1.1 Basic assumptions on the growth of filamentary structures.

Testo completo

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ON THE GROWTH OF FILAMENTARY STRUCTURES IN PLANAR MEDIA

by

D. Andreucci(1), M. A. Herrero(2) and J. J. L. Vel´azquez(2)

(1) Dipartimento di Metodi e Modelli Matematici, Universit`a di Roma La Sapienza, via Scarpa 16, 00161 Roma, Italy.

E-mail: andreucci@dmmm.uniroma1.it

(2) Departamento de Matem´atica Aplicada, Facultad de Matem´aticas, Universidad Complutense, 28040 Madrid, Spain.

E-mail: herrero@mat.ucm.es; JJ Velazquez@mat.ucm.es.

Appeared in Mathematical Methods In The Applied Sciences, 27 (2004), pp.1935–1968.

Copyright c John Wiley & Sons, Ltd. 2004.

Short title: GROWTH OF FILAMENTARY STRUCTURES

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Summary: We analyze a mathematical model for the growth of thin fil- aments into a two dimensional medium; typical applications are in biology.

More exactly we focus on a certain reaction/diffusion system, based on the interaction of two chemicals: an activator, which promotes elongation of the filament, and an antagonist (usually called inhibitor). Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, when some of the pa- rameters in the system are very large, and some very small, thereby allowing the onset of different time scales. Namely, we describe the motion of the tip of a filament, and the changes in the relevant chemical species in the nearby.

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On the growth of filamentary structures in planar media.

1 Introduction.

1.1 Basic assumptions on the growth of filamentary structures.

1.2 An asymptotic model. A first glance at the nature of solutions.

2 Preliminary results.

2.1 Stationary solutions: asymptotic estimates.

2.2 Linear stability of stationary solutions.

2.3 Asymptotic behaviour of the eigenvalues of the adjoint operator A.

3 The motion of a tip of the net.

3.1 An equation for tip motion.

3.2 Estimating time scales.

3.3 Refined estimates on the width of the filament.

3.4 Estimating the growth factor near a filament tip.

• References.

• Appendix 1: A bound on eigenvalues of the linearized operator A.

• Appendix 2: The asymptotics of eigenfunctions of the adjoint operator A.

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ON THE GROWTH OF FILAMENTARY STRUCTURES IN PLANAR MEDIA

by

D. Andreucci1, M. A. Herrero2 and J. J. L. Vel´azquez2

1 INTRODUCTION:

This work is concerned with the analysis of a mathematical model to simu- late the growth of a two-dimensional network of thin moving filaments which expand into a surrounding medium. A motivation for dealing with that question is provided by the consideration of a long-standing biological prob- lem. This consists in understanding the way in which living beings develop complex-shaped organs, each being made of long, branching filaments, that grow within a surrounding organic matrix which is thereby supplied with oxygen, water, nutrients and information in an efficient way. Typical ex- amples of such organs are the blood vessels, the trachea of insects, and the nervous system of vertebrates, to mention but a few.

When one tries to understand the way in which a filamentary network unfolds, a key question consists in ascertaining a mechanism responsible for:

1 The location of the origin of any branch, and the direction in which it will grow.

2 The size and shape of each branch, and

3 When and where in a given filament a new generation of branches will appear.

To the best of our knowledge, no mathematical model has been derived that could satisfactorily account for all these points. However, it has been observed that relatively simple reaction-diffusion systems can be derived,

1Dipartimento di Metodi e Modelli Matematici, Universit`a di Roma La Sapienza, via Scarpa 16, 00161, Italy.

2Departamento de Matem´atica Aplicada, Facultad de Matem´aticas, Universidad Com- plutense, 28040 Madrid, Spain.

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which are based on a reduced number of assumptions, and whose numerical simulation yields patterns which resemble some qualitative features of an expanding filamentary net. We shall be concerned here with one of these models, to be described below, and will focus on a particular aspect of point 1 above, namely, the analysis of the local behaviour around the tip of a filament. This we shall do under the key assumption that the process under consideration involves different time scales. We intend to address question 3 above in a subsequent paper, where much of the material contained here will be used as a starting point.

1.1 Basic assumptions on the growth of filamentary structures.

Following Hans Meinhardt (see for instance [17], [19]) the following hypothe- ses will be made:

H1 A local signal for filament elongation is generated by local self-enhancement of a substance (activator) and long-range diffusion of an activator an- tagonist, henceforth denoted as inhibitor.

H2 The signal mentioned above produces an elongation in each filament by accretion of newly differentiated cells at the tip of that filament. Once such differentiation is achieved, it will be preserved for all subsequent times.

H3 Filaments grow in a surrounding medium (for instance, a tissue) which drives the net by producing a growth factor that is removed by the expanding filaments. Therefore the highest concentration of the growth factor will occur in those regions less supplied by the unfolding net.

Bearing in mind these assumptions, the following model has been pro- posed in [17] and [19] to describe the growth of vein or filamentary structures in a two-dimensional medium:

∂a

∂t = Da∆a + ρa2s

h − µaa + σay , (1.1)

∂h

∂t = Dh∆h + ρ0a2s − µhh + σhy , (1.2)

∂s

∂t = Ds∆s − µss − νsy + σs , (1.3)

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∂y

∂t = y2

1 + κy2 − µyy + σya . (1.4) Here a = a (x1, x2, t) (respectively h = h (x1, x2, t)) denotes the concen- tration of a chemical which is termed as activator (respectively, inhibitor), s = s (x1, x2, t) is a growth factor (or substrate, as is sometimes termed) con- centration, and y = y (x1, x2, t) denotes a cell variable that accounts for cell differentiation. This last can be thought of as reflecting the incorporation of new cells into the net, which can thus be roughly characterised as being the region in the plane where y achieves (and maintains) a sufficiently large value.

It is interesting to examine the way in which the previous system is related to assumptions (H1)-(H3) above. A quick glance at (1.1)-(1.4) reveals that these equations describe a dynamics in which an autocatalytic activator a is produced, which is the actual driving force for cell differentiation (cf. (1.4)).

Also, the activator produces an antagonist, h which counteracts a. As to the growth factor s, it in turn enhances the production of both a and h, as seen from equations (1.1), (1.2), and so does the incorporation of new cells into the net (represented by y), as illustrated by the first two equations in the system.

The particular form of function g (y) = 1+kyy2 2 − µyy, appearing on the right of (1.4), has been selected so that the evolution equation ˙y = g (y) had only two stable roots y = 0 and y = m for some m > 0, this last corresponding to the value at which differentiation occurs.

In system (1.1)-(1.4) constants Da, Dh, Ds are the respective diffusion coefficients of variables a, h and s. The corresponding substances are thus assumed to undergo random motion in the surrounding medium. Thus no ac- tive, directed transport mechanism (as, for instance, convection) is supposed to act at the molecular level. As to the constants ρ, ρ´, µa, σa, µh, σh, σs, µs, ν, κ, µy

and σy, these are positive parameters related to the various physical and chemical aspects (decay of activity, molecular reaction,...) that are taken into account in (1.1)-(1.4).

In this paper, we shall consider a particular form of (1.1)-(1.4) in which the diffusion coefficients Da, Dh and Ds have quite different numerical values (thus allowing for the appearance of fast and slow time scales, as required for instance in assumption (H1) above). On the other hand, most of the remaining parameters will be merely assumed to be of order one (further details will be provided later). However, as we shall see presently, some coefficients in equations (1.3) and (1.4) will be supposed to be small, thus

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allowing for a clear-cut description of the behaviour of solutions as done in the following sections. Rather than arguing that these assumptions can be justified in some situations, our aim is instead to provide in that case a picture of the local evolution of an individual branch by analytical means.

The type of behaviour thus being identified is shortly described below.

1.2 An asymptotic model. A first glance at the nature of solutions.

Let us make precise the mathematical problem to be dealt with in this paper.

We shall be concerned with the following system:

∂a

∂t = ε∆a + a2s

h − a + Γ1y , (1.5)

∂h

∂t = 1

ε∆h +a2s − h+ Γ2y , (1.6)

∂s

∂t = ∆s + αε (1 − s) − αsy , (1.7)

∂y

∂t = β y2

1 + κy2 − y + γε2a

!

, (1.8)

where:

0 < ε << 1 , (1.9)

and:

Constants Γ1, Γ2, α, κ, β and γ are of order one. (1.10) Notice that the term αε in the right of (1.7) corresponds to slow produc- tion of the growth factor s over the region where filaments expand. On the other hand, the quantity ε2a in the right of (1.8) will be shown to provide a source of order one in that equation (cf. (1.15) below).

The goal of this work can now be stated as follows:

To describe by means of formal asymptotic methods the behaviour of solutions to (1.5)-(1.8) along any individual filament, under (1.11) assumptions (1.9), (1.10) above.

More precisely, following a classical approach in asymptotic analysis, we shall first state as an ansatz the behaviour of variables a, h, s and y over

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a slowly moving filament when 0 < ε << 1, and then check that these behaviours are consistent with the dynamics encoded in system (1.5)-(1.8).

The picture thus obtained describes solutions for which a (x1, x2, t) remains concentrated, and attains comparatively large values (of order ε12), at the tip of thin straigth filaments, with width of order √

ε, length of order 1ε, and which move slowly (with tip speed of order ε2). Here and henceforth, these estimates are to be understood up to logarithmic corrective terms, (u.l.t) for short, a notation to be retained henceforth. On its turn, the inhibitor variable h will decrease over a much larger region than that where the activator a achieves its peak. Functions a and h displaying such behaviour are often termed as spikes. These have been the subject of considerable interest, mostly in the case of scalar equations, but also for reaction-diffusion systems related to the so-called activator-inhibitor system, introduced by Gierer and Meinhardt in [5]. This last corresponds to equations (1.5), (1.6) with s = 1 and y = 0 there. See for instance [2], [13], [21], [22], [26], [27], [28], [29]; a survey of results on spikes behaviour is provided at the Introduction of [2]. See also [20] for a description of related results.

While the activator-inhibitor system referred to above provides a key building block for the branch dynamics to be described here, in our case the motion of the corresponding spikes will be driven by the growth factor s.

Actually, its concentration will remain of order one everywhere, but will be slightly depleted over the net. On the other hand, the form of (1.8) indicates that variable y will jump from zero to a value of order one in regions where ε2a becomes of order one. This in turn has the effect of diminishing the value of s on these regions, which turns off the production of a in (1.5), thus signalling the displacement of the spike to regions with higher concentration of s. This scenario is summarized in Figure 1.

More precisely, in line with the previous picture, we shall assume that, in the region where the spike-solutions (a, h) are placed, these can be described to the lowest order by the stationary Gierer-Meinhardt system:

ε∆a +a2

h − a = 0 , (1.12)

1

ε∆h + a2− h = 0 . (1.13)

Throughout this paper, we will consider systems (1.5)-(1.8) and (1.12)- (1.13) over the whole plane −∞ < x1, x2 < ∞, a natural setting for a situation in which filaments are sparsely distributed. Assuming then that all

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t

x1 x2

y

a

h x ∼√

ε y ∼ 1, s ∼ s0 > 0

0 < y  1, s ∼ s1 > s0

Figure 1: A magnification of a region around a tip of an isolated filament, showing the expected distribution of functions a, h, s and y there.

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terms in (1.12) are of same order of magnitude and that the origin is initially located at a particular tip filament, to be subsequently analysed, we readily obtain that:

a ∼ h, r2 = x21+ x22 ∼ ε , (1.14) up to logarithmic terms (u.l.t), which will be eventually reckoned with when necessary. A similar analysis in (1.13) provides, again (u.l.t):

h ∼ a ∼ 1

ε2 as ε → 0 . (1.15)

Estimates (1.14) and (1.15) suggest at once the scaling:

a = ˜a

ε2, h = ˜h

ε2, x =√ εz,

In such a way, an inner length scale, corresponding to distances over which spikes are concentrated, has been identified. It is given by:

x = (x1, x2) = O

ε, i.e, z = O (1) , (u.l.t) . (1.16) We next consider those distances where h undergoes considerable varia- tions. Namely, we impose in (1.6) the balance:

1

ε∆h ∼ h , This happens at distances:

x = O 1

√ε

!

. (1.17)

In order to describe the dynamics of solutions to (1.5)-(1.8) in those regions, we define an outer scale ˜x as follows:

˜

x = εz , (so that ˜x =√

εx) (1.18)

As it turns out, the relevant quantity in our analysis is not so much s as the difference (1 − s) with respect to the saturation value s = 1. We then define:

G = 1 − s . (1.19)

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Putting all these remarks together and writing for simplicity a, h instead of ˜a, ˜h, the inner and outer systems corresponding to the previous scaling can be rewritten in the form:

∂a

∂t = ∆za + a2(1 − G)

h − a + Γ1ε2y , (1.20) ε2∂h

∂t = ∆zh +a2(1 − G) − ε2h+ Γ2ε4y , (1.21)

∂G

∂t = 1

ε∆zG − αεG + αy (1 − G) , (1.22)

∂y

∂t = β y2

1 + κy2/ε − y + γa

!

, (1.23)

and:

∂a

∂t = ε2x˜a + a2(1 − G)

h − a + Γ1ε2y , (1.24)

∂h

∂t = ∆x˜h + a2(1 − G) ε2 − h

!

+ Γ2ε2y , (1.25)

∂G

∂t = ε ∆x˜G − αG + αy (1 − G) ε

!

, (1.26)

∂y

∂t = β y2

1 + κy2/ε− y + γa

!

. (1.27)

As we shall see in the sequel, systems (1.20)-(1.23) and (1.24)-(1.27) will be respectively used to obtain suitable expansions for the variables along any filament, as well as to estimate the far-field structure of solutions away from any such branch.

We conclude this Introduction by describing the plan of the paper. To begin with, a number of auxiliary results (many of which are known) are gathered in Section 2 below. These include estimates on the asymptotics of stationary solutions to Gierer-Meinhardt system under our current assump- tions on the corresponding diffusivities, and a discussion on the linear sta- bility of such solutions. This Section also includes a result on the analysis of the eigenfunctions of the adjoint of a linearised operator, which is introduced to discuss the linear stability of the spike-like stationary solutions previously considered. This tool is instrumental in deriving some basic estimates in the rest of the paper. For convenience of the reader, some of the technical points involved are recalled in two Appendixes at the end of this work.

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The main results in this article are contained in Section 3, which is in turn divided into several subsections. In Subsection 3.1, an equation for the tip motion is obtained under the assumption that filaments move slowly to gain stability (in other words, motion proceeds so that the contribution to the dynamics arising from neutral eigenvalues is cancelled). We therefore assume that filaments are already present, and proceed to analyse the way in which they move. A discussion on the events leading to filament sprouting in a related context can be found in [15], [16]. In these papers, a mathematical model for angiogenesis is derived, in which the onset of new vessel formation is characterised as a critical aggregation of endothelial cells around some points in the preexisting vasculature.

The equation thus derived for the motion of the tip (cf. (3.14) below) does not provide by itself sharp estimates on the actual speed at which any filament expands (this last is expected to be of order O (ε2) in our approach).

To check this last statement on equation (3.14), one needs to bound the various terms arising there, and to this end, the crucial step consists in estimating the manner in which the growth factor variable s stabilizes over each filament towards its homogeneous value s0 near the tip. Therefore, understanding the evolution of (s − s0) becomes the main step to characterise the motion of a single branch.

To proceed further, we first derive preliminary information on the stabi- lization times of variables a, h, s and y. This is done in Subsection 3.2, where it is shown that the stabilization of s to its stationary value occurs over the whole net in times t of order 1ε (u.l.t) . In particular, after such times one also has that s ∼ s0 near the tip of any filament. These times are relatively long, but much smaller than those required in our picture for the branch to expand a distance of order one, this last being of order ε12 (u.l.t) . Bearing this fact in mind, we then set out to analyse the differential equation satisfied by (s − s0) (actually, we find it more convenient to deal with G − G0 with G = 1 − s, but this is not important for our current description). As it turns out, such equation has the cell variable y as a coefficient, and far away from the tip, the corresponding term can be replaced by a Dirac mass, which allows us to explicitly solve that equation, as far as the fluctuations in the width of the branch (where y ∼ 1) remain negligible. We then have to check that, as the motion proceeds, changes in that width are indeed small, and this is done in Subsection 3.3. This fact is used in deriving in Subsection 3.4 an inner expansion for (s − s0) , valid on regions of width x ∼ √ε. The tip equation (3.14) is then shown to yield an estimate for the speed of order O (ε2) (up to

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logarithmic terms), and the consistency of our approach is established.

Acknowledgements:The authors have been partially supported by DGES Project BFM2000-0605 and by TMR European Contract HPRN-CT-2000- 00105.

2 Preliminary results

In this Section we gather a number of properties of solutions of a suitable auxiliary system, namely:

∂A

∂t = ε2x˜A + A2

H − A , (2.1)

∂H

∂t = ∆˜xH +A2

ε2 − H . (2.2)

where ˜x ∈ IR2, t > 0 and 0 < ε << 1. Equations (2.1), (2.2) constitute an example of an activator-inhibitor system (as those considered in [5], [18]), in which the respective diffusion coefficients are of different orders of magnitude.

It clearly corresponds to the first two equations in the outer system (1.24)- (1.27) when one sets s = 1 and y = 0 there. System (2.1), (2.2) has drawn considerable attention in the literature (cf. for instance [23], [26], [27], [28], [29] and references therein, as well as [25] for early existence and uniqueness results in bounded domains). As a consequence, a large part of the facts to be recalled below can be considered as of a standard nature. However, for convenience of the reader, we have decided to list them below in a way suitable for our purposes in this work.

2.1 Stationary solutions: asymptotic estimates

Consider now the equations:

ε2∆A +A2

H − A = 0 , (2.3)

∆H + A2

ε2 − H = 0 , (2.4)

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where 0 < ε << 1, A = A (˜x) , H = H (˜x) , and we have dropped the corre- sponding subscripts in the laplacean operator for simplicity. When written in terms of the inner length (cf. (1.16)), one obtains:

zA +A2

H − A = 0 , (2.5)

zH + A2− ε2H = 0 . (2.6)

Recalling the expected behaviour of solutions that has been sketched at the end of Section 1, we shall be interested in solutions of (2.5), (2.6) which are such that:

A and H should be bounded in IR2, and decay exponentially

when |z| → ∞. (2.7)

Bearing this fact on mind, and denoting by ∆−1z h the Poisson kernel corresponding to a finite distribution of mass h, it follows from (2.6) that when |z| >> 1, to the leading order one has that:

H (z) = Hε− ∆−1z

A2∼ Hε− 1 2π

Z

IR2A2dz



log (|z|) , (2.8) for some constant Hε depending on ε. Notice that (2.8) actually holds when z is in the range of values 1 << |z| << 1ε. In terms of the outer variable

˜

x = εz, (2.8) reads:

H (x) ∼ Hε+ 1 2π

Z

IR2A2dz



log (ε) − 1 2π

Z

IR2A2dz



log (|x|) , (2.9) where here and henceforth we are replacing ˜x by x for notational convenience, a convention that we shall keep to, whenever no risk of confusion arises.

Notice that, far away from each filament, it makes sense to replace Aε2 by cεδ (x) for some cε > 0, where δ (x) denotes Dirac’s delta centered at the tip of the filament. Thus, in the limit ε → 0, it follows from (2.4) that, in the far field approximation, H satisfies to the leading order:

−∆H + H =

Z

IR2A2dz



δ (x) .

As |x| → 0, solutions of this equation behave in the form:

H ∼



− 1

2πlog (|x|) − γ + log (2)



·

Z

IR2 A2dz ,

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where γ is Euler’s constant. Matching this and (2.9), we deduce at once that to the leading order:

Hε∼ − 1 2π

Z

IR2A2dz



(log (ε) + O (1)) when 0 < ε << 1 . (2.10) Before describing more in detail the behaviour of H for large values of its argument, a similar local analysis will be performed for A. To begin with, (2.8) implies that when z is of order one:

H (z) ∼ Hε .

It is then natural to scale A in (2.5) as follows:

A = HεΦ, (2.11)

in which case (2.5) reads to the leading order:

zΦ + Φ2− Φ = 0 when z ∈ IR2 . (2.12) Equation (2.12) has been extensively analysed in the literature (see for instance [4], [6], [7], [12],...). In particular, global, nonnegative and bounded solutions of (2.12) are radial (cf. [7]). Therefore, they satisfy:

Φ000

r + Φ2− Φ = 0 when 0 < r = |z| < ∞ . (2.13) We are interested in bounded solutions of (2.13) such that:

Φ0(0) = 0, Φ (r) decays exponentially as r → ∞ . (2.14) It has been shown in [4] (see also [12]) that (2.13), (2.14) has a unique positive solution. Moreover, standard asymptotic methods (see for instance [3], [4]) yield that:

Φ (r) ∼ K

√re−r as r → ∞ for some K > 0 . (2.15) Putting together (2.10) and (2.11), we now obtain that, to the leading order:

Hε ∼ 2π

R

IR2Φ2dz · 1

|log (ε)| for 0 < ε << 1 . (2.16)

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Let us now return to equation (2.6). If we rewrite it in terms of the outer variable ˜x (again we drop the superscript for notational convenience), we obtain:

xH − H + A2

ε2 = 0. (2.17)

Since we are looking for radial solutions, recalling (2.7) and (2.11) and the fact that Aε22 is expected to be concentrated around x = 0, it turns out that for r = |z| sufficiently large it makes sense to replace (2.17) by:

−H00(r) − H0(r)

r + H (r) = Hε2

Z

IR2(Φ (z))2dz



δ (r) , (2.18) where δ (r) stands for the Dirac’s delta centered at the origin. Equation (2.18) is to be supplemented with the following matching condition arising from (2.9) and (2.11):

H (r) ∼ −Hε2

Z

IR2(Φ (z))2dz



log (r) as r → 0 . (2.19) The solution of (2.18) that satisfies (2.19) and decays exponentially as r → ∞ is:

H (r) = Hε2

Z

IR2(Φ (z))2dz



K0(r) , (2.20)

where K0(r) is the modified Bessel function of third order (cf. [1], [14], pp. 108-111). Summing all these results up, we have obtained the following estimates for functions A and H. At the inner zone, where |z| ∼ 1, there holds:

A ∼ HεΦ (z) , Φ as in (2.13),(2.14),(2.15), (2.21) where Hε ∼ 2π

Z

IR2(Φ (z))2dz

−1

(|log (ε)|)−1 as ε → 0.

On the other hand, as we are leaving the inner region we have that:

H ∼ Hε−Hε2

Z

IR2(Φ (z))2dz



log (r) for 1 << |z| << 1

ε , (2.22) to eventually obtain that (2.20) holds as we enter the outer region where ˜x becomes of order one.

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For completeness, we state here the asymptotic behaviour of A when

|z| → ∞. This is given by:

A (z) ∼ HεK

q|z|e−|z|, for some positive constant K.

2.2 Linear stability of stationary solutions

Our next step consists in discussing the stability of the solutions to (2.5), (2.6) that have been recalled in our previous paragraph. These shall be denoted byA, ¯¯ Hfrom now on. As it will become apparent in the sequel, it suffices to perform the corresponding study in the inner variable z, in which case system (2.1), (2.2) reads:

∂A

∂t = ∆zA +A2

H − A , (2.23)

ε2∂H

∂t = ∆zH + A2− ε2H . (2.24) We now set:

A = ¯A + ζ, H = ¯H + θ . (2.25) Plugging (2.25) into (2.23), (2.24) and retaining only linear terms, we readily see that ζ and θ satisfy:

∂ζ

∂t = ∆zζ + 2 ¯A H¯ − 1

!

ζ − A¯ H¯

!2

θ , (2.26)

ε2∂θ

∂t = ∆zθ + 2 ¯Aθ − ε2θ . (2.27) If we now write:

ζ = eλtψ , θ = eλtU , (2.28) we eventually obtain from (2.26), (2.27) and (2.28) the following eigenvalue problem:

A ψ

U

!

= λ ψ

U

!

, (2.29)

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where:

A ψ

U

!

=

z+ 2 ¯H¯A− I , −HA¯¯

2 2 ¯A

ε2 , ε2z − I

ψ U

!

. (2.30)

Discussing the linear stability of solutions A, ¯¯ H is thus tantamount to describing the spectral properties of operator A given above. This operator is not self-adjoint in its natural domain, which prevents its spectral analysis from being entirely straightforward. It is possible, however, to characterize its spectrum in a way suitable for our purposes here. To this end, we use the following result:

Lemma 2.1 For any α > 0 there exists Cα > 0 such that, if λ belongs to the spectrum of A and Re (λ) > 0, there holds:

|λ| ≤ Cαε−α . (2.31)

To keep the flow of the main arguments here, we shall describe the main arguments in the proof of Lemma 2.1 in Appendix A at the end of the paper, and continue.

We now obtain refined estimates for the spectrum of A. To this end, we take advantage of (2.11) to replace in (2.29), (2.30) the term HA¯¯ by Φ (z) . To the leading order, one thus obtains the eigenvalue equations:

zψ + (2Φ − 1) ψ − Φ2U = λψ , (2.32)

zU − ε2(λ + 1) U = −2HεΦψ , (2.33) where Hε is as in (2.10). Local analysis of such equations reveals that the eigenfunctions ψ and U decay exponentially fast as |z| → ∞. Bearing in mind (2.31), it turns out that the second term in the left of (2.33) can be neglected when 0 < ε << 1, even at distances z of order one. It then follows from the resulting simplified equation that:

U ∼ Uε− ∆−1z (2HεΦψ) ∼ Uε− Hε

π

Z

IR2Φ (z) ψ (z) dz



log (|z|) , for some constant Uε and 1 << |z| << 1ε. Setting x = εz, this statement can be recast as follows:

U ∼ Uε+Hε π

Z

IR2Φ (z) ψ (z) dz



log (ε)−Hε π

Z

IR2Φ (z) ψ (z) dz



log (|x|) .

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The first two terms in the expansion above should cancel each other, thus avoiding their contribution to become unbounded when x is of order one and ε is small enough. Recalling (2.16), this yields:

Uε ∼ 2RIR2Φ (z) ψ (z) dz

R

IR2(Φ (z))2dz .

Plugging this into (2.32) gives (to the leading order) that:

λψ = ∆zψ + (2Φ − 1) ψ − 2RIR2Φ (z) ψ (z) dz

R

IR2(Φ (z))2dz Φ2 . (2.34) Let us examine the basic eigenvalue equation (2.34). Suppose first that:

Z

IR2Φ (z) ψ (z) dz = 0 . (2.35) Then (2.34) reduces to:

A0 ≡ ∆zψ + (2Φ − 1) ψ = λψ . (2.36) The spectral properties of A0 are well known (see for instance [9], [21], [28],...). For convenience of the reader, these are summarised in the following:

Lemma 2.2 Operator A0 is self-adjoint in L2IR2 Its spectrum consists of a positive eigenvalue ν1, a zero eigenvalue for which the corresponding eigenspace has dimension two (and whose corresponding eigenfunctions have the angular dependence {cos (θ) , sin (θ)}), a negative eigenvalue ν2 such that

−0.8 < ν2 < −0.75, and a continuous part, consisting in all real numbers ν satisfying ν ≤ −1. The eigenfunctions corresponding to the eigenvalues ν1, ν2 are radial.

As a matter of fact, the bounds on ν2 are not explicitly given in the references just mentioned, but can be readily obtained by a combination of numerical exploration and classical Sturmian theory.

In particular we have that:

Corollary 2.3 Equation (2.35) holds if λ = 0, which is an eigenvalue of our linearised problem (2.32), (2.33).

When λ 6= 0, the following result holds:

Corollary 2.4 ([28], Section 5). Let λ 6= 0 be a solution of the eigenvalue equation (2.34). Then Re (λ) < 0.

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2.3 Asymptotic behaviour of the eigenfunctions of the adjoint operator A

We conclude this Section by reviewing some properties of the adjoint A of the linear operator A defined in (2.30). These will be later used to describe the motion of the spikes at the tip of each filament. We recall that, if we denote as h·, ·i the standard inner product in L2IR2× L2IR2, operator A is defined by:

* W V

!

, A R Z

!+

=

*

A W V

!

, R

Z

!+

. It is then readily seen that A is given by:

A W V

!

=

z+ 2 ¯H¯A− I 2 ¯εA2

HA¯¯

2 z

ε2 − I

W V

!

. (2.37)

Since we already know that A has a doubly degenerate zero eigenvalue, it follows at once that the same result happens for A (cf. [11], p. 184).

Furthermore, the following result holds.

Lemma 2.5 The number λ = 0 is a doubly degenerate eigenvalue of A. For i = 1, 2, the corresponding eigenfunctions i, ˜Vi

 are such that when 0 < ε << 1 :

i− ˜α∂Φ

∂zi

= gε(r) e−µr , (2.38)

where gε(r) = o (1) as ε → 0, uniformly on r > 0 , and:

i(r) ≤ C |˜α| ε2e−Γεrmin {1, r} , (2.39) for some C > 0, Γ > 0, ˜α and any r. Moreover, the derivatives of ˜Wi and V˜i satisfy the asymptotic behaviours obtained by formally differentiating in (2.38), (2.39).

The proof of this Lemma will be postponed to Appendix B at the end of the paper.

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3 THE MOTION OF A TIP OF THE NET

In this Section we address the question of describing how single filaments expand. To this end, we analyse in detail the motion of isolated tips in the network. We shall proceed in several steps, and begin as follows.

3.1 An equation for tip motion.

In line with the remarks made at the Introduction, we shall check in the sequel that when ε → 0, system (1.5), (1.8) has branches that behave in the following manner:

Any filament consists (roughly) in a rectangular thin strip, capped by a circular tip, whose center x = λ (t) = (λ1(t) , λ2(t)) moves slowly in time, approximately along a straight line with speed ˙λ (t) = O (ε2) (u.l.t). The width of any such strip is of order |z| ∼ 1 (u.l.t), and the branch may reach a length of order |z| ∼ 1ε (u.l.t) before secondary branching takes place.

This last phenomenon, that will be considered in detail elsewhere, happens at such distances since this is the characteristic length for inhibitor decay (see (1.13)). Over any such strip, the growth factor s remains approximately constant, and is not completely depleted; see Figure 2.

In the following we will check that the previous picture is actually compat- ible with our model system (1.20)-(1.23) (in inner variables) or (1.24)-(1.27).

To this end, we argue as follows:

We will show in detail later (cf. Subsection 3.2) that G stabilizes to an steady state locally near the tip in times of order 1ε, as can be seen from (1.22). Let us denote by G0 the constant value of this steady state at the tip x = λ (t) . We now try an expansion of the form:

a = A¯ x − λ (t)˜ ε

!

+ R x − λ (t)˜ ε , t

!

, (3.1)

h = (1 − G0) H¯ x − λ (t)˜ ε

!

+ Z x − λ (t)˜ ε , t

!!

, (3.2)

where A, ¯¯ H is the stationary solution of (2.3), (2.4) already discussed in Subsection 2.1, ˜x is the outer variable defined in (1.18), and R and Z are small corrective terms. Setting ξ = ˜x−λ(t)ε , and plugging (3.1), (3.2) into

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λ(t) z ∼ 1

z ∼ 1/ε

Figure 2: A close-up image of an isolated filament. Scales are represented (u.l.t.).

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(1.20), (1.21), one readily obtains to the leading order that:

−˙λ ε

ξA + ∇¯ ξR+∂R

∂t (3.3)

= ∆ξR + 2 ¯A H¯ − 1

!

R −

2

2 Z − (G − G0) (1 − G0)

2

H¯ + Γ1ε2y 1 − G0

−˙λ ε

ξH + ∇¯ ξZ+ ∂Z

∂t (3.4)

= 1

ε2ξZ +2 ¯AR

ε2 − Z + A¯ ε

!2G0− G 1 − G0



+ Γ2ε2y 1 − G0

Since R and Z are assumed to be small terms, we will neglect the quan- tities ε˙λξR and ε˙λξZ in equations (3.3), (3.4). It then turns out that, to the lowest order, R and Z satisfy:

−˙λ ε

ξ

ξ

!

+ ∂

∂t R Z

!

= A R

Z

!

+

(A¯)2

H¯

G0−G 1−G0

+Γ1−G1ε2y0

¯

A ε

2

G0−G 1−G0

+Γ1−G2ε2y0

, (3.5) where A is the linear operator defined in (2.30), and

˙λ · ∇ξA = ˙λ¯ 1∂ ¯A

∂ξ1 + ˙λ2∂ ¯A

∂ξ2 .

To proceed further, we adapt in a suitable way a classical approach (the so-called projection method) which has been used to describe spike motion in the Gierer-Meinhardt system (see for instance [13] and [29]). To this end, and as in the statement of Lemma 2.5, we denote by i, ˜Vi with i = 1, 2, the eigenfunctions corresponding to the zero eigenvalue of the adjoint operator A given in (2.37). If we continue to write h·, ·i to denote the standard scalar product in L2IR2× L2IR2, and observe that:

* i

i

!

, A R Z

!+

=

*

Ai

i

!

, R

Z

!+

= 0 , we then derive from (3.5) that:

* ii

!

, ˙λ ε

ξ

ξ

!+

+

* ii

!

, ∂

∂t R Z

!+

=

* ii

!

,

(A¯)2

H¯

G0−G 1−G0

+Γ1−G1ε2y0

¯

A ε

2

G0−G 1−G0

+ Γ1−G2ε2y0

+

. (3.6)

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We now observe that in the moving frame ξ = x−λ(t)˜ ε one has that:

* i

i

!

, ∂

∂t R Z

!+

= ∂

∂t

* i

i

!

, R

Z

!+!

(3.7)

*

∂t W˜i

i

!

, R

Z

!+

,

we now select λ (t) = (λ1(t) , λ2(t)) in such a way as to cancel the first term in the rigth of (3.7). In doing so, λ (t) is such that the contribution associated to the neutral eigenvalue of A to R

Z

!

will be eliminated, and therefore this corrective term will approach exponentially to zero. On the other hand, recalling the estimates for functions W˜i

i

!

obtained in Lemma 2.5, we have that:

∂ ˜Wi

∂t

≤ C ˙λi

ε

i

,

∂ ˜Vi

∂t

≤ C ˙λi

ε

i

, (3.8)

for some C > 0. We now claim that:

*

∂t W˜i

i

!

, R

Z

!+

<<

* i

i

!

, ˙λ ε

ξ

ξ

!+

as ε → 0 . (3.9)

To check (3.9), we first notice that, in view of (3.8):

*

∂t W˜i

i

!

, R

Z

!+

≤ C ˙λi

ε

Z

IR2



iR + iZ dz , (3.10) whereas on the other hand:

* i

i

!

, ˙λ ε

ξ

ξ

!+

˙λ ε

Z

IR2

iξA + ˜¯ Viξdz

(3.11)

˙λ ε

Z

IR2iξAdz¯

.

The last statement in (3.11) can be obtained as follows. By (2.8) and (2.21) we have that ¯H ∼ Hε− (Hε)2−1z2) . Using the bound:

z

−1z Φ2 ≤ C 1 + |z| ,

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for some C > 0, and taking advantage of estimate (2.39), one readily sees that there exists σ > 0 such that:

Z

IR2

i · z dz ∼ Oε2(Hε)2

Z

IR2

e−εσ|z|

1 + |z|dz = Oε2(Hε)2 , (3.12) whereas on the other hand:

Z

IR2izAdz¯

∼ |˜α|

Hε

Z

IR2

∂ ¯A

∂zi

!2

dz = |˜α| Hε

Z

IR2

∂Φ

∂zi

!2

dz . (3.13) Putting together (3.12) and (3.13), the desired conclusion follows in (3.11).

To proceed further, we now observe that, since we are assuming that |R| <<

as ε → 0, we readily obtain that:

Z

IR2

i

|R| dz ∼ |˜α|

Z

IR2

∂Φ

∂zi

|R| dz <<

|˜α|

Z

IR2

∂Φ

∂zi

dz ∼ |˜α| Hε

Z

IR2Φ

∂Φ

∂zi

dz ,

so that the first term on the right of (3.10) is negligible compared with the quantity which has been estimated in (3.11). As a matter of fact, a similar result can be derived for RIR2

i · |Z| dz, namely:

Z

IR2

i

· |Z| dz <<

Z

IR2

i

· ¯Hdz ≤ Cε2Hε

Z

IR2e−εσ|z|dz ≤ C |˜α| Hε, whereupon (3.9) follows. Recalling now that:

Z

IR2

∂Φ

∂z1

!2

dz =

Z

IR2

∂Φ

∂z2

!2

dz = 1 2

Z

IR2(∇Φ)2dz ,

we then deduce from (3.6), (3.7) and (3.9) that, to the lowest order, there holds:

−˙λiHεα˜ 2ε

Z

IR2(∇Φ)2dz =

* i

i

!

,

(A¯)2

H¯

G0−G 1−G0

+Γ1−G1ε2y

0

¯

A ε

2

G0−G 1−G0

+ Γ1−G2ε2y0

+

, (3.14)

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for i = 1, 2 and 0 < ε << 1, which yields an evolution equation for the tip center λ (t) = (λ1(t) , λ2(t)) such that the first term on the right of (3.7) cancels out.

Equation (3.14) is our basic startpoint to describe the motion of any fila- ment. To proceed further, we need to estimate the various terms appearing in the right of (3.14). This is the task to be performed in the following paragraphs.

3.2 Estimating time scales

To begin with, we observe that the corrections R Z

!

become negligible in times t = O (1) , since they are driven by a negative eigenvalue. On the other hand, as we have already observed (cf. equations (1.23) and (1.24)), the dynamics of variable y is governed by the equation:

∂y

∂t = β y2

1 + y2/ε − y + γa

!

≡ β (gε(y) + γa) . (3.15) For sufficiently small ε > 0, the graph of gε(y) defined in (3.15) is depicted in Figure 3.

A quick glance at Figure 3 reveals that, for small values of y, ∂y∂t becomes positive as soon as function a attains values of order ε. On the other hand, when a becomes of order one, y undergoes variations of order one too.

Consider now equations (1.20)-(1.23) or (1.24)-(1.27). An inspection of the first two equations of either system reveals that over distances z of order one across the tip of each filament, stabilization of a towards ¯A (respectively of h towards ¯H) occurs (u.l.t) in times or order t ∼ 1. Therefore, stabilization of y towards the value y = 1 occurs over each filament in times t ≈ 1 (u.l.t) . Moreover, in view of our previous analysis of equation (3.15) and the be- haviour of a described above, we have that G approaches towards a stationary solution of (1.22) in times of order 1ε.

Indeed, consider for instance equation (1.26), and let Ge be a solution of:

xGe− αGe+α (1 − Ge) y

ε = 0 , (3.16)

where, as we have repeatedly done before, we are dropping the superscript in the outer variable ˜x, thus denoting it by x. Substracting (3.16) from (1.26),

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y gε(y)

1/4 1 1

ε/4

ε/2

Figure 3: An approximate plot of gε(y) when 0 < ε  1. The values explicitly mentioned (ε/2, ε/4, . . . ) are to be understood asymptotically as ε → 0.

we readily see that:

∂ (εt)(G − Ge) = ∆x(G − Ge) −



α +αy ε



(G − Ge) (3.17) A quick glance at (3.17) reveals that, as (εt) increases, (G − Ge) decays to zero even faster than the corresponding solution to a homogeneous heat equation would do. This yields at once the stabilization claim.

For later purposes we observe that we may further simplify (3.16) by noting that, since y ∼ 1 over each filament, and any of these has a width of order ε in the x−scale (u.l.t) , it is natural to approximate (3.16) by:

xGe− αGe+ βε(1 − Ge) δΓ(t)(x) = 0 , x = ˜x ∈ IR2 , (3.18) where Γ (t) denotes the filament under consideration, δΓ(t)(x) is the length measure of Γ (t) , and we are making the approximation:

αy

ε ∼ βεδΓ(t)(x) . (3.19)

for distances much larger than ε (u.l.t) from the net. In other words, in the terminology of matched asymptotic expansions theory, (3.18) can be thought

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of as being an outer approximation to (3.16). Notice that βε represents the width of the filament when measured in terms of the inner length z.

Concerning equation (3.18), a remark is in order. Namely one has that:

0 < Ge < 1 , (3.20)

and therefore the value S = 1 − Ge achieved by the growth factor s over the net never vanishes completely. To check (3.20), we merely observe that, by (3.18), function W = −S = Ge− 1 satisfies:

−∆xW + αW + βεδΓ(t)(x) W = −α < 0 . (3.21) Set now W+= W, when W ≥ 0 and W+= 0 otherwise. It then turns out that:

xW · sgn W ≤ ∆x(W+) ,

where the last inequality above is to be understood in the sense of distribu- tions. Then, on multiplying both sides of (3.21) by sgn W we arrive at:

−∆xW++ αW+< 0 ,

also in the sense of distributions. Since α > 0, it then follows from by comparison (cf. for instance [8]) that W < 0, whereupon (3.20) follows.

3.3 Refined estimates on the width of a filament

Let us summarize a bit. We want to show that equation (3.14) is compatible with an estimate on the tip speed where ˙λ (t) is of order ε2 (u.l.t) . To this end, we need to obtain suitable bounds for (G − G0) in (3.14). However, as illustrated by (3.18) and (3.19) the values of Ge (whence also those of G0) depend on the width of the filament under consideration. This has been assumed so far to be of order one (u.l.t) in the z variable. We need to check now that such width cannot change much in times t ∼ 1ε which are those in which G stabilizes to Ge.

To this end, let us consider again equation (3.15) and Figure 3 at the beginning of paragraph 3.2. From these, one readily sees that y switches from y = 0 to y of order one as soon as γa becomes larger than ε4, say:

γa ≥ ε

4+ ε1+σ , (3.22)

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for some σ with 0 < σ < 1. On the other hand (cf. (3.1)), we are taking:

a = ¯A + R = HεΦ + R . (3.23) where

R << HεΦ . (3.24)

Then, recalling (2.11), (2.15), it follows from (3.23) and (3.24) that (3.22) occurs, to the lowest order, when:

γHεKe−|z|

√z ∼ ε

4+ ε1+σ , (3.25)

whenever |z| ∼ zε, where:

γHεKe−zε

√zε ∼ ε

4 or zε∼ |log (ε)| . (3.26) where for convenience we are changing slightly the definition in (1.18) and we will denote as z the variable ˜x−λ(t)ε . Notice that this yields at once the estimate

βε∼ |log (ε)| as ε → 0. (3.27)

where βε is given in (3.19). As a matter of fact, we may readily compute the correction due to the term ε1+σ in (3.22). Indeed, on setting z = zε+ ∆zε

(with ∆zε << zε), it follows from (3.22) and (3.26) that:

ε 4



1 − ∆zε+ O

∆zε

zε



= ε

4 + ε1+σ , whence ∆zε = −εσ.

We have to show yet that y becomes of order one over distances of order

|log (ε)| on the z variable around the tip center in times smaller than those required for the activator peak to leave that region. Indeed, as long as we remain within the circle |z| ≤ zε− |∆zε| , we have that:

γa ≥ γHεKe−zε

√zε



1 − ∆zε

2



≥ ε

4+ ε1+σ

2 , (3.28)

Notice that, whenever (3.28) holds, the graph of (gε(y) + γa) looks as depicted in Figure 4.

figura

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