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MATHEMATICAL
PERGAMON Mathematical and Computer Modelling 29 (1999) 53-66
COMPUTER MODELLING
Diffusion Driven Instability in an Inhomogeneous Circular Domain
A. MAY, P. A. FIRBY AND A. P. BASSOM* School of Mathematical Sciences
University of Exeter, North Park Road Exeter, Devon, EX4 4QE, U.K.
(Received and accepted July 1998)
Abstract-classical reaction-diffusion systems have been extensively studied and are now well understood. Most of the work to date has been concerned with homogeneous models within one- dimensional or rectangular domains. However, it is recognised that in most applications nonhomo- geneity, as well ss other geometries, are typically more important. In this paper, we present a two chemical reaction-diffusion process which is operative within a circular region and the model is made nonhomogeneous by supposing that one of the diffusion coefficients varies with the radial variable. Linear analysis leads to the derivation of a dispersion relation for the point of onset of instability and our approach enables both sxisymmetric and nonaxisymmetric modes to be described. We apply our workings to the standard Schnsckenberg s&w&or-inhibitor model in the case when the variable diffusion coefficient takes on a step-function like profile. Some fully nonlinear simulations show that the linear analysis captures the essential details of the behaviour of the model. @ 1999 Elsevier Science Ltd. All rights reserved.
Keywords-Two chemical reaction, Schnsckenberg model, Step-like diffusivity.
1. INTRODUCTION
The seminal work by Turing [l] was the first to propose that reaction-diffusion systems might
be suitable candidates as theories for explaining the mechanisms behind pattern formation in a
variety of biological phenomena. In this paper, Turing showed that under favourable conditions
diffusive effects could actually promote patterning rather than acting as a suppressor. Although
this is somewhat counter-intuitive, the relevance of reaction-diffusion mechanisms to a wide class
of biological systems is now almost universally accepted. In consequence, here we do not propose
to discuss the details of classical reaction-diffusion problems, but instead direct the interested
reader to the reviews [2] or [3].
Although Turing’s theory provides a plausible route to biological patterning, it has been pointed
out [4] that it presents certain difficulties. The crux of these drawbacks is that the original
formulation was concerned with homogeneous backgrounds (that is, all parameters in the model
are constant across the domain) and this means that solutions are inevitably symmetric and fill
*Author for correspondence. We are indebted to an anonymous referee whose comments led to substantial im- provements in the presentation of this work. AM was funded by an EPSRC research studentship while undertaking this work.
0895-7177/99/$ - see front, matter @ 1999 Elsevier Science Ltd. All rights reserved
PII: s0895-7177(99)00039-4
54 A. MAY et al.
the whole of the available area. Maini et al. [4] also discuss other inadequacies including the fact
that solutions of Turing models are sensitive to initial conditions, can sometimes only lead to
certain observable patterns through a complex sequence of particular bifurcations and are scale
dependent. In response, they speculated that some of these problems could be circumvented
by examination of models based upon inhomogeneous background conditions. There are several
examples in the biological world of mechanisms controlled by spatial variations in underlying
diffusion coefficients (see for example [5]) and despite numerous computational investigations
into the effect of spatial inhomogeneity in various reaction-diffusion systems there is relatively
little analytical work in this area.
In a sequence of papers [4,6,7] two species reaction-diffusion models in one space dimension were
considered in which one of the diffusion coefficients was nonconstant. To ease the analysis this
variation was taken to be of a simple step-function type. Even so this proved to be of sufficient
complexity to provide valuable insight into the problem of pattern generation in backgrounds
with smoothly varying inhomogeneity.
Our objective here is to extend the ideas described in the aforementioned papers along two
main directions. For many biological applications it is clear that one-dimensional or rectangular
domains are too simplistic to be viable as realistic models and so we focus here on circular patches
instead. While this is still admittedly very idealised, it opens up the long-term possibility of ap
plying reaction-diffusion schemes to other classes of observed phenomena. Like Maini et al. [4] we
account for spatial inhomogeneity by allowing one of the diffusion coefficients to take a steplike
dependence, this time in the radial direction. We describe the possibilities for pattern devel-
opments using both linear theory and numerical simulations of the full nonlinear problem. In
concert, we also conduct a theoretical analysis into the effect of inclusion of a second spatial di-
mension into the biological patterning by allowing solutions to acquire dependence on the usual
polar angle. While fully two-dimensional solutions are in practice often associated with some
two-dimensional variation in the underlying diffusion characteristics, the findings of Maini et al.
illustrate how useful information can be forthcoming from even grossly simplified models in which
the diffusion is a one-dimensional function.
The remainder of this work is organised as follows. In the next section, we define our inhomo-
geneous system and formulate the conditions that must be satisfied in order that diffusion-driven
instability is possible. Numerical solutions under these linearised conditions are undertaken in
Section 3 and are based on the Schnackenberg activator-inhibitor model for the chemical reac-
tions. Some solutions of the full nonlinear system are computed in Section 4 and the paper closes
with a short discussion.
2. FORMULATION AND THE LINEAR PROBLEM
Since the basic Turing mechanism has been discussed extensively in the literature, we restrict
ourselves here to a quick reminder of the main elements. Suppose we have two chemical species u
and v which react and diffuse in a circular domain. Relative to the standard (T, 0) polar coordi-
nates the ubiquitous reaction-diffusion equations may be nondimensionalised to the forms
(2.la)
(2.lb)
where t is the time and f and g denote the reaction kinetics for the two species. The diffusion
coefficient for the morphogen u has been taken to be constant (and scaled to unity) while that
for v is assumed to be a function of the radial coordinate T. As has already been mentioned, the
system of interest here is the one for which D( T is nonhomogeneous and in this first study we )
Diffusion Driven Instability 55
use the simple step function type behaviour
D(r) = Din, QIr<E,
Dout, t < r 5 1, (24
where the values Din and Dout are constants and the position of the step r = 6 is assumed given.
Standard arguments (e.g., [2]) may b e used to determine criteria which are necessary for the
diffusionless versions of (2.1) to be linearly stable. These requirements are merely expressed in terms of partial derivatives of the reaction terms evaluated at some homogeneous steady state u = 210, 21 = 210 say. In particular,
a+d<O and ad-bc>O, (2.3)
wherea=$$,b=$,c=g,andd=$$ are all evaluated at (~0, ~0). To analyse the stability
of this state when diffusion is restored we seek solutions of (2.1) with
21 = 210 + 6eime+XtXu(r), 21 = ‘~0 + f5eime+xtXv(r), (2.4
where 6 < 1 and it is noted that some angular dependence has been incorporated when m # 0. Substitution of (2.4) in (2.1) leads to the coupled linear equations
r”Xi + rXL + (r2(a - A) - m2) X, + br2X, = 0,
r (rDXL)’ + (r2(d - X) - Dm2) Xv + cr2X, = 0,
(2.5a)
(2.5b)
where primes denotes differentiation with respect to r and the constants a to d are as before. Equations (2.5) are most easily solved in regions where D is constant (= Din for r < { and Dout for r > <) by adding (si/D) multiples of (2.5b) to (2.5a) (i = 1,2): here si and s2 are the roots of the quadratic
cs2 + (D(a - X) - (d - X))s - bD = 0.
This reduces (2.5) to equations for X, + six, of the form
(2.6)
r2(Xu + s?X,)” + r(Xu + six,)’ + (r2czi2 - m2) (Xu + six,) = 0, (2.7)
where cy( = (a - X + CS~/D)“~. We notice that this is just a scaled version of Bessel’s equation with general solution Ai& + BiY,( ro. %) f or constants Ai and Bi and where J,,, and Y, are the usual Bessel functions of order m (see [8]). These solutions for X,, + six, and X, + s2Xv enable us to deduce solutions for the individual components X,, and X, and we have separate expressions for the inner and outer portions of the disc (due to the different values of 0).
We are now left with the two pairs of second order equations (2.5): one pair relating to the inner domain where D = Din and the other to the outer region for which D = Dout. To solve the system we need to specify appropriate boundary and matching conditions across r = <. The latter are determined by the physically sensible property that the individual concentrations X,,
and X,, together with the morphogen fluxes XL and DXL all be continuous across the diffusion
step. In biological modelling there is always some debate as to the precise boundary conditions that
ought be applied in any particular circumstance. We note that in the overwhelming majority of studies zero-flux (Neumann) requirements are appropriate. Moreover, these are precisely the constraints that have been imposed in all the related previous works referred to above and, motivated by these facts, we too demand that the fluxes XL and XL vanish on r = 1. However, we are still short of two conditions on system (2.5); after all this is essentidly an eighth-order problem and so far we have four matching conditions across the diffusion jump plus the two
56 A. MAY et al.
boundary conditions now specified for the edge of the disc. In the case of one-dimensional or
rectangular geometries this apparent deficit of boundary conditions does not arise as there are
inevitably two outer boundaries on which the zero-flux requirements are to be applied: here
we have just one. However, it is a routine exercise to show that there are only two linearly
independent solutions of (2.5) which remain bounded at the centre of the disc T = 0 as the other
two are singular there. It is then obvious that for biological realism we can only admit the well-
behaved solutions and those components that are singular must be rejected. Therefore, our lack
of outer boundary constraints are compensated for by the need for regularity of the solution at
the origin: a feature which is absent in earlier studies.
System (2.5) can now be solved using the method described by (2.6),(2.7). The imposition of
the boundary, matching and regularity conditions mentioned above shows, after much tedious
algebra, that our governing system only admits a nontrivial solution if
p(4ru+Q(4rv =o, (2.8a)
R(X)l?, +s(x)r, =o, (2.8b)
where we have denoted X, (<) E rU and XV (<) = rV. Furthermore, the coefficients here are
given by
Q (A> = $‘&” (T;I” - Tp)
in SjLn - s1
+ $“tsyt (Tyt _ Tyt) out ’
32 - ,pt
R(X) G Din (Tp _ Tp) + D”“t (Tyt _ T;Ut)
in St - .sl out s2
_qt ’
s (A> 3
and
T+ 3 @‘t YA (aput) J,!,, (&zp”“) - J,!,, (uyt) YA (<a;““) J, @put) Yh (uyt) - Y, (&put) J,!,, (aput) > ’
(2.9a)
(2.9b)
(2.9c)
(2.9d)
(2.10a)
(2.10b)
In writing these we have used the superscripts in and Out in the obvious way so as to refer to
values of variables in the regions r < < and r > <, respectively. Clearly, (2.8) admits only the
trivial solution unless
F(X) z P(X)S(X) - Q(X)R(X) = 0, (2.11)
and this eigenrelation yields complex values for A.
We are now able to clarify the circumstances under which our system is susceptible to diffusion
driven instability. To recap, first we need to ensure that in the absence of any diffusive effects
model (2.1) is linearly stable: all this requires is that the partial derivatives of the reaction
kinetics terms satisfy (2.3). In addition, it is necessary that the full problem be unstable and
this occurs if and only if the solution X of (2.11) has a positive real part. These two constraints,
when satisfied, ensure that diffusion is a destabilising mechanism.
3. NUMERICAL SOLUTIONS OF THE LINEARISED DISPERSION RELATION
In order to assess the properties of (2.3),(2.11) it is necessary to prescribe some underIying
chemical kinetics model. Over the years a large number of such models have been forwarded and
Diffusion Driven Instability 57
here we elect to use the auto-catalytic, activator-inhibitor sequence as suggested by Schnacken- berg. The governing kinetics of this reaction are given by the relatively simple forms
f(u,v)=~(A-u+u2v), 9(% u> = Y (B - u2u) 7 (3.1)
where 7 is an appropriate scale factor and A and B are constants whose values are at our disposal.
The homogeneous steady state of system (2.1) is then ‘1~ = ZLO = A + B, v = vg = B/(A + B)2 and with the choices A = 0.1, B = 0.9 it is guaranteed that conditions (2.3) are satisfied so that diffusion-driven instability is possible. Of course other values of A and/or B would also suffice, but for all the computations henceforth these choices were always made: it is most unlikely that other pairs (A, B) could lead to substantial changes in the qualitative behaviours of the solutions.
Our first considerations are for the axisymmetric problem: that is we consider forms of 21 and u that depend only on r (and hence m = 0). Eigenrelation (2.11) was solved for various diffusion coefficients D’” and Dout with the diffusion step taken at < = l/2. For diffusion-driven instability we are only interested in those regions of parameter space for which %(A) > 0 and the results sketched in Figure 1 show that the only region in Di”/Dout space for which this is not the case is almost rectangular in shape. Instability is, therefore, possible outside this domain and it is noted that this Turing space is similar in both size and form to that predicted by [4] in regard to instability within a rectangular area.
6-
5-
4 5 6 7 8 9 10
D’” Figure 1. The Turing space for sxisymmetric modes associated with Schnackenberg kinetics (3.1) with 7 = 1040, A = 0.1, B = 0.9, and the step in the diffusion coefficient located at r = 6 = l/2. Instability can be induced by diffusion for combinations of Din and DoUt lying above and to the right of the curve.
Unstable solutions for the functions Xu(r) and X,,(T) are illustrated in Figure 2 for selected
pairs of diffusion coefficients. The eigenfunctions are chosen so as to show the variety in possible structures and there appears to be two distinct types of behaviours. (We remark that as this is a linear analysis there is arbitrary scaling of the eigenfunctions: our results have been plotted so
58 A. MAY et al.
-1.0 / I I I I I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(a) Din = 7, Dout = 10.
-1.0 1’ I I I I I I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(b) Din = 10, Put = 10.
Figure 2. Representative solutions X,,(T) (bold line) and X,(r) (dotted) of the linearised Schnackenberg system. Other parameter values are 7 = 1040, c = l/2, A = 0.1, and B = 0.9. All solutions have been scaled to have a maximum fluctuation of unity from the steady state.
that the maximum value taken either by X,, or X, across 0 < T 5 1 is unity. Furthermore, it must be remembered that what is shown is a small deviation from the steady state (~c,2rc), see (2.4), so negative values of the eigensolutions are physically sensible.) The first family of solutions occurs when both diffusion coefficients are of appreciable (but different) size: the eigensolutions then fill the whole of the domain although they are distinctly larger in one half of the disc than the other. Figures 2a and 2b show these types of behaviour (with the largest values in different parts of the disc). Figures 2c and 2d together show the form of the second type of eigensolution
Diffusion Driven Instability 59
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(c) D’” = 1, Put = 10.
0.4- ‘Y,,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(d) Din = 10, Dout = 4.
Figure 2. (cont.)
which is much more localised and tends to be isolated in one part of the disc. It is remarked that although X,, XV and Xl are all continuous across the step at r = <, XL is not (except in the
degenerate case Din = Dout) in view of the continuity of flux imposed there. (This discontinuity is particularly evident in Figure 2c.) Obviously, there is no definitive distinction between the two fundamental forms of solution. However, the fact that a range of possibilities exists is of importance since this behaviour is precluded in the standard Turing theory for a homogeneous
medium. One question of interest is the effect the positioning of the diffusion step may have on the struc-
tures of unstable modes. Figure 3 shows some linear&d eigensolutions when the discontinuity is
60 A. MAY et al.
-1.0 y I I I 1 I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(a) Din = 16, Dout = 1, e = l/4.
l.O-
0.5-
o.o- . . . . . . . .
-0.5-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(b) Din = 16, Dout = 2, < = 314.
Figure 3. Solutions of the linearised model with zero flux across the outer boundary showing the effect of changing the location of the diffusion step at T = E. As before other parameter valuea are y = 1040, A = 0.1, B = 0.9, and the functions X,, and X, are represented by the bold and dotted lines respectively.
either moved closer to the centre of the disc or towards the boundary. The same generic types of solution are observed as before--some localised forms are illustrated in Figures 3a and 3b while examples of the global variety can be seen in Figures 3c and 3d. No radically different solutions were found during a thorough investigation into ranges of Din, Dout, and <; this lends credence to our earlier assertion that the location of the discontinuity will affect the quantitative details of solution behaviours, but will not influence broad qualitative features.
Diffusion Driven Instability 61
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(qin = 16, DO”t = 12, < = l/4.
I I I I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r
(d) Din = 16, Dout = 16, < = 3/4.
Figure 3. (cont.)
Next we switch attention to nonaxisymmetric modes. In Figure 4, we map out the Turing spaces for linear instability associated with a range of mode numbers m. The somewhat unexpected observation is the very tiny effect m has on the geometry of the domain (at least for small values of m). For a wide selection of biological and physical phenomena there is often some mechanism at work which leads to preferential amplification of some particular mode: often the axisymmetric one. Here there seems to be no such favouritism and indeed, for Put N 9, as D’” is gradually increased, so nonaxisymmetric modes are the first to lose stability. Conversely, for Din N 9, as Dout grows so sxisymmetric structures are observed first. We also note that only when m 2 5
62 A. MAY et al.
lo-
8-
6-
5-
;I :I
i
;I \ ;I i 'iI i
iI i
iI i
;I i i! i : I ;I
!
;I \
4 i I I I I I I
4 5 6 7 8 9 10 11 D'"
Figure 4. The Turing space for nonaxisymmetric modes associated with Schnack- enberg kinetics with 7 = 1040, { = l/2, A = 0.1, and E = 0.9. Shown are the curves delimiting the stable and unstable regions of Din/Dout parameter space for azimuthal wavenumbers m = 0, 1,5,7, and 10.
are there any significant changes to the stability boundary. Moreover, it appears that the upper
part of the boundary is almost completely unaltered by variations in m while the right-hand side
is more markedly affected. In particular, it can be tentatively concluded that for modes with m
greater than about ten the value of Din is largely irrelevant in determining whether diffusion-
driven instability is possible. There is no obvious mathematical feature of the dispersion relation,
nor any physically sensible property, which would explain why these two different behaviours
occur. If the diffusion step is located at positions other than < = l/2 then the corresponding
Turing spaces are similar to those illustrated in Figure 4. Thus, broadly similar conclusions may
be drawn, although some of the quantitative details will naturally be different.
Examples of solutions for the functions X,, (T) and X,, (r) for m = 5 and m = 10 can be
seen in Figure 5. We note that for small m (in particular m = 5 in Figures 5a and 5b) we
obtain qualitatively similar solutions to those from the axisymmetric case, including both isolated
solutions and solutions of varying amplitude. As m is increased though, see Figures 5c and 5d
for m = 10, so it becomes much harder to produce solutions that are not isolated in the outer
part of the domain. Further investigations into solutions of ever larger values of m have not been
studied as it has already been noted that these are of relatively little significance when compared
to either axisymmetric solutions or solutions of small m.
4. DISCUSSION
The objective here has been to extend results concerning the effects of spatial dependence
of diffusion coefficients in two-chemical reaction-diffusion processes from rectangular domains
to circular patches. As in Maini et aE. [4] this spatial inhomogeneity has been modelled by a
Diffusion Driven Instability 63
l.O-
-0.5-
-1 .o
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(a) Din = 1, Dout = 12, and m = 5.
0.6-
0.4-
-0.8 -
r
(b) Din = 12, Put = 12 and m = 5.
Figure 5. Nonaxisymmetric solutions of the linear&d model subject to zero flux boundary conditions. Graphs show the functions X,, (bold line) and X,, (dotted). Other parameter values are 7 = 1040, 6 = l/2, A = 0.1, and B = 0.9.
jump in one of the relevant diffusion coefficients and a variety in the solution types has been noted. Apart from the different geometry, a key component of our study has been the attempt to explore modes with dependence in two spatial dimensions and not just to restrict attention to purely axisymmetric modes.
Of course while the determination of linear instability parameter regimes is of interest, such an analysis is unable to guide us as to which mode might be the primary one in any particular application. Such questions can only be resolved by recourse to numerical experiments on the
64 A. MAY et al.
I I I I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(c) Din = 1, Dout = 12, and m = 10.
l.O-
0.6-
0.6-
0.4-
-0.4-
-0.6-
-0.6 I I I I I I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
r
(d) Din = 12, Dout = 12 and m = 10.
Figure 5. (cont.)
full nonlinear system and in general it will be the case that the evolution of the chemical concen- trations are sensitive to the details of the initial conditions. In the absence of such initial-value calculations, which would represent a formidable task for the two-dimensional modes discussed here, we take the opportunity to show in Figure 6 some representative solutions of the nonlinear steady state equations (2.1) for the axisymmetric case. These solutions demonstrate that the nonlinear forms of solutions are sensibly approximated by the linearised forms described earlier: our nonlinear axisymmetric solutions shown in Figures 6a and 6b are clearly of the global and localised types, respectively. (Note that once again we are plotting deviations from the steady
Diffusion Driven Instability 65
0.4-l
0.3-
0.2-
-0.2-
-0.3-
-0.4 , 0.0 011 012 013 014 015 0:s 017 oI9
-I 9.9 1.9
r
(a) Din = 7, Dout = 10.
-0.4 1 I I I I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.0
r
(b) D’” = 1, Dout = 10.
Figure 6. Axisymmetric solutions of the time-independent form of the full nonlinear system (2.1) solved subject to zero flux boundary conditions. Graphs show the deviations in the concentration of morphogens u (bold line) and TJ (dotted) from their homogeneous steady state values ug and ~0, Other parameter values are 7 = 1040,
< = l/2, A = 0.1, and B = 0.9.
state values so negative values of X, or XV are permissible.) In passing, it is observed that the wavelength of the oscillations in the chemical concentrations is remarkably constant (cf. the other figures). Thus, even though appropriate choices of Din, Put , and < can lead to wide variations in the solution amplitudes in different parts of the domain, the ‘frequency’ of the oscillations is largely outside our control.
66 A. MAY et al.
The patterns described here are not obtainable via the standard Turing mechanism. Circular,
or near circular domains occur in many biological instances, one example of which being a suitable
model with which to study the healing of open wounds. This initial mathematical study could
lead to the analysis of various biologically significant phenomena where spatial inhomogeneity of
the background is an important ingredient. However, various theoretical extensions could also
be contemplated. Included in these we mention investigation into the effect of a two-dimensional
variation in diffusion coefficients (i.e., D = D(r, 0)) and the computational solution of the nu-
merical initial value problem. Although here the spatial dependence of D has been assumed to
be very simplistic, the robustness of our solution families over the ranges of Din, Dout, and <
parameters suggests that our results are probably not atypical of those relevant to physically mo-
tivated cases. Perhaps, the most surprising outcome of our studies has been the demonstration
that the relevant Turing spaces for axisymmetric and nonaxisymmetric modes are so similar, at
least for smallish azimuthal wavenumbers. Whether there is a more dramatic preference for one
mode over another in problems of real biological concern remains an open issue, but is one of
obvious importance.
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