ~ ] and appropriately smooth, then there exists a unique global positive solution (a(t, x), h(t, x)) in (t, x) ~ (0, +oo) x 1.
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
247
Remark 3.1. L e m m a 3.1 and T h e o r e m 3.1 can be directly extended to higher dimensional spaces. W h e n e = 0 and the diffusivity is highly effective, one could expect that the solutions for (2.1) become spatially homogeneous as t ~ oo. This p r o b l e m was studied by, for instance, Conway, H o f f and Smoller (1978).
Theorem 3.2. Consider the initial-boundary value problem (2.1) with 8 = O. Suppose that
I MI M i n (da, dn) < 72 + Ao' where ~to is the smallest positive eigenvalue o f the eigenvalue problem d2~
ax~ = A ~ , x e I
and
d~ d d ~ ( o ) = ~ (1) = 0,
and M1 = M a x { [ d ( f o ( f - h), g - vh)] for (a, h) ~ C}. Then the solution (a(t, x), h( t, x ) ) becomes spatially homogeneous with exponential order as t -+ ~ . Proof. See T h e o r e m 3.1 in Conway, H o f f and Smoller (1978). Remark 3.2. Consider the stationary p r o b l e m of (2.1) with e = 0. I f Min (da, dh) > 89 + M1/)to, then there are no non-constant steady state solutions which lie in the interior o f the rectangle C for any x E i. This result is obtained as a direct consequence of T h e o r e m 3.2. F o r simplicity, we rewrite the system (2.1) as
Oa 02a 0-7 = da ~ + fo(a, h){fe(a) - h) (3.8), Oh 0ah 0-7 = du ~ + ge(a) - vh. O u r discussions will be carried out under the following assumptions: (3.8)0 has a unique positive constant solution, say (t~,/~), that is, it satisfies fo(~) = /~ and
h = f~
g~
= h,
(H.1)
has the positive gradient at a = ~,
(H.2)
and
d(fo(a, h){f~
- h}, g~
- vh) is a stable matrix at a = ~ and h = h. (H.3)
248
M. Mimura and Y. Nishiura
l
h : f]a) =
o
9
a
Fig. 1. Functional forms of f ~
and g~
These requirements f o r f ~ and gO can be also expressed as
2cp~ /~(1 + K~2)2
> I~,
2cpfi
2cp~
h(1 + Kt~2)2 < / ~ + v
and
2cp~
/~(1 + K~2) + tz > /~(1 + Kfi2)2"
T h e non-linearities o f f ~ and gO satisfying (H.I), (H.2) and (H.3) are drawn in Figure 1.
4. Perturbed Bifurcating Steady States (Local Theory) F o r the basic system (3.8)0, the analysis o f bifurcation from a constant steady state (&/~) can be done, for instance, as in M i m n r a , Nishiura and Y a m a g u t i (1977). In this section, we consider h o w bifurcated steady state solutions change when e # 0. T r a n s f o r m i n g to u = a - t~ and v = h - / ~ , we write (3.8)~ as follows _ 02U y0U f = D-~-~x2 + BU + H(U) + eG(U), (t,x)~(O, +oo) x I,
(4.1.1),
OU(t'O)o--~= -~x (aU't,l) = 0,
(4.1.2)8
U(0, x) = Uo(x),
t e ( 0 , +oo),
x ~L
(4.1.3),
where U = t(u, v), D is a diagonal matrix with the elements dl = da and d2 = dh, B = {b~s}~.j=l,2 is the J a c o b i a n at (if, h), H is a smooth non-linear term defined in some neighborhood o f U = 0 and G satisfies G(0) # 0. We restate (H.2) and (H.3) as follows:
BsatisfiesdetB>O, trB O, b12 O and
b22 1), we have the set of bifurcation points with respect to D, which is represented by the following hyperbolic curves {C,} in R~ : bzabaa/(Tna) 2 b2a C.: d2 = dl - b,z/(yn 2) + ~-'~'
n = 1, 2 . . . . .
R e m a r k 4.2. All the curves {C.} are tangent to the line d2 = m d l ;
m =
det B - btab21 + 2 ~ / - b12b21 det B b~l
We define Qs by the set {(dl, da)[(dl, d2) s R2+ and Re (A) < 0 for all n = 1, 2 . . . . . }, where A is a root of (4.5). Then f~s is called the stable region with respect to the trivial solution V = 0. The boundary of ~s and its upper side are called the bifurcation curve F and the unstable region, respectively (see Figure 2). Here I~ is represented by F
= 0
C,,, C,, = {(dl, d2)~C.lP,,
~< dl < Pn-1},
n=l
where P0 = b11/7 and P.(n >1 1) is an abscissa of the intersecting point of (7. and C.+ 1- {t~} satisfy t~. # ~ and ~'. r~ C,. = ~ when n # m. Suppose that the bifurcation parameter D varies along a fixed path D = D(o) starting from the stable region and going into the unstable region with the following properties: D: Io --> •a+ is a smooth mapping, where lo is an open interval in R 1, which contains 0. (H.5) D(0) = (d ~ d ~ lies on some t~, and D intersects transversally with r at D = D(0), (H.6)
and D(0) is not an intersecting point of two curves of {6',}. d2
C2 C.
(H.7)
Ci /d~ =md~
\ \ \ \
det B
[
\ P~
Po
dl
Fig. 2. Schematic bifurcation curve in (dl, at=) space
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
251
Thus our problem will be discussed within the framework of 'Bifurcation theory at simple eigenvalues'. We recast the system (4.2)+ into the form of an operator equation in (L2(I)) ~, which has a real parameter o, L(o)V + H ( V ) + eG(V) = O, o e Io,
(4.7L
where L(o) = D(c~)(d2/dx 2) + B and D(a) satisfies (H.5), (H.6) and (H.7). The domain of L(~) is
~(L(o)) = ~(Lo) = (HP,(t)) 2, (Lo = L(0)). H ( V ) and G(V) are both smooth non-linear operators from ~(Lo) t3 6 into (L2(I)) 2 with H(0) = Hv(O) = 0 and G(0) ~ 0, where 6 is an open neighborhood of 0 in (L2(I))L Expand the non-linear term H ( V ) as H(V) = Ho(V, V) + Ho(V) + HR(V),
(4.8)
where H o, Hc and HR denote the quadratic part, the cubic part and the remainder of H, respectively, and then define =,/3 and o~ by
= - v n ~ ( x ~ . , ~*) > 0, 13 = (Hc(qb.) -- Ho(~., KTHo(~., ~b.)) - HQ(KTHQ(d~., ~.), ~.), ~*), and
o~ = ( c O ) , ~*), respectively. Here ~ . is the normalized eigenvector corresponding to the zero eigenvalue of Loaf~ = A~F~ d~Fo d~O dx ( 0 ) = ~ ( l )
x ~/, =0
and the definitions of X, K, T a n d q)* are found in Lemmas A.1, A.2 and (A.5) in the Appendix. Now we can show two theorems. The proofs are relegated to the Appendix.
Theorem 4.1. (Local existence). Suppose that/3 # 0 and co # O. There exist positive constants %, ~o and eo such that for each fixed e satisfying 0 < I~l < to, (4.2)~ has a unique one parameter family of solutions (cr('q; e), V(r/; e)) e ( - %, %) x {(H~(I)) 2 n O}for 0 < ~ql < I'q] < ~o, where ~t is a constant depending on e. Here V0]; e) = ~/q~. + 1~(~(+/; 8), n),
(4.9)
where l?(~(-q; e), +7) satisfies the inequality
l[ r
~), ,7)112 ~< c{I,71(I,,(,~; +)l + 1,71) + I~l}
(4.10)
252
M. Mimura and Y. Nishiura BO
B>O LuO
w 0
./'or some positive constant c independent of ~, ~ and 8. The relation among ~, ~ and 8, that is the perturbed bifurcation diagram, is determined by the following scalar equation: ~
+ ~v8 + o,8 + r
7, 8) = 0,
(4.11)
where ~:(cr,"1, 8) is a higher order term with respect to ~, ~ and 8. From each fixed small e > 0, (4.11) gives the relation between ~ and ~7, which is drawn in Figure 3. From the above theorem, we found that two different types of solutions exist. One is of ordinary type, the other is of snap-like type. We call the former S-type and the latter U-type. Next we study the stability of V(-q; e). It can be discussed by making use of (4.11) with 8 = 0. Theorem 4.2. (Linearized stability). I f ~ < 0, then the S-type solutions are stable in the linearized sense, i.e., all eigenvalues of the linearized operator of (4.7)~ at the S-type solution have negative real parts, l f fl > O, then they are unstable. Remark 4.3. One can also discuss the stability of U-type solutions and the nonlinear stability of both types (see Kielh6fer (1974) and Mimura, Nishiura and Yamaguti (1977)). Thus, it is found that the sign of fl plays an important role to determine stability. In general, however, it requires fairly intricate calculations to know its sign. For some cases a useful criterion is given in Mimura, Nishiura and Yamaguti (1977). Using the above results, we intend to study the basic property of pattern formation that the polarity-pattern of the tissue is determined by the forms of source densities p(x) and p'(x). For a simple biological example, they are both assumed to be monotone decreasing functions. In this case, the region of higher density distribution, the terminal x = 0, is regarded as the head part of the tissue. Here the following problem occurs: Starting from even distributions of activator and inhibitor, can shallow gradients of p(x) and/or p'(x) produce a striking pattern in activator with higher concentration at the head part x = 0 ? This phenomenon was already confirmed numerically (Gierer and Meinhardt (1972)) (see Figure 4). We will investigate it by studying the stable perturbed bifurcating solutions V(V; ~) with
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
253
Fig. 4. Stable distributions of activator ( ) and inhibitor ( - - - - ) with a shallow gradient in source densities p(x) = p'(x) (^ ^ A A). (From Gierer and Meinhardt (1972)) fundamental mode (n = 1). We note here that fl < 0 holds for n = 1 under the condition d~ ,~ dh which was assumed by Gierer and Meinhardt (1972) (see Mimura, Nishiura and Yamaguti (1977)). To clarify the discussion, we specify that e is positive and the first c o m p o n e n t o f q~l and q~* are both m o n o t o n e decreasing on i. Thus, (4.9) implies that if 7/ > 0 (resp. < 0), the end x = 0 (resp. x = l) becomes head part. Therefore, from the perturbed bifurcation diagram (see Figure 3), we find as a direct consequence o f T h e o r e m 4.1,
Proposition 4.1.
Consider stable S-type solutions with n = 1 o f the problem (4.2)8. I f oJ > 0 (resp. < 0), then x = 0 (resp. x = l) is the head part. We discuss a particular case for p(x) and p'(x) (see Figure 4).
Proposition 4.2.
Suppose that p(x) = p'(x). I f p(x) is monotone decreasing on i, then
~o>0. Proof. F r o m simple calculations:
a~ ' G(0) = t(p(x)p0 +/~(1cp(x) + Ka2)
c,p,(x)a~)
and t
q)* = --7-
d~
+ v, h2(1 + K~2)/ cos -7-'
where k* ( > 0) is a normalizing constant such that [Ir ~o = (G(0), qb*) i> k*po
fOIp(x)
II =
1. Thus we have
cos ~X 7 " dx
and hence co > 0, which completes the proof.
[]
Let us restate this proposition in a biological language. W h e n a nearly flat gradient (0(e)) is introduced into source distributions such that the end x = 0 is head part, prominent stable patterns o f activator and inhibitor (0(3x/~)) can be recognized with higher concentrations at the end x = 0, which is in agreement with numerical results in Figure 4.
254
M. Mimura and Y. Nishiura
Remark 4.4. Even if slight random fluctuations are introduced in strictly monotone source distributions we can obtain similar results to that of Proposition 4.2, because the quantity co is defined by the inner product between G(0) and ~*.
5. A Global Continuum of the Perturbed Bifurcating Branch
In this section we show the global existence of a continuum of the perturbed bifurcating branch obtained in Section 4. First we define the global path of the parameter D = (d~, d2) in R2+. For simplicity, we consider a path parameterized by dl only. Definition 5.1. We say D = D(dl) = (d~, d2(d0): R~+ ~ R2+ is a global path in the parameter space Ra+ when it satisfies: d2 = d2(dz): R~+ --> ~+ is a smooth function.
(H.8)
D intersects transversally with I' at only one point and it is not an intersection point of two curves o f {C,}. (H.9) D contains a point satisfying the inequality in Remark 3.2.
(H.IO)
Remark 5.1. We note that (H.10) always holds when D has the property that d2(dl) ~ + oo as dl ~ -t-~. So far we needed only local properties in the vicinity of a constant solution to study the perturbed bifurcating branch. To study the global structure of (4.7)~, we need global properties of the system. Thus we assume in addition to (H.4), There exists an invariant rectangle C in which there is a unique constant stationary solution V = 0, and on OC, the vector field corresponding to the non-linear term of (4, 7)0 is always directed strictly inward. (H.11) We find, in Section 3, that (1.2) satisfies these conditions. Before showing the main theorem, we introduce some notations: 5~ denotes the closure of the set of solutions of (4.7)~ in R~+ x (H~) 2 and Proj denotes the projection onto the first component of R~+ x (Hg) 2, that is, Proj (dl, V) --- d~ for (d~, V) e
R~+ x (H~,)~. The global existence theorem of a continuum along the path defined above is the following: Theorem 5.1. Let ~ denote the component (a maximal closed connected set) in 50 which contains the S-type branch (resp. U-type branch) when fl < 0 (resp. fl > 0). Then there exists a positive constant eo such that for each fixed e with 0 < It[ < e0
g n {dl} x ((H~) 2 :~ C)} ~ c~ for any dl e R~+, that is, Proj (8) = (0, + oo).
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
255
Remark 5.2. The branch near d ~ is obviously contained in the interior of C. Suppose that the component g comes into contact with the boundary of C when dx continuously varies from d ~ Defining the solution in this situation by U*(x), we know that there exists an x* satisfying U*(x*)E aC. Using (H.11), it is seen that U* cannot be a solution of (4.2)~, which is a contradiction. Thus, we find that any solution in g is contained in the interior of C. The proof of Theorem 5.1 essentially depends on the result of Remark 3.2, that is, non-existence of non-homogeneous solutions of (4.7)0 for large diffusion coefficients and on the method of Leray-Schauder degree which was developed in the important paper by Rabinowitz (1971). We begin by showing five lemmas so as to prove Theorem 5.1. Hereafter we fix a global path D such that it intersects with C, of I" at D O = (d~ d2(d~ and denote by Do the part of this global path for dl /> d ~ + 8 with a positive constant 3. Lemma 5.1. For any fixed 8 > O, the operator norm of
from (LZ)z into (H~) 2 is uniformly bounded on Dn. Proof. See Appendix. /_,emma 5.2. For any 8 > O, there exist positive constants c = c(8) and e = 8(8) such that for D(dl) E Do and e with lel < e(8), there exists a unique solution V(dl, e) of the problem (4.7)~ in l~c~o). V(dl, e) is continuous in (H~) z with respect to dl and e for O(dl) ~ Do and lel < e(8). Proof. Operating w i t h / ( i on (4.7),, we obtain the equation V + K I H ( V ) + eKlG(V) = 0
(5.1)
in (H~) 2. It is noted that (V, dl, e) = (0, d~, 0) is a trivial solution of(5.1) and that the Frech& derivative of the left-hand side of (5.1) with respect to V at (0, dl, 0) with D(d~) ~ Do is the identity operator. Therefore, applying the implicit function theorem to (5.1), we find that there exist positive constants Co and e0 such that for D(d~) ~ Do and [e I < e0, Eq. (5.1) has a unique solution V(d~, e) in /~c0 which depends smoothly on d~ and e (see, for instance, Crandall and Rabinowitz (1971)). Since the operator norm of/{1 is uniformly bounded on Do, it is seen that e0 and Co depend only on 3, which completes the proof. [] The solutions obtained in the above lemma are called the perturbed trivial branch. Lemma 5.3. (A priori estimate). Let V be any solution of(4.7)~ in (H~) 2 n C. Then
the following estimate holds:
where M2 is a positive constant depending on C and the coefficients of(4.7)..
256
M. Mimura and Y. Nishiura
Proof. Since the existence of C leads to L~~ on V, it is easy to obtain the result by using the equation (4.7),. So we omit the details. [] Lemma 5.4. (Uniqueness in the large). Suppose that the diffusion coefficients D =
(dl, d2) satisfy Min (dl, d2) > ~1 + M1
(see Remark 3.2).
Then there exists a positive constant e1 such that for e satisfying l,l < ~1, (4.7), has a unique solution in (H~) 2 n C. Proof. From Lemma 5.2, we found that (4.7), has a unique solution in the neighborhood of the origin, say in /~_~.Therefore it suffices to prove the non-existence of solutions of (4.7)~ outside this neighborhood when e is sufficiently small. Suppose that there are solutions of (4.7)~ in (H~) 2 n C - /~ for any small e. Then we extract a sequence of solutions { V J of (4.7), w i t h , = e,, which satisfy
< LIv..ll2
e
for a positive constant ~ independent of e, where e, --~ 0 as n -+ + oo. Thus, we can extract a weakly convergent subsequence {V~.,} in (H~) 2, which also converges strongly in (H~) 2. Since V,, is a solution of (4.7)~ and H~ = C~ {V~,} is indeed a strongly convergent sequence in (H~) 2. Therefore, it is seen that the limit function V* of {V,.,} is a solution o f (4.7)o in (H~) 2 n C, and that
_c
IIV*ll
< o.
On the other hand, from the assumption of this lemma and (H.11), Eq. (4.7)o has only one solution V = 0 in (H~) 2 n C. This is a contradiction, which completes the proof. [] Lemma 5.5. For appropriately small ~, the perturbed trivial branch is connected to the
S-type branch (resp. U-type branch), if fi < 0 (resp. fl > 0). Proof. The proof is obtained as a direct consequence of the uniqueness of the perturbed bifurcating branch in Theorem 4.1 and Lemma 5.2. [] Let us prove Theorem 5.1 by using the above lemmas. For the proof the homotopy invariance of Leray-Schauder degree plays an essential role as in Rabinowitz (1971). First, by operating with Kz =
d2 ) -1 -- D~xx2 + ,8"
(5.2)
on (4.7),, it is converted into an operator form
V = ~(dl, V),
(5.3)
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
257
where fr V) = K2(B + J ) V + K2H(V) + eK2G(V). It is noted that fr is a compact continuous operator from R~ x (H~) z into (H~) z. Let us define ~ ( d l , V) by ~ ( d l , V) = V -
if(d1, V).
(5.4)
From (H.10), we can choose dl such that D(dl) satisfies the inequality in Remark 3.2. Then from Lemma 5.2 and Lemma 5.4, it is easily seen that deg (o~f'(d~, .),/}_~) :r 0
(5.5)
for an appropriate positive constant _c. We denote go and 8' 1 by 8 n ( 0 , dl] x (H~) 2 = 8o
and
8n[dl,
+ ~ ) x (H~) 2 = 81,
respectively. P r o o f o f Theorem 5.1. Since 81 contains the perturbed trivial branch from Lemma
5.5, it follows Proj (81) = [dl, + ~ ) . Therefore it suffices to prove Proj (80) = (0, dl].
(5.6)
Suppose that there exists a positive constant s such that Proj (80) = [s, dl], then we have a contradiction in view of the following: We note that 80 is compact in R~- • (H~) 2 from Lemma 5.3 and the compactness of re. Here we use the following lemma given by Rabinowitz (1971). Lemma 5.6. There exists a bounded open set C) in (s/2, + oo) x ((H~) 2 c~ C) such that 80 = 6~ 8(~al ) c~ 5~ = ~ f o r any dl ~ (O dl] and ~3~_c ~o~, where 6~a~ = {U~ (H~)~l(dl, U) ~ 0} and 8(0a~) is its boundary.
From this lemma, deg ( ~ ( d l , -), ~~ is well-defined for dx e (0, all]. The homotopy invariance of the Leray-Schauder degree implies that deg (W(da, .), d)al) = const. = c for dl E (0, d~].
(5.7)
From Lemma 5.4 and (5.5), we obtain deg (~(d~, .), d?~,) = deg ( ~ ( d l , -),/~_) ~ 0.
(5.8)
This implies c ~ 0. On the other hand, deg(~,~(dl, .),d?d~)=0
for d~ ~ (0, 2).
This contradicts (5.7) and (5.8), which completes the proof.
[]
258
M. Mimura and Y. Nishiura
6. Concluding Remark This paper is devoted to two problems. One is the determination of the polarity in the tissue, which is interesting from a biological point of view. It is found that slightly asymmetric source distributions produce striking stable patterns depending on the asymmetry. Our analytical results are in agreement with Gierer and Meinhardt's numerical experiments. The other is the existence of a continuum of perturbed bifurcating solutions when one of diffusion coefficients is adopted as a bifurcation parameter in (0, + oo). We confined our study to Gierer and Meinhardt's model. However, there is apparently no such restriction in our arguments. In fact, from Section 3 on, we carried out our discussion in a general form JDVxx + F(x, V) = O,
V = (v~, v~).
The global continuum, in essence, can be obtained if the following conditions hold: For local properties, 1) there exists one constant solution, say V = 0, 2) the system satisfies Turing's condition at V = 0 (see (H.4)), and 3) F(x, O) # O. For global properties, 4) there exists an invariant set which contains the point V = 0 in R a. Finally, our global uniqueness result (see Lemma 5.4) leads to the interesting fact that the global branch which contains unstable perturbed bifurcating solutions (or bifurcating ones when e = 0) must return to the opposite direction (see Figure 5 when 8 = 0).
o-
stable unstable
Pig. 5. Recovery of stability of the unstable branch with e=0
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
259
Appendix Proofs o f Theorem 4.1. and Theorem 4.2.
We first state two lemmas without proofs (see, for instance, Sather (1973)). Lemma A.1. (1) Lo: ~(Lo) --~ (L2) ~ is a linear Fredholm operator. (2) The null space N(Lo) o f Lo and the null space N(L *) of i t s adjoint operator L* are both one-dimensional and they are spanned by the vector a~ = k.$~ cos mrx --7-
and
(I)* = k'd,* cos nrrx 7""
respectively, where ~ = ~(d~ 2 - b22, b21) and
$* = t(d~ 2 - b22, blz)
and k~ and k* are normalizing constants such that 11(I)~[I = ]1(I)* 1[ = 1 (for definiteness, we take kn > 0 and k* > 0). (3) Lo is a one-to-one mapping of ~(Lo) n N(Lo)" onto ~(Lo) (range o f Lo) so that K = (Lo]~(Lo)~mLo)')-1 is well defined, and K T is a linear compact operator defined on (L2) 2. Here T is defined in the next lemma.
Lemma A.2. Let P be the orthogonal projection operator of(L2) 2 onto N(Lo) and T be that o f (LZ) 2 onto ~(Lo). Then it is true that Pw = ( w , *~)*~ KLow = (I - e ) w
and
(I - T)w = (w, dp*)**,
f o r all w e ~(Lo),
(1) (2) (3)
LoKTw = Tw for all w ~ (L2) 2.
Using the projection P, we write V as (A.1)
V = P V + (I - P ) V = ~ap~ + 17, where ~/is a real parameter. Substituting (A. 1) into (4.7)~, we obtain
LoI7 + A(q)(V(I)~ + l?)x~ + H(~7(I)~ + I~) + ,G(v(I). + l?) = 0, where A ( a ) = D ( q ) -
(A.2)
D(0). Using the projection T, the following system is t
equivalent to (A.2) Lo I? + TA(a)(~(I). + l?)~ + TH(71ap~ + I7) + eTG(~*~ + l?) = 0
(A.3.1)
and (I-
T){A(~r)(~*~ + I7)~ + H(~7. n + l?) + ,G07dp, + I7)} = 0.
(A.3.2)
The property (2) of Lemma A.2 reduces (A.3.1) to + KTA(~)(,Ta,~ + ~ ) ~
+ KTH(n*~ + ~) + ,KT6(n*~ + ~') = 0.
(A.4)
From the assumption (H.6), A(cr) can be represented by a(q) = (rX + 0(~),
(A.5)
260
M. M i m u r a a n d Y. Nishiura
where x is a diagonal matrix with the elements {x~}~=~.2and (x~, x2) is not parallel to the tangent of (', at D(0), and O(a) is a higher order term. Then (A.4) is rewritten as
r + aKTx07r + KTH(~r
+ r + KTO(a)(Vr + 1~)~ + 17") + eKTG(~7r + 12) = O.
(A.6)
Applying a standard implicit function theorem in Banach space (see, for instance, Crandall and Rabinowitz (1971), Dieudonnd (1960)) to (A.6), we can get the unique solution l?(a, ~7, *) of (A.6). The form of the principal part of l?(a, V, 8) for sufficiently small or, ~ and e is represented by
r
~, 8) = o~Tn2KTxCb~ - ~?2KTHo(aP,, 0 , ) - ,KTG(O) + V(a, .,7, e),
(A.7)
where V(a, ~7, e) is a higher order term. Substituting the solution r ~7, e) into (A.3.2), we obtain the scalar equation with respect to a, ~ and e, which is called the
perturbed bifurcation equation ( I --
T)(A(a)(V~P, + 12(a, zl, e))xx + H(71cb,~ + 12(a, 7/, e)) + ~G(~r + r ~, ~))) = 0.
(A.8)
Using Lemma A.2 and (A.5), Eq. (A.8) can be written as
(aX(~?aP~ + IT(a, ",1,e))xx + O(a)(~cb~ + If(a, ~7, *))xx + H(~aP~ + 9(a, '7, *)) + eG(~/r + 12(a, ~/, e)), 0 " ) = 0. (A.9) By making use of the expansion of H ( V ) and (A.7), we have
-- awn2(xcb,, r
+ ~2(HQ(r On), r - HQ(KTHQ(Cbn, r
+ ,/S(Hc(~P~) - HQ(r KTHQ(rb,, r r r + ,(G(0), r + $(a, n, e) = 0, (A.IO)
where r ~7, *) is a smooth higher-order term with respect to a, ~7 and 8. Noting (HQ(ap~, qb), dp*) = 0, we obtain the perturbed bifurcation Eq. (4.11). Here we set ,~ =
O*),
-~,n2(xO,,,
fl = (Hc(r
- Ho(r
KTHo(fb,, r
- Ho(KTHQ(r
r
r
r
and
o, = ( 6 ( 0 ) , ~*). Wc sec a > 0 from the definition of the path D = D(a). From now on we assume fl:/:0
and
oJ#0.
(A.11)
These are generic conditions, so almost all non-linearities satisfy (A. 11). In fact, there hardly appears the case/3 = 0 and/or oJ = 0 in applications.
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
261
Applying a smooth change of coordinates from (a, ~7, e) to (#, 9, ~) (we can take ~/ = 9) to (4.11), whose Jacobian at (a, 7, e) = (0, 0, 0) is 1, we get the following normal polynomial form of (4.11): ~e~ + flcfl + oJ~ = 0.
(A.12)
From (A.12), the picture of the relation between a and -q for any fixed small e is given by Figure 3. For details we refer to Nirenberg (1973). Theorem 4.1 is the direct consequence of the above results. Proof of Theorem 4.2 First we note that (4.7)~ can be converted into the compact operator form (5.3). Using a similar method as in the proof of Lemma 3.2 in Sattinger (1971), it is seen that the linearized operator at V(~7;e) corresponding to the solution of (4.11) with cr'(~) # 0(' = dido) and ~7 # 0 is invertible. Therefore the linearized operator along the S-type solution is invertible for all sufficiently small ~. For e = 0 and cr < 0, the spectrum of the trivial solution is of stable type, i.e., all the eigenvalues of (4.3) have negative real parts. The Continuous dependency of an eigenvalue with respect to implies that the spectrum at the S-type solution for ,r < 0 and sufficiently small E is of stable type. Combining this result with the invertibility along the S-type solution, we get the linearized stability of the S-type solutions. [] Proof of Lemma 5.1 For the proof, it suffices to show the existence of a constant K which only depends on 3 such that for any F ~ (L2)2 with HFI[ ~< 1, there exists a solution V~ (H~) 2 of D)-~x2 + B V = F,
(A.13)
which satisfies
II vii2
(A.14)
K
for arbitrary D ~ Do. This will be done by applying Fourier-cosine series expansion to (A.13). Inserting the expressions V=
~ ~o V. cos .=o
l'
9
F=
F. cos •=0
l
into (A. 13), we obtain the system of equations for V. and F., M . V . = F.,
(n = 0, 1, 2,),
(A.15)
where ~ - d w n 2 + bll M~ = \
b21
b12
)
- d 2 y n 2 + b22 "
262
M. M i m u r a a n d Y. N i s h i u r a
Since D is in the stable region ft. (see Section 4), M , is non-singular for all n. Therefore it follows V~ = M y tF~,
(A.16)
where
1 ( - d a y n a + baa M~-x = det--~ \ -bax
-b~a ) - d w n ~ + b~x
and det M,, = (b~t - dx~,na)(b~ - d~,n ~) - bx~b~x. To obtain the estimate (A.14), it suffices to show the following inequalities
I V.I < r
(A.17)
and vn~lV.I 1 t*).
From the definition of fl~, it is easily seen that for any 8, D~ is contained in tl~ for sufficiently small #. For any D e tl~, we obtain the following estimates:
I(bxl -
yn2(b2z - d2yn') t I.L]d2yn 2 -
b221 >
~lb=~l. (A.21)
O n the other h a n d , the left-hand side of (A.20) is equal to {Tn2{(d~ -
b~/yn2)(d2
-
b 2 2 / y n 2) -
b~2b2~/(yn2)2}} -1.
Therefore (A.21) implies the required estimate (A.20). F r o m the form o f M g 1 of (A.16), the estimates (A. 18)-(A.20) lead to the second inequality (A. 17). Thus the p r o o f is completed. [ ]
References Auchmuty, J. F. G., Nicolis, G.: Bifurcation Analysis of Nonlinear Reaction-Diffusion Equation-L Evolution Equations and the Steady State Solutions, Bulletin Math. Biol. 37, 323-365 (1975) Chueh, K. H., Conley, C. C., Smoller, J. A." Positively Invariant Regions for Systems of Nonlinear Diffusion Equations, Indiana Univ. Math. J. 26, 373-392 (1977) Conway, E., Hoff, D., Smoller, J. A. : Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations, SIAM J. Appl. Math. 35, 1-16 (1978) Crandall, M. G. Rabinowitz, P. H.: Bifurcation from Simple Eigenvalues, J. Func. Anal. 8, 321-340 (1971) Dieudonn6, J. : Foundation of Modern Analysis, Academic Press, New York, 1960 Friedman, A.: Partial Differential Equations of Parabolic Type, Prentice-Hail, Englewood Cliffs, N.J., 1964 Gierer, A., Meinhardt, H.: A Theory of Biological Pattern Formation, Kybernetik 12, 30-39 (1972) Granero, M. I., Porati, A., Zanacca, D.: A Bifurcation Analysis of Pattern Formation in a Diffusion Governed Morphogenetic Field, J. Math. Biol. 4, 21-27 (1977) Kielh/Sfer, H. : Stability and Semilinear Evolution Equations in Hilbert Space, Arch. Rat. Mech. Anal. 57, 150-165 (1974) Mimura, M., Nishiura, Y., Yamaguti, M.: Some Diffusive Prey and Predator Systems and Their Bifurcation Problems, to appear in the Annals of the New York Academy of Sciences (1977) Nirenberg, L. : Topics in Nonlinear Functional Analysis, Lecture Notes, Courant Institute of Mathematical Sciences, New York University, 1973 Rabinowitz, P. H. 9Some Global Results for Nonlinear Eigenvalue Problems, J. Func. Anal. 7, 487-513 (1971) Sather, D. : Branching of Solutions of Nonlinear Equations, Rocky Mount. J. Math. 3, 203-250 (1973) Sattinger, D. H.: Stability of Bifurcating Solutions by Leray-Schauder Degree. Arch. Rat. Mech. Anal. 43, 154-166 (1971) Turing, A. M. : The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. 13237, 32, 37-72 (1952) Received July 5~Revised October 16, 1978
J. Math. Biology7, 243-263 (1979)
Illalkmal
l
by Springer-Verlag 1979
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis Masayasu Mimura 1 and Yasumasa Nishiura 2 x Department of Applied Mathematics, Konan University, Kobe Department of Computer Sciences, Kyoto Sangyo University, Kyoto, Japan
Summary. A certain interaction-diffusion equation occurring in morphogenesis is considered. This equation is proposed by Gierer and Meinhardt, which is introduced by Child's gradient theory and Turing's idea about diffusion driven instability. It is shown that slightly asymmetric gradients in the tissue produce stable striking patterns depending on its asymmetry, starting from uniform distribution of morphogens. The tool is the perturbed bifurcation theory. Moreover, from a mathematical point of view, the global existence of steady state solutions with respect to some parameters is discussed. Key words: Bifurcation.
Morphogenesis--Spatial
patterns--Non-linear
diffusion--
1. Introduction
Since Turing's famous paper (1952) has been published, many interesting mathematical models, based on Turing's idea, have been proposed in various fields to explain spatial patterns. Most of the models are described by OU _ 02U -oi- = ~ T ~ x ~ + F ( U ) .
(1.D
The variable U represents quantities such as temperature, concentrations and densities. T h e most interesting problems for (1.1) are related to the analysis of the qualitative behavior. The main tool to study the formation of spatial patterns of (1.1) is the theory of bifurcation (see, e.g., Auchmuty and Nicolis (1975)). Gierer and Meinhardt (1972) proposed several models to interpret cell differentiation. One of their models takes the form Oa . O2a cp(x)a ~ - ~ = a,, ~ + p(X)po + h(1 + Ka2) Oh 02h O~ = dn ~ + c'p'(x)a 2 -
pa
(1.2) vh.
0303-6812/79/0007/0243/$04.20
244
M. Mimura and Y. Nishiura
Here a(t, x) and h(t, x) are the concentrations of a short range activator and a long range inhibitor at time t and position x, respectively, p(x) and o'(x) are the source density distributions of activator and inhibitor, which generally depend on position x. The difference between (1.1) and (1.2) is that the interaction terms are generally functions o f x and U. The reason is that Child's gradient theory is introduced into Turing's idea. The x-dependence of p(x) and p'(x) reflect the appearance of polarity in the tissue, p0, c, K,/z, c' and v are assumed to be positive constants, da and dh are the diffusion constants of activator and inhibitor, respectively (for the biological meaning, see Gierer and Meinhardt (1972)). These authors showed numerically that slightly asymmetric source density distributions produce stable striking patterns starting from uniform distributions of activator and inhibitor. These numerical experiments show the importance of the polarity. From an analytical point of view, Granero, Porati and Zanacca (1977) studied the stationary problem of (1.2) under zero flux boundary conditions when K = 0 and p(x) and p'(x) are both constants. They showed the explicit form of small amplitude steady state solutions by making use o f the Poincarr-Lindstedt method. The aim o f this paper is to discuss the effect of non-constant p(x) and p'(x) on spatial structures of steady state solutions of (1.2) and to interpret Gierer and Meinhardt's numerical experiments analytically. Moreover, we show some global structure of the set of the steady state solutions. The main mathematical tools used here are the perturbed bifurcation theory and the Leray-Schauder degree. In Section 2, we describe the model discussed in this paper. In Section 3, we show the uniform boundedness of a solution (a(t, x), h(t, x)) of the initial-boundary value problem for (1.2) under-zero flux boundary conditions and hence obtain global existence theorem. In Section 4, we discuss the existence of small amplitude steady state solutions by using the Lyapunov-Schmidt method and their stability. The solutions are of two different types, classified as ordinary type and snap-like type. In Section 5, we study a certain continuum which contains the solutions stated in Section 4. Finally we conclude with some comments on our results. We shall use the following notations throughout the paper: 1) R~ = (UtU ~ ~'~ and each component is positive}. 2) L2(I) = space of functions which are square summable on I = (0, l). We denote its norm and inner product by II" II and (,), respectively. 3)
4) 5) 6) 7) 8)
Hg(I)=
mrx~~~ in the usual Sobolev space Hk(/), whose closure of _-~c~ "--l-_,=o
norm is denoted by I1" Ilk{Hg(I)} 2 = Hz~(I) x Hz~(1). We use the same notation I1 Ilk for the norm of this space as in 3). ck(I) = space of k-times cotinuously differentiable functions oh L Bc = {VIV~(H~) 2, IlVl12 ~< c). Bc = interior of Be. J = identity operator. o
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
245
2. Description of the Model
The system treated here is the model (1.2) introduced in Section 1 ; we rewrite it as Oa 02a 07 = d~ ~ + fo(a, h, x){f(a, x ) - h} _
(2.1.1) Oh 02h 0-7 = dh ~ + g(a, x ) -- vh,
wherefo, f a n d g are described by fo(a, h, x ) = at~ - p(X)po h '
f(a, X) =
ep(x)a 2 (1 + Ka2)(atL -- p(x)po)
and g(a, x ) = c'p'(x)a 2.
Here we assume that p(x) and p'(x) are slightly inhomogeneous in the following sense: p(x) = p + ep(x)
and
p'(x) = p' + ,p'(x),
where p and p" are both positive constants, e is a small number compared with p and p', and #(x) and #'(x) are appropriately smooth and bounded functions. The parameters tz, c, v, c', po and K are all positive constants and P0 is sufficiently small. We consider the initial-boundary value problem for (2.1.1) in (t, x) ~ (0, +oo) x I, where I = (0, l), subject to the boundary conditions "t
Oa (t, O) = Oa O/ Ux ~x (t, I) =
ol ,
Oh (t, O)
Ux
Oh
Ux (t, 1)
t>O
(2.1.2)
and the initial conditions a(O, x ) = ao(x)'~
h(O,x)
ho(x)J'
x
(2.1.3)
EZ
For simplicity, we assume Max lp(x)] ~< 1 and XEi
Ma_x ]p'(x)l ~< 1. X~I
246
M. Mimura and Y. Nishiura
3. Global Existence
We show the uniform bound of solutions (a(t, x), h(t, x)) of the problem (2.1). Define the rectangle C in R ~ as follows: C = {(a,
h)ls=
~< a ~< A
and
8n ~< h ~< H},
where e=, A, 8n and H are positive constants satisfying 8d~ - (p + 8)00 > 0,
c'(0'_-- 8) {(0--r 1/
8~
>
(3.1)
> -
"
8h~
c(o + .)A s (1 + KA~){AV -- ( 0 + 8)00}'
c'(p' + 8)A =
< H,
(3.2)
(3.3)
(3.4)
V
and
c(p - 0(.=) ~ > H. {1 + x(sa)z}{eat L - (p - 8)p0}
(3.5)
Lemma 3.1. Consider the initial-boundary value problem (2.1). I f (ao(x ), h0(x)) is in C for x ~ 1, then the solution (a(t,x), h(t, x)) is confined to C for any (t, x) ~ [0, + ~ )
xi. Proof. It suffices to show that the vector field (fo(a, h, x) {f(a, x) - h}, g(a, x) - vh) on the boundary of C is always directed inward for any x E i. Because, with the aid of this result, the maximum principle leads to the proof of Lemma 3.1 (see, for instance, Chueh, Conley and Smoller (1977)). It is easy to see
f(e=, x) > h > f(A, x)
for x ~ i,
(3.6)
if 8n g(a, x) > wh for x e L
(3.7)
by making use of (3.2) and (3.4). Thus, noting that fo(a, h, x) is positive for (a, h, x) ~ C x L we see that the vector field ( f o ( f - h), g - vh) on 0C is directed inward. This completes the proof. [] Using this result and the local existence theorem of a solution (see, for instance, Friedman (1964)), we have immediately Theorem 3.1. Consider the initial-boundary value problem (2.1). I f (ao(x ), ho(x )) is in
C for x ~ ] and appropriately smooth, then there exists a unique global positive solution (a(t, x), h(t, x)) in (t, x) ~ (0, +oo) x 1.
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
247
Remark 3.1. L e m m a 3.1 and T h e o r e m 3.1 can be directly extended to higher dimensional spaces. W h e n e = 0 and the diffusivity is highly effective, one could expect that the solutions for (2.1) become spatially homogeneous as t ~ oo. This p r o b l e m was studied by, for instance, Conway, H o f f and Smoller (1978).
Theorem 3.2. Consider the initial-boundary value problem (2.1) with 8 = O. Suppose that
I MI M i n (da, dn) < 72 + Ao' where ~to is the smallest positive eigenvalue o f the eigenvalue problem d2~
ax~ = A ~ , x e I
and
d~ d d ~ ( o ) = ~ (1) = 0,
and M1 = M a x { [ d ( f o ( f - h), g - vh)] for (a, h) ~ C}. Then the solution (a(t, x), h( t, x ) ) becomes spatially homogeneous with exponential order as t -+ ~ . Proof. See T h e o r e m 3.1 in Conway, H o f f and Smoller (1978). Remark 3.2. Consider the stationary p r o b l e m of (2.1) with e = 0. I f Min (da, dh) > 89 + M1/)to, then there are no non-constant steady state solutions which lie in the interior o f the rectangle C for any x E i. This result is obtained as a direct consequence of T h e o r e m 3.2. F o r simplicity, we rewrite the system (2.1) as
Oa 02a 0-7 = da ~ + fo(a, h){fe(a) - h) (3.8), Oh 0ah 0-7 = du ~ + ge(a) - vh. O u r discussions will be carried out under the following assumptions: (3.8)0 has a unique positive constant solution, say (t~,/~), that is, it satisfies fo(~) = /~ and
h = f~
g~
= h,
(H.1)
has the positive gradient at a = ~,
(H.2)
and
d(fo(a, h){f~
- h}, g~
- vh) is a stable matrix at a = ~ and h = h. (H.3)
248
M. Mimura and Y. Nishiura
l
h : f]a) =
o
9
a
Fig. 1. Functional forms of f ~
and g~
These requirements f o r f ~ and gO can be also expressed as
2cp~ /~(1 + K~2)2
> I~,
2cpfi
2cp~
h(1 + Kt~2)2 < / ~ + v
and
2cp~
/~(1 + K~2) + tz > /~(1 + Kfi2)2"
T h e non-linearities o f f ~ and gO satisfying (H.I), (H.2) and (H.3) are drawn in Figure 1.
4. Perturbed Bifurcating Steady States (Local Theory) F o r the basic system (3.8)0, the analysis o f bifurcation from a constant steady state (&/~) can be done, for instance, as in M i m n r a , Nishiura and Y a m a g u t i (1977). In this section, we consider h o w bifurcated steady state solutions change when e # 0. T r a n s f o r m i n g to u = a - t~ and v = h - / ~ , we write (3.8)~ as follows _ 02U y0U f = D-~-~x2 + BU + H(U) + eG(U), (t,x)~(O, +oo) x I,
(4.1.1),
OU(t'O)o--~= -~x (aU't,l) = 0,
(4.1.2)8
U(0, x) = Uo(x),
t e ( 0 , +oo),
x ~L
(4.1.3),
where U = t(u, v), D is a diagonal matrix with the elements dl = da and d2 = dh, B = {b~s}~.j=l,2 is the J a c o b i a n at (if, h), H is a smooth non-linear term defined in some neighborhood o f U = 0 and G satisfies G(0) # 0. We restate (H.2) and (H.3) as follows:
BsatisfiesdetB>O, trB O, b12 O and
b22 1), we have the set of bifurcation points with respect to D, which is represented by the following hyperbolic curves {C,} in R~ : bzabaa/(Tna) 2 b2a C.: d2 = dl - b,z/(yn 2) + ~-'~'
n = 1, 2 . . . . .
R e m a r k 4.2. All the curves {C.} are tangent to the line d2 = m d l ;
m =
det B - btab21 + 2 ~ / - b12b21 det B b~l
We define Qs by the set {(dl, da)[(dl, d2) s R2+ and Re (A) < 0 for all n = 1, 2 . . . . . }, where A is a root of (4.5). Then f~s is called the stable region with respect to the trivial solution V = 0. The boundary of ~s and its upper side are called the bifurcation curve F and the unstable region, respectively (see Figure 2). Here I~ is represented by F
= 0
C,,, C,, = {(dl, d2)~C.lP,,
~< dl < Pn-1},
n=l
where P0 = b11/7 and P.(n >1 1) is an abscissa of the intersecting point of (7. and C.+ 1- {t~} satisfy t~. # ~ and ~'. r~ C,. = ~ when n # m. Suppose that the bifurcation parameter D varies along a fixed path D = D(o) starting from the stable region and going into the unstable region with the following properties: D: Io --> •a+ is a smooth mapping, where lo is an open interval in R 1, which contains 0. (H.5) D(0) = (d ~ d ~ lies on some t~, and D intersects transversally with r at D = D(0), (H.6)
and D(0) is not an intersecting point of two curves of {6',}. d2
C2 C.
(H.7)
Ci /d~ =md~
\ \ \ \
det B
[
\ P~
Po
dl
Fig. 2. Schematic bifurcation curve in (dl, at=) space
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
251
Thus our problem will be discussed within the framework of 'Bifurcation theory at simple eigenvalues'. We recast the system (4.2)+ into the form of an operator equation in (L2(I)) ~, which has a real parameter o, L(o)V + H ( V ) + eG(V) = O, o e Io,
(4.7L
where L(o) = D(c~)(d2/dx 2) + B and D(a) satisfies (H.5), (H.6) and (H.7). The domain of L(~) is
~(L(o)) = ~(Lo) = (HP,(t)) 2, (Lo = L(0)). H ( V ) and G(V) are both smooth non-linear operators from ~(Lo) t3 6 into (L2(I)) 2 with H(0) = Hv(O) = 0 and G(0) ~ 0, where 6 is an open neighborhood of 0 in (L2(I))L Expand the non-linear term H ( V ) as H(V) = Ho(V, V) + Ho(V) + HR(V),
(4.8)
where H o, Hc and HR denote the quadratic part, the cubic part and the remainder of H, respectively, and then define =,/3 and o~ by
= - v n ~ ( x ~ . , ~*) > 0, 13 = (Hc(qb.) -- Ho(~., KTHo(~., ~b.)) - HQ(KTHQ(d~., ~.), ~.), ~*), and
o~ = ( c O ) , ~*), respectively. Here ~ . is the normalized eigenvector corresponding to the zero eigenvalue of Loaf~ = A~F~ d~Fo d~O dx ( 0 ) = ~ ( l )
x ~/, =0
and the definitions of X, K, T a n d q)* are found in Lemmas A.1, A.2 and (A.5) in the Appendix. Now we can show two theorems. The proofs are relegated to the Appendix.
Theorem 4.1. (Local existence). Suppose that/3 # 0 and co # O. There exist positive constants %, ~o and eo such that for each fixed e satisfying 0 < I~l < to, (4.2)~ has a unique one parameter family of solutions (cr('q; e), V(r/; e)) e ( - %, %) x {(H~(I)) 2 n O}for 0 < ~ql < I'q] < ~o, where ~t is a constant depending on e. Here V0]; e) = ~/q~. + 1~(~(+/; 8), n),
(4.9)
where l?(~(-q; e), +7) satisfies the inequality
l[ r
~), ,7)112 ~< c{I,71(I,,(,~; +)l + 1,71) + I~l}
(4.10)
252
M. Mimura and Y. Nishiura BO
B>O LuO
w 0
./'or some positive constant c independent of ~, ~ and 8. The relation among ~, ~ and 8, that is the perturbed bifurcation diagram, is determined by the following scalar equation: ~
+ ~v8 + o,8 + r
7, 8) = 0,
(4.11)
where ~:(cr,"1, 8) is a higher order term with respect to ~, ~ and 8. From each fixed small e > 0, (4.11) gives the relation between ~ and ~7, which is drawn in Figure 3. From the above theorem, we found that two different types of solutions exist. One is of ordinary type, the other is of snap-like type. We call the former S-type and the latter U-type. Next we study the stability of V(-q; e). It can be discussed by making use of (4.11) with 8 = 0. Theorem 4.2. (Linearized stability). I f ~ < 0, then the S-type solutions are stable in the linearized sense, i.e., all eigenvalues of the linearized operator of (4.7)~ at the S-type solution have negative real parts, l f fl > O, then they are unstable. Remark 4.3. One can also discuss the stability of U-type solutions and the nonlinear stability of both types (see Kielh6fer (1974) and Mimura, Nishiura and Yamaguti (1977)). Thus, it is found that the sign of fl plays an important role to determine stability. In general, however, it requires fairly intricate calculations to know its sign. For some cases a useful criterion is given in Mimura, Nishiura and Yamaguti (1977). Using the above results, we intend to study the basic property of pattern formation that the polarity-pattern of the tissue is determined by the forms of source densities p(x) and p'(x). For a simple biological example, they are both assumed to be monotone decreasing functions. In this case, the region of higher density distribution, the terminal x = 0, is regarded as the head part of the tissue. Here the following problem occurs: Starting from even distributions of activator and inhibitor, can shallow gradients of p(x) and/or p'(x) produce a striking pattern in activator with higher concentration at the head part x = 0 ? This phenomenon was already confirmed numerically (Gierer and Meinhardt (1972)) (see Figure 4). We will investigate it by studying the stable perturbed bifurcating solutions V(V; ~) with
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
253
Fig. 4. Stable distributions of activator ( ) and inhibitor ( - - - - ) with a shallow gradient in source densities p(x) = p'(x) (^ ^ A A). (From Gierer and Meinhardt (1972)) fundamental mode (n = 1). We note here that fl < 0 holds for n = 1 under the condition d~ ,~ dh which was assumed by Gierer and Meinhardt (1972) (see Mimura, Nishiura and Yamaguti (1977)). To clarify the discussion, we specify that e is positive and the first c o m p o n e n t o f q~l and q~* are both m o n o t o n e decreasing on i. Thus, (4.9) implies that if 7/ > 0 (resp. < 0), the end x = 0 (resp. x = l) becomes head part. Therefore, from the perturbed bifurcation diagram (see Figure 3), we find as a direct consequence o f T h e o r e m 4.1,
Proposition 4.1.
Consider stable S-type solutions with n = 1 o f the problem (4.2)8. I f oJ > 0 (resp. < 0), then x = 0 (resp. x = l) is the head part. We discuss a particular case for p(x) and p'(x) (see Figure 4).
Proposition 4.2.
Suppose that p(x) = p'(x). I f p(x) is monotone decreasing on i, then
~o>0. Proof. F r o m simple calculations:
a~ ' G(0) = t(p(x)p0 +/~(1cp(x) + Ka2)
c,p,(x)a~)
and t
q)* = --7-
d~
+ v, h2(1 + K~2)/ cos -7-'
where k* ( > 0) is a normalizing constant such that [Ir ~o = (G(0), qb*) i> k*po
fOIp(x)
II =
1. Thus we have
cos ~X 7 " dx
and hence co > 0, which completes the proof.
[]
Let us restate this proposition in a biological language. W h e n a nearly flat gradient (0(e)) is introduced into source distributions such that the end x = 0 is head part, prominent stable patterns o f activator and inhibitor (0(3x/~)) can be recognized with higher concentrations at the end x = 0, which is in agreement with numerical results in Figure 4.
254
M. Mimura and Y. Nishiura
Remark 4.4. Even if slight random fluctuations are introduced in strictly monotone source distributions we can obtain similar results to that of Proposition 4.2, because the quantity co is defined by the inner product between G(0) and ~*.
5. A Global Continuum of the Perturbed Bifurcating Branch
In this section we show the global existence of a continuum of the perturbed bifurcating branch obtained in Section 4. First we define the global path of the parameter D = (d~, d2) in R2+. For simplicity, we consider a path parameterized by dl only. Definition 5.1. We say D = D(dl) = (d~, d2(d0): R~+ ~ R2+ is a global path in the parameter space Ra+ when it satisfies: d2 = d2(dz): R~+ --> ~+ is a smooth function.
(H.8)
D intersects transversally with I' at only one point and it is not an intersection point of two curves o f {C,}. (H.9) D contains a point satisfying the inequality in Remark 3.2.
(H.IO)
Remark 5.1. We note that (H.10) always holds when D has the property that d2(dl) ~ + oo as dl ~ -t-~. So far we needed only local properties in the vicinity of a constant solution to study the perturbed bifurcating branch. To study the global structure of (4.7)~, we need global properties of the system. Thus we assume in addition to (H.4), There exists an invariant rectangle C in which there is a unique constant stationary solution V = 0, and on OC, the vector field corresponding to the non-linear term of (4, 7)0 is always directed strictly inward. (H.11) We find, in Section 3, that (1.2) satisfies these conditions. Before showing the main theorem, we introduce some notations: 5~ denotes the closure of the set of solutions of (4.7)~ in R~+ x (H~) 2 and Proj denotes the projection onto the first component of R~+ x (Hg) 2, that is, Proj (dl, V) --- d~ for (d~, V) e
R~+ x (H~,)~. The global existence theorem of a continuum along the path defined above is the following: Theorem 5.1. Let ~ denote the component (a maximal closed connected set) in 50 which contains the S-type branch (resp. U-type branch) when fl < 0 (resp. fl > 0). Then there exists a positive constant eo such that for each fixed e with 0 < It[ < e0
g n {dl} x ((H~) 2 :~ C)} ~ c~ for any dl e R~+, that is, Proj (8) = (0, + oo).
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
255
Remark 5.2. The branch near d ~ is obviously contained in the interior of C. Suppose that the component g comes into contact with the boundary of C when dx continuously varies from d ~ Defining the solution in this situation by U*(x), we know that there exists an x* satisfying U*(x*)E aC. Using (H.11), it is seen that U* cannot be a solution of (4.2)~, which is a contradiction. Thus, we find that any solution in g is contained in the interior of C. The proof of Theorem 5.1 essentially depends on the result of Remark 3.2, that is, non-existence of non-homogeneous solutions of (4.7)0 for large diffusion coefficients and on the method of Leray-Schauder degree which was developed in the important paper by Rabinowitz (1971). We begin by showing five lemmas so as to prove Theorem 5.1. Hereafter we fix a global path D such that it intersects with C, of I" at D O = (d~ d2(d~ and denote by Do the part of this global path for dl /> d ~ + 8 with a positive constant 3. Lemma 5.1. For any fixed 8 > O, the operator norm of
from (LZ)z into (H~) 2 is uniformly bounded on Dn. Proof. See Appendix. /_,emma 5.2. For any 8 > O, there exist positive constants c = c(8) and e = 8(8) such that for D(dl) E Do and e with lel < e(8), there exists a unique solution V(dl, e) of the problem (4.7)~ in l~c~o). V(dl, e) is continuous in (H~) z with respect to dl and e for O(dl) ~ Do and lel < e(8). Proof. Operating w i t h / ( i on (4.7),, we obtain the equation V + K I H ( V ) + eKlG(V) = 0
(5.1)
in (H~) 2. It is noted that (V, dl, e) = (0, d~, 0) is a trivial solution of(5.1) and that the Frech& derivative of the left-hand side of (5.1) with respect to V at (0, dl, 0) with D(d~) ~ Do is the identity operator. Therefore, applying the implicit function theorem to (5.1), we find that there exist positive constants Co and e0 such that for D(d~) ~ Do and [e I < e0, Eq. (5.1) has a unique solution V(d~, e) in /~c0 which depends smoothly on d~ and e (see, for instance, Crandall and Rabinowitz (1971)). Since the operator norm of/{1 is uniformly bounded on Do, it is seen that e0 and Co depend only on 3, which completes the proof. [] The solutions obtained in the above lemma are called the perturbed trivial branch. Lemma 5.3. (A priori estimate). Let V be any solution of(4.7)~ in (H~) 2 n C. Then
the following estimate holds:
where M2 is a positive constant depending on C and the coefficients of(4.7)..
256
M. Mimura and Y. Nishiura
Proof. Since the existence of C leads to L~~ on V, it is easy to obtain the result by using the equation (4.7),. So we omit the details. [] Lemma 5.4. (Uniqueness in the large). Suppose that the diffusion coefficients D =
(dl, d2) satisfy Min (dl, d2) > ~1 + M1
(see Remark 3.2).
Then there exists a positive constant e1 such that for e satisfying l,l < ~1, (4.7), has a unique solution in (H~) 2 n C. Proof. From Lemma 5.2, we found that (4.7), has a unique solution in the neighborhood of the origin, say in /~_~.Therefore it suffices to prove the non-existence of solutions of (4.7)~ outside this neighborhood when e is sufficiently small. Suppose that there are solutions of (4.7)~ in (H~) 2 n C - /~ for any small e. Then we extract a sequence of solutions { V J of (4.7), w i t h , = e,, which satisfy
< LIv..ll2
e
for a positive constant ~ independent of e, where e, --~ 0 as n -+ + oo. Thus, we can extract a weakly convergent subsequence {V~.,} in (H~) 2, which also converges strongly in (H~) 2. Since V,, is a solution of (4.7)~ and H~ = C~ {V~,} is indeed a strongly convergent sequence in (H~) 2. Therefore, it is seen that the limit function V* of {V,.,} is a solution o f (4.7)o in (H~) 2 n C, and that
_c
IIV*ll
< o.
On the other hand, from the assumption of this lemma and (H.11), Eq. (4.7)o has only one solution V = 0 in (H~) 2 n C. This is a contradiction, which completes the proof. [] Lemma 5.5. For appropriately small ~, the perturbed trivial branch is connected to the
S-type branch (resp. U-type branch), if fi < 0 (resp. fl > 0). Proof. The proof is obtained as a direct consequence of the uniqueness of the perturbed bifurcating branch in Theorem 4.1 and Lemma 5.2. [] Let us prove Theorem 5.1 by using the above lemmas. For the proof the homotopy invariance of Leray-Schauder degree plays an essential role as in Rabinowitz (1971). First, by operating with Kz =
d2 ) -1 -- D~xx2 + ,8"
(5.2)
on (4.7),, it is converted into an operator form
V = ~(dl, V),
(5.3)
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
257
where fr V) = K2(B + J ) V + K2H(V) + eK2G(V). It is noted that fr is a compact continuous operator from R~ x (H~) z into (H~) z. Let us define ~ ( d l , V) by ~ ( d l , V) = V -
if(d1, V).
(5.4)
From (H.10), we can choose dl such that D(dl) satisfies the inequality in Remark 3.2. Then from Lemma 5.2 and Lemma 5.4, it is easily seen that deg (o~f'(d~, .),/}_~) :r 0
(5.5)
for an appropriate positive constant _c. We denote go and 8' 1 by 8 n ( 0 , dl] x (H~) 2 = 8o
and
8n[dl,
+ ~ ) x (H~) 2 = 81,
respectively. P r o o f o f Theorem 5.1. Since 81 contains the perturbed trivial branch from Lemma
5.5, it follows Proj (81) = [dl, + ~ ) . Therefore it suffices to prove Proj (80) = (0, dl].
(5.6)
Suppose that there exists a positive constant s such that Proj (80) = [s, dl], then we have a contradiction in view of the following: We note that 80 is compact in R~- • (H~) 2 from Lemma 5.3 and the compactness of re. Here we use the following lemma given by Rabinowitz (1971). Lemma 5.6. There exists a bounded open set C) in (s/2, + oo) x ((H~) 2 c~ C) such that 80 = 6~ 8(~al ) c~ 5~ = ~ f o r any dl ~ (O dl] and ~3~_c ~o~, where 6~a~ = {U~ (H~)~l(dl, U) ~ 0} and 8(0a~) is its boundary.
From this lemma, deg ( ~ ( d l , -), ~~ is well-defined for dx e (0, all]. The homotopy invariance of the Leray-Schauder degree implies that deg (W(da, .), d)al) = const. = c for dl E (0, d~].
(5.7)
From Lemma 5.4 and (5.5), we obtain deg (~(d~, .), d?~,) = deg ( ~ ( d l , -),/~_) ~ 0.
(5.8)
This implies c ~ 0. On the other hand, deg(~,~(dl, .),d?d~)=0
for d~ ~ (0, 2).
This contradicts (5.7) and (5.8), which completes the proof.
[]
258
M. Mimura and Y. Nishiura
6. Concluding Remark This paper is devoted to two problems. One is the determination of the polarity in the tissue, which is interesting from a biological point of view. It is found that slightly asymmetric source distributions produce striking stable patterns depending on the asymmetry. Our analytical results are in agreement with Gierer and Meinhardt's numerical experiments. The other is the existence of a continuum of perturbed bifurcating solutions when one of diffusion coefficients is adopted as a bifurcation parameter in (0, + oo). We confined our study to Gierer and Meinhardt's model. However, there is apparently no such restriction in our arguments. In fact, from Section 3 on, we carried out our discussion in a general form JDVxx + F(x, V) = O,
V = (v~, v~).
The global continuum, in essence, can be obtained if the following conditions hold: For local properties, 1) there exists one constant solution, say V = 0, 2) the system satisfies Turing's condition at V = 0 (see (H.4)), and 3) F(x, O) # O. For global properties, 4) there exists an invariant set which contains the point V = 0 in R a. Finally, our global uniqueness result (see Lemma 5.4) leads to the interesting fact that the global branch which contains unstable perturbed bifurcating solutions (or bifurcating ones when e = 0) must return to the opposite direction (see Figure 5 when 8 = 0).
o-
stable unstable
Pig. 5. Recovery of stability of the unstable branch with e=0
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
259
Appendix Proofs o f Theorem 4.1. and Theorem 4.2.
We first state two lemmas without proofs (see, for instance, Sather (1973)). Lemma A.1. (1) Lo: ~(Lo) --~ (L2) ~ is a linear Fredholm operator. (2) The null space N(Lo) o f Lo and the null space N(L *) of i t s adjoint operator L* are both one-dimensional and they are spanned by the vector a~ = k.$~ cos mrx --7-
and
(I)* = k'd,* cos nrrx 7""
respectively, where ~ = ~(d~ 2 - b22, b21) and
$* = t(d~ 2 - b22, blz)
and k~ and k* are normalizing constants such that 11(I)~[I = ]1(I)* 1[ = 1 (for definiteness, we take kn > 0 and k* > 0). (3) Lo is a one-to-one mapping of ~(Lo) n N(Lo)" onto ~(Lo) (range o f Lo) so that K = (Lo]~(Lo)~mLo)')-1 is well defined, and K T is a linear compact operator defined on (L2) 2. Here T is defined in the next lemma.
Lemma A.2. Let P be the orthogonal projection operator of(L2) 2 onto N(Lo) and T be that o f (LZ) 2 onto ~(Lo). Then it is true that Pw = ( w , *~)*~ KLow = (I - e ) w
and
(I - T)w = (w, dp*)**,
f o r all w e ~(Lo),
(1) (2) (3)
LoKTw = Tw for all w ~ (L2) 2.
Using the projection P, we write V as (A.1)
V = P V + (I - P ) V = ~ap~ + 17, where ~/is a real parameter. Substituting (A. 1) into (4.7)~, we obtain
LoI7 + A(q)(V(I)~ + l?)x~ + H(~7(I)~ + I~) + ,G(v(I). + l?) = 0, where A ( a ) = D ( q ) -
(A.2)
D(0). Using the projection T, the following system is t
equivalent to (A.2) Lo I? + TA(a)(~(I). + l?)~ + TH(71ap~ + I7) + eTG(~*~ + l?) = 0
(A.3.1)
and (I-
T){A(~r)(~*~ + I7)~ + H(~7. n + l?) + ,G07dp, + I7)} = 0.
(A.3.2)
The property (2) of Lemma A.2 reduces (A.3.1) to + KTA(~)(,Ta,~ + ~ ) ~
+ KTH(n*~ + ~) + ,KT6(n*~ + ~') = 0.
(A.4)
From the assumption (H.6), A(cr) can be represented by a(q) = (rX + 0(~),
(A.5)
260
M. M i m u r a a n d Y. Nishiura
where x is a diagonal matrix with the elements {x~}~=~.2and (x~, x2) is not parallel to the tangent of (', at D(0), and O(a) is a higher order term. Then (A.4) is rewritten as
r + aKTx07r + KTH(~r
+ r + KTO(a)(Vr + 1~)~ + 17") + eKTG(~7r + 12) = O.
(A.6)
Applying a standard implicit function theorem in Banach space (see, for instance, Crandall and Rabinowitz (1971), Dieudonnd (1960)) to (A.6), we can get the unique solution l?(a, ~7, *) of (A.6). The form of the principal part of l?(a, V, 8) for sufficiently small or, ~ and e is represented by
r
~, 8) = o~Tn2KTxCb~ - ~?2KTHo(aP,, 0 , ) - ,KTG(O) + V(a, .,7, e),
(A.7)
where V(a, ~7, e) is a higher order term. Substituting the solution r ~7, e) into (A.3.2), we obtain the scalar equation with respect to a, ~ and e, which is called the
perturbed bifurcation equation ( I --
T)(A(a)(V~P, + 12(a, zl, e))xx + H(71cb,~ + 12(a, 7/, e)) + ~G(~r + r ~, ~))) = 0.
(A.8)
Using Lemma A.2 and (A.5), Eq. (A.8) can be written as
(aX(~?aP~ + IT(a, ",1,e))xx + O(a)(~cb~ + If(a, ~7, *))xx + H(~aP~ + 9(a, '7, *)) + eG(~/r + 12(a, ~/, e)), 0 " ) = 0. (A.9) By making use of the expansion of H ( V ) and (A.7), we have
-- awn2(xcb,, r
+ ~2(HQ(r On), r - HQ(KTHQ(Cbn, r
+ ,/S(Hc(~P~) - HQ(r KTHQ(rb,, r r r + ,(G(0), r + $(a, n, e) = 0, (A.IO)
where r ~7, *) is a smooth higher-order term with respect to a, ~7 and 8. Noting (HQ(ap~, qb), dp*) = 0, we obtain the perturbed bifurcation Eq. (4.11). Here we set ,~ =
O*),
-~,n2(xO,,,
fl = (Hc(r
- Ho(r
KTHo(fb,, r
- Ho(KTHQ(r
r
r
r
and
o, = ( 6 ( 0 ) , ~*). Wc sec a > 0 from the definition of the path D = D(a). From now on we assume fl:/:0
and
oJ#0.
(A.11)
These are generic conditions, so almost all non-linearities satisfy (A. 11). In fact, there hardly appears the case/3 = 0 and/or oJ = 0 in applications.
Spatial Patterns for an Interaction-Diffusion Equation in Morphogenesis
261
Applying a smooth change of coordinates from (a, ~7, e) to (#, 9, ~) (we can take ~/ = 9) to (4.11), whose Jacobian at (a, 7, e) = (0, 0, 0) is 1, we get the following normal polynomial form of (4.11): ~e~ + flcfl + oJ~ = 0.
(A.12)
From (A.12), the picture of the relation between a and -q for any fixed small e is given by Figure 3. For details we refer to Nirenberg (1973). Theorem 4.1 is the direct consequence of the above results. Proof of Theorem 4.2 First we note that (4.7)~ can be converted into the compact operator form (5.3). Using a similar method as in the proof of Lemma 3.2 in Sattinger (1971), it is seen that the linearized operator at V(~7;e) corresponding to the solution of (4.11) with cr'(~) # 0(' = dido) and ~7 # 0 is invertible. Therefore the linearized operator along the S-type solution is invertible for all sufficiently small ~. For e = 0 and cr < 0, the spectrum of the trivial solution is of stable type, i.e., all the eigenvalues of (4.3) have negative real parts. The Continuous dependency of an eigenvalue with respect to implies that the spectrum at the S-type solution for ,r < 0 and sufficiently small E is of stable type. Combining this result with the invertibility along the S-type solution, we get the linearized stability of the S-type solutions. [] Proof of Lemma 5.1 For the proof, it suffices to show the existence of a constant K which only depends on 3 such that for any F ~ (L2)2 with HFI[ ~< 1, there exists a solution V~ (H~) 2 of D)-~x2 + B V = F,
(A.13)
which satisfies
II vii2
(A.14)
K
for arbitrary D ~ Do. This will be done by applying Fourier-cosine series expansion to (A.13). Inserting the expressions V=
~ ~o V. cos .=o
l'
9
F=
F. cos •=0
l
into (A. 13), we obtain the system of equations for V. and F., M . V . = F.,
(n = 0, 1, 2,),
(A.15)
where ~ - d w n 2 + bll M~ = \
b21
b12
)
- d 2 y n 2 + b22 "
262
M. M i m u r a a n d Y. N i s h i u r a
Since D is in the stable region ft. (see Section 4), M , is non-singular for all n. Therefore it follows V~ = M y tF~,
(A.16)
where
1 ( - d a y n a + baa M~-x = det--~ \ -bax
-b~a ) - d w n ~ + b~x
and det M,, = (b~t - dx~,na)(b~ - d~,n ~) - bx~b~x. To obtain the estimate (A.14), it suffices to show the following inequalities
I V.I < r
(A.17)
and vn~lV.I 1 t*).
From the definition of fl~, it is easily seen that for any 8, D~ is contained in tl~ for sufficiently small #. For any D e tl~, we obtain the following estimates:
I(bxl -
yn2(b2z - d2yn') t I.L]d2yn 2 -
b221 >
~lb=~l. (A.21)
O n the other h a n d , the left-hand side of (A.20) is equal to {Tn2{(d~ -
b~/yn2)(d2
-
b 2 2 / y n 2) -
b~2b2~/(yn2)2}} -1.
Therefore (A.21) implies the required estimate (A.20). F r o m the form o f M g 1 of (A.16), the estimates (A. 18)-(A.20) lead to the second inequality (A. 17). Thus the p r o o f is completed. [ ]
References Auchmuty, J. F. G., Nicolis, G.: Bifurcation Analysis of Nonlinear Reaction-Diffusion Equation-L Evolution Equations and the Steady State Solutions, Bulletin Math. Biol. 37, 323-365 (1975) Chueh, K. H., Conley, C. C., Smoller, J. A." Positively Invariant Regions for Systems of Nonlinear Diffusion Equations, Indiana Univ. Math. J. 26, 373-392 (1977) Conway, E., Hoff, D., Smoller, J. A. : Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations, SIAM J. Appl. Math. 35, 1-16 (1978) Crandall, M. G. Rabinowitz, P. H.: Bifurcation from Simple Eigenvalues, J. Func. Anal. 8, 321-340 (1971) Dieudonn6, J. : Foundation of Modern Analysis, Academic Press, New York, 1960 Friedman, A.: Partial Differential Equations of Parabolic Type, Prentice-Hail, Englewood Cliffs, N.J., 1964 Gierer, A., Meinhardt, H.: A Theory of Biological Pattern Formation, Kybernetik 12, 30-39 (1972) Granero, M. I., Porati, A., Zanacca, D.: A Bifurcation Analysis of Pattern Formation in a Diffusion Governed Morphogenetic Field, J. Math. Biol. 4, 21-27 (1977) Kielh/Sfer, H. : Stability and Semilinear Evolution Equations in Hilbert Space, Arch. Rat. Mech. Anal. 57, 150-165 (1974) Mimura, M., Nishiura, Y., Yamaguti, M.: Some Diffusive Prey and Predator Systems and Their Bifurcation Problems, to appear in the Annals of the New York Academy of Sciences (1977) Nirenberg, L. : Topics in Nonlinear Functional Analysis, Lecture Notes, Courant Institute of Mathematical Sciences, New York University, 1973 Rabinowitz, P. H. 9Some Global Results for Nonlinear Eigenvalue Problems, J. Func. Anal. 7, 487-513 (1971) Sather, D. : Branching of Solutions of Nonlinear Equations, Rocky Mount. J. Math. 3, 203-250 (1973) Sattinger, D. H.: Stability of Bifurcating Solutions by Leray-Schauder Degree. Arch. Rat. Mech. Anal. 43, 154-166 (1971) Turing, A. M. : The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. 13237, 32, 37-72 (1952) Received July 5~Revised October 16, 1978
