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MUTUALLY EXCLUSIVE SPIKY PATTERN AND SEGMENTATION MODELLED BY

THE FIVE-COMPONENT MEINHARDT-GIERER SYSTEM

JUNCHENG WEI AND MATTHIAS WINTER

Abstract. We consider the five-component Meinhardt-Gierer model for mutually exclusive patterns and seg-

mentation which was proposed in [11]. We prove rigorous results on the existence and stability of mutually

exclusive spikes which are located in different positions for the two activators. Sufficient conditions for existence

and stability are derived, which depend in particular on the relative size of the various diffusion constants. Our

main analytical methods are the Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical

results are confirmed by numerical simulations.

1. Introduction

We analyze the five-component Meinhardt-Gierer system whose components are two activators and one

inhibitor as well as two lateral activators. It has been introduced and very successfully used in various modeling

aspects by Meinhardt and Gierer [11]. In particular, it can explain the phenomenon of mutual exclusion and

handle segmentation in the simplest case of two different segments. This model has been reviewed and its many

implications have been discussed in detail by Meinhardt in Chapter 12 of [10].

The most important features of this system can be highlighted as lateral activation of mutually exclusive

states. To each of the local activators a lateral activator is associated in a spatially nonlocal and time-delayed

way. The consequence of the presence of the two lateral activators in the system is the possibility to have

stable patterns which for the two activators are mutually exclusive, or in other words, the patterns for the two

activators are located in different positions. It is clear that mutually exclusive patterns are not possible for a

three-component system with only two activators and one inhibitor since mutually exclusive patterns for the

two activators could destabilize each other in various ways. Therefore the lateral activators are needed.

Numerical simulations of mutually exclusive patterns have been performed in [11], [10]. Many interesting

features have been discovered and explained but those works do not give analytical solutions and they are not

mathematically rigorous. To obtain mathematically rigorous results, in this study we show the existence and

stability of mutually exclusive spikes in such a system.

The overall feedback mechanism of the system can be summarized as follows: Lateral activation is coupled

with self-activation and overall inhibition. We will explain this in more detail after the system has been

formulated quantitatively.

A widespread pattern in biology is segmentation. The mutual exclusion effect described in this paper

is a special case of segmentation where only two different segments are present.

segmentation are the body segments of insects or the segments of insect legs. The segments usually resemble

each other strongly, but on the other hand they are different from each other. Segments may for example

have an internal polarity which is often visible by bristles or hairs. This internal pattern within a segment

depends on the position of the segment within the sequence in its natural state. In some biological cases a

good understanding of how segment position and internal structure are related has been obtained. One famous

example are surgical experiments on insects, e.g. for cockroach legs. Creating a discontinuity in the normal

neighborhood of structures by cutting a leg and pasting one piece to the end of another partial leg creates a

discontinuity in the segment structure as some segments are missing their natural neighbors. This forces the

emergence of new stable patterns in the cockroach leg such that all segments get back their natural neighbors.

However, the resulting pattern can be very different from any naturally occurring pattern.

For example for cockroach legs, if the normal sequence of structures within a segment is 123...9, a combi-

nation of a partial leg 12345678 to which the piece 456789 is added first leads to the structure 12345678456789.

Note the presence of the jump discontinuity in this sequence between the numbers 8 and 4. Now segment regu-

lation adds the piece 765 which removes the discontinuity and leads to the final structure 12345678765456789.

This is different from the original natural structure but nevertheless each segment has the same neighbors as

in the natural situation.

In this example which was experimentally verified by Bohn [1], it is not the natural sequence but the normal

neighborhood which is regulated. It is exactly this neighboring structure which can be modelled mathematically

Examples for biological

1991 Mathematics Subject Classification. Primary 35B35, 92C15; Secondary 35B40, 92D25.

Key words and phrases.Pattern Formation, Mutual Exclusion, Stability, Steady states.

1

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2 JUNCHENG WEI AND MATTHIAS WINTER

using the system from [11] which is considered here and this paper can be the starting point to a rigorous

understanding of more complex segmentation phenomena.

Now we give a sociological application of mutual exclusion (see[11]): Consider two families. They can hardly

live in exactly the same house as this would lead to overcrowding and is therefore less preferable. But if

they live in the same street or neighborhood they can support, nurture and benefit each other. Thus this

collaborative behavior can lead to a rather stable situation. Indeed, stable coexisting states with concentration

peaks remaining close but keeping a certain characteristic distance from each other are typical phenomena

which are observed in quantitative models of systems modelling mutual exclusion and they obviously resemble

real-world behavior in this example very well.

This feedback mechanism of lateral activation coupled with overall inhibition can be quantified by formulating

the effects of “activation”, “lateral activation” and “inhibition” using the language of molecular reactions and

invoking the law of mass action. Now we are going to discuss this in a quantitative manner. We will introduce

the resulting model system first and then explain how these feedback mechanisms are represented by the terms

in the model.

The original system from [11] (after re-scaling and some simplifications) can be stated as follows:

g1,t= ?2g1,xx− g1+cs2g2

τrt= Drrxx− r + cs2g2

τs1,t= Dss1,xx− s1+ g1,

1

r

1+ cs1g2

,g2,t= ?2g2,xx− g2+cs1g2

2,

τs2,t= Dss2,xx− s2+ g2.

2

r

,

(1.1)

Here 0 < ? << 1, Dr > 0 and Ds > 0 are diffusion constants, c is a positive reaction constant and τ is

nonnegative time-relaxation constant (in [11] the choice τ = 1 was made).

The x-indices indicate spatial derivatives. We will derive results for the system (1.1) on a bounded interval

Ω = (−L,L) for L > 0 with Neumann boundary conditions. Some results for the system on the real line

(L = ∞) will also be established and they will be compared with the bounded interval case.

The first two components, the activators g1and g2activate themselves locally which is due to the terms g2

and g2

The lateral activators are introduced in (1.1) by the fourth and fifth components s1and s2as follows: To

both the activators, gi, i = 1,2, there are nonlocal and delayed versions si. Now s1acts as an activator to g2

and s2acts as in activator to g1due to the terms s2in the first and s1in the second equation which have a

positive feedback. The expression lateral activation is used since giactivate g3−ilaterally through its nonlocal

counterpart sirather than locally through giitself.

Lateral activation is finally coupled with overall inhibition as follows: The third component r acts as an

inhibitor to both g1and g2due to the term r in the first and second equations which has a negative feedback.

Note also that both the local and the nonlocal activators have a positive feedback on r due to the terms s2g2

and s1g2

This feedback mechanism is a generalization of the well-known Gierer-Meinhardt system [6] which has one

local activator coupled to an inhibitor. We recall that the classical Gierer-Meinhardt system as well as the

five-component system considered here are both Turing systems [13] as they allow spatial patterns to arise out

of a homogeneous steady state by the so-called Turing instability. (Some analytical results for the existence

and stability of spiky Turing pattern for the Gierer-Meinhardt system have been obtained for example in [3],

[4], [5], [9], [12], [14], [17], [18], [19].)

Now we state our rigorous results on the existence and stability of stationary, mutually exclusive, spiky

patterns for the system (1.1).

We prove the existence of a spiky pattern with one spike for g1and one spike for g2which are located in

different positions under the following conditions:

(i) the diffusivities of the two lateral activators are large compared to the inhibitor diffusivity and

(ii) the inhibitor diffusivity is large compared to the diffusivities of the two (local) activators.

We summarize the two main conditions (i), (ii) which guarantee the existence of mutually-exclusive spike

patterns for (1.1) in the following:

1

2, respectively, in the first two equations.

1

2in the third equation.

We assume that

?2<< C1Dr≤ Ds

for some constant C1> 0.

(1.2)

We also prove the stability of these mutually exclusive spiky patterns provided certain conditions are met

which are of the type (1.2) with C1replaced by some new constant C2.

In this paper we consider a pattern displaying one spike for g1and one for g2which are located in different

positions.

In particular, we prove the existence of a mutually exclusive two-spike solution to the system (1.1) if Ds/Dr>

4. We show that this solution is stable if (i) Ds/Dr> 43.33 for L = ∞, or in general if (5.3) holds (condition

for O(1) eigenvalues) and if (ii) Ds/Dr> 4 (condition for o(1) eigenvalues).

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MEINHARDT-GIERER SYSTEM 3

The main results will be stated in Theorem 1 (Section 3) on the existence of solutions and in Theorem 2

(Section 5) as well as Theorem 3 (Section 6) on the large and small eigenvalues of the linearized problem at

the solutions, respectively.

What do these results tell us about segmentation? As a first step, we have proved that in the case of two

segments which we call 1 and 2 the sequence 12 can exist and be stable, and we have found sufficient conditions

for this effect to happen.

The case of n > 2 components will lead to a system with 2n+1 components which is very large and not easy

to handle. Even in the case n = 2 for the five-component system investigated in this paper the analysis becomes

rather lengthy. We expect that, following our approach, we will be able to prove existence and stability of n

spikes in n different locations. We do not see any major obstacle, only the proofs become more technical. We

are currently working on this issue.

The outline of the paper is as follows: In Section 2, we compute the amplitudes. In Section 3, we locate

the spikes and show the existence of solutions. In Section 4, we first derive the eigenvalue problem. Then we

compute the large (i.e. O(1)) eigenvalues and we derive sufficient conditions for the stability of solutions with

respect to these. In Section 5, we solve a nonlocal eigenvalue problem which has been delayed from Section

4. In Section 6, we give the most important steps and state the main result on the stability of solutions with

respect to small (i.e. o(1)) eigenvalues. Sufficient conditions for this stability are derived. The technical details

of the analysis of small eigenvalues is delayed to the appendices. Finally, in Section 7, our results are confirmed

by numerical simulations.

Acknowledgements: The work of JW is supported by an Earmarked Grant of RGC of Hong Kong. The

work of MW is supported by a BRIEF Award of Brunel University. MW thanks the Department of Mathematics

at CUHK for their kind hospitality.

2. Computing the Amplitudes

We construct steady states of the form

g1(x) = t1w

?x − x1

?

?

(1 + O(?)),g2(x) = t2w

?x − x2

?

?

(1 + O(?)),

where w(y) is the unique positive and even homoclinic solution of the equation

wyy− w + w2= 0(2.1)

on the real line decaying to zero at ±∞. Here we assume that the spikes for g1and g2have the same amplitude,

i.e. t1= t2. We often use different notations for the two amplitudes as this will be important later when we

consider stability since there could be an instability which breaks the symmetry of having the same amplitudes.

The analysis will show that t1, t2and x1, x2depend on ? but to leading order and after suitable scaling are

independent of ?. To keep notation simple we will not explicitly indicate this dependence.

All functions used throughout the paper belong to the Hilbert space H2(−L,L) and the error terms are

taken in the norm H2(−L,L) unless otherwise stated. After integrating (2.1), we get the relation

?

which will be used frequently, often without explicitly stating it. We denote

?x − x1

?

Note that g1and g2are small-scale variables, as ? << 1, and r, s1, and s2are large-scale (with respect to

the spatial variable). For steady states, using Green functions, these slow variables, to leading order, can be

expressed by an integral representation.

To get this representation, g1in the last three equations of (1.1) can be expanded as

??

where δx1(x) = δ(x − x1) is the Dirac delta distribution located at x1. Similarly, for g2we have

g2(x) = t2?

Rw

Using the Green function GD(x,y) which is defined as the unique solution of the equation

Rw(y)dy =

?

Rw2(y)dy

(2.2)

w1(x) = w

?

,w2(x) = w

?x − x2

?

?

.

(2.3)

g1(x) = t1?

Rw

?

δx1(x) + O(?2),g2

1(x) = t2

1?

??

Rw2

?

δx1(x) + O(?2),

??

?

δx2(x) + O(?2),g2

2(x) = t2

2?

??

Rw2

?

δx2(x) + O(?2).

D∆GD(x,y) − GD(x,y) + δy(x) = 0,

−L < x < L,GD,x(−L,y) = GD,x(L,y) = 0,

(2.4)

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4JUNCHENG WEI AND MATTHIAS WINTER

we can represent s1(x) by using the fourth equation of (1.1) as

s1(x) = t1?

??

Rw

?

GDs(x,x1) + O(?2).

(2.5)

An elementary calculation gives

GD(x,y) =

θ

sinh(2θL)coshθ(L + x)coshθ(L − y),

θ

sinh(2θL)coshθ(L − x)coshθ(L + y),

−L < x < y < L,

−L < y < x < L

(2.6)

with θ = 1/√D. Note that

GD(x,y) =

1

2√De−|x−y|/√D− HD(x,y),

(2.7)

where HDis the regular part of the Green function GD. In particular, for L = ∞, we have

GD(x1,x2) =

1

2√De−|x−y|/√D=: KD(x1,x2).

(2.8)

In the same way, we derive

s2(x) = t2?

??

Rw

?

GDs(x,x2) + O(?).

(2.9)

Now we compute the last two terms on the r.h.s. of the third equation of (1.1) as follows:

??

and, similarly,

cs2g2

1(x) = cs2(x1)t2

1?

Rw

?

δx1(x) + O(?2) = ct2

1t2?2

??

Rw

?2

δx1(x)GDs(x1,x2) + O(?3)

cs1g2

2(x) = ct1t2

2?2

??

Rw

?2

δx2(x)GDs(x1,x2) + O(?3).

Now, using the third equation of (1.1), we can represent r(x) by the Green function GDr

??

Going back to the first equation in (1.1), we get

?2∆g1− g1+cs2g2

r

To have the same amplitudes of the two contributions in (2.11), we require

r(x) = ct1t2?2

Rw

?2

GDs(x1,x2)(t1GDr(x,x1) + t2GDr(x,x2)) + O(?3).

(2.10)

1

= t1(?2∆w1− w1) +cs2t2

1w2

r

1

+ O(?) = t1

?cs2t1

r

− 1

?

w2

1+ O(?).

(2.11)

cs2(x1)t1

r(x1)

= 1 + O(?).

(2.12)

Now we rewrite (2.12), using (2.9) and (2.10):

cs2(x1)t1

r(x1)

=

1

?(?

Rw)(t1GDr(x1,x1) + t2GDr(x1,x2))+ O(?).

(2.13)

Thus, (2.12), for x = x1, gives

t1GDr(x1,x1) + t2GDr(x1,x2) =

1

Rw+ O(1).

??

(2.14)

In the same way, from the second equation in (1.1), we get

t1GDr(x1,x2) + t2GDr(x2,x2) =

1

Rw+ O(1).

??

(2.15)

The relations (2.14), (2.15) are a linear system for the amplitudes t1, t2of the spikes if their positions state that

the amplitudes x1, x2are known. Note that the amplitudes depend on the positions in leading order as also

the Green function GDrdepends on its arguments in leading order. We say that the amplitudes are strongly

coupled to the positions.

Note that the system (2.14), (2.15) has a unique solution t1, t2since by (2.6)

GDr(x1,x1)GDr(x2,x2) − (GDr(x1,x2))2=

×[coshθr(L + x1)coshθr(L − x2) − coshθr(L − x1)coshθr(L + x2)] > 0

θ2

r

sinh2(2θrL)coshθr(L − x1)coshθr(L + x2)

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MEINHARDT-GIERER SYSTEM 5

for −L < x2< x1< L, where θr= 1/√Dr.

By symmetry, for x1= −x2, we have t1= t2. This is the case we are interested in. But we have not shown

that there are such positions x1, x2, yet. This will be done in the next section.

For the special case L = ∞, we have GDr(x1,x2) =

by

t1+ t2e−|x1−x2|/√Dr=2√Dr

??

Lemma 1. Assume that ? > 0 is small enough. Then for spike-solutions of (1.1) of the type

?x − x1

?

where w(y) is the unique positive and even solution of the equation

1

2√Dre−|x−y|/√Drand (2.14), (2.15) in this case are given

t2+ t1e−|x1−x2|/√Dr=2√Dr

??

Rw,

Rw.

Finally, we summarize the main result of this section

g1(x) = t1w

?

(1 + O(?)),g2(x) = t2w

?x − x2

?

?

(1 + O(?)),

wyy− w + w2= 0

on the real line decaying to zero at ±∞, the amplitudes t1and t2are given as the unique solution of the system

t1GDr(x1,x1) + t2GDr(x1,x2) =

??

where GDis the Green function defined in (2.4).

1

Rw+ O(1),t1GDr(x1,x2) + t2GDr(x2,x2) =

1

Rw+ O(1),

??

3. Existence of Mutually Exclusive Spikes

In this section, we use the Liapunov-Schmidt reduction method to rigorously prove the existence of mutually

exclusive spikes. We will get a sufficient condition on the locations of the spikes.

The problem here is that the linearization of the r.h.s. of the first equation in (1.1) around w1 has an

approximate nontrivial kernel. This comes from the fact that a derivative of the equation (2.1) with respect to

y gives

(wy)yy− wy+ 2wwy= 0.

Thus, wybelongs to the kernel of the linearization of (2.1) around w. Note that the function wyrepresents the

translation mode. Therefore a direct application of the implicit function theorem is not possible, but one has

to deal with this kernel first. This is the goal in this section.

Recall that for given g1,g2∈ H2

in H2(Ω?) satisfying the Neumann boundary condition, by the fourth equation of (1.1) s1is uniquely determined,

by the fifth equation s2is uniquely determined, and finally by the third equation r is uniquely determined.

Therefore, the steady state problem is reduced to solving the first two equations.

We are looking for solutions which satisfy

?x − x1

?

with g1(x) = g2(−x)(x1 > 0). By this reflection symmetry the problem is reduced to determining just one

function: g1(x) = t1w1(x) + v.

We are now going to determine this function in two steps. Denoting the r.h.s. of the first equation of (1.1)

by S?[t1w1+v], which is well-defined for steady states, our problem can be written as follows: S?[t1w1+v] = 0,

where S? : H2

First Step. Determine a small v ∈ H2(Ω?) with

S?[t1w1+ v] = β?dw1

N(Ω?), where Ω?= (−L/?,L/?) and H2

N(Ω?) denotes the space of all functions

g1(x) = t1w

?

(1 + O(?)),g2(x) = t1w

?x + x1

?

?

(1 + O(?))

N(Ω?) → L2(Ω?).

?

Ωvdw1

dxdx = 0 such that

dx.

(3.1)

Second Step. Choose x1such that

β = 0.

(3.2)

We begin with the first step. To this end, we need to study the linearized operator

˜L?,x1: H2(Ω?) → L2(Ω?)

where S

We define the approximate kernel and co-kernel, respectively, as follows:

?

dx

defined by

˜L?,x1:= S

?

?[t1w1]φ,

?

?[t1w1] denotes the Frechet derivative of the operator S?at t1w1.

K?,x1:= span

?dw1

?

⊂ H2(Ω?),

C?,x1:= span

?

?dw1

dx

?

⊂ L2(Ω?).