Rotating waves in scalar equations with neumann boundary conditions

MATHEMATICAL COMPUTER MODELLING PERGAMON

Mathematical

and Computer

Modelling

37 (2003) 767-778 www.elsevier.com/locate/mcm

Rotating Waves in Scalar Equations with Neumann Boundary Conditions D. SCHLEY Department of Medical Physics and Bioengineering Southampton General Hospital, Southampton, Hampshire SO16 6YD, U.K. David.SchleyOsuht.swest.nhs.uk

(Received June 2000; revised and accepted August 2002) Abstract-Rotating and spiral waves occur in a variety of biological and chemical systems, such as the Belousov-Zhabotinskii reaction. Existence of such solutions is well known for a number of different coupled reaction-diffusion models, although analytical results are usually difficult to obtain. Traditionally, rotating and spiral waves have been thought to arise only in coupled systems of equations, although recent work has shown that such solutions are also possible in scalar equations if suitable boundary conditions are imposed. Such spiral or rotating boundaries are not. however, physically realistic. In this paper, we consider a class of scalar equations incorporating a discrete delay. We find that the time delay can bring about rotating wave solutions (through a Hopf bifurcation from a nontrivial spatially uniform equilibrium) in equations with homogeneous Neumann boundary conditions. @ 2003 Elsevier Science Ltd. All rights reserved.

Keywords-Reaction-diffusion,

Rotating

waves, Time-delay,

Hopf bifurcation

1. INTRODUCTION Rotating tams.

and They

spiral

waves

have been

the Belousov-Zhabotinskii see [1,2], respectively, Zhabotinskii system ing and meandering

occur

naturally

commonly

observed

(chemical) for visual

in a large

variety

in a number

reaction

of biological

and the social amoebas

demonstrations.

Recent

and

of two-dimensional

chemical

systems,

Dictyostelium

experimental

sys-

such as

discoideum.

work on the Belousov-

includes the initiation of rotating waves [3], the transitions spirals [4], the influence of electric fields on such patterns

between rotat[5] and three-

dimensional scrolling waves [6]. Research on D. discoideum includes pattern selection and formation [7-121, spiral motion [13], cell movement [14], scrolling waves [15], and three-dimensional waves [16]. Other work includes rotating vortices in models of cardiac tissue [17] and numerical studies in excitable

media [18].

For a discussion of spiral waves in biochemistry, patterns in biochemistry, Rotating

see [19, p. 3471.

For a review of periodic

see [20].

and spiral waves have been investigated

for a number of different reaction-diffusion

models (see, for example, [al-23]), but the analytical results obtainable are usually limited by the complexity of the problem. Spiral waves have been investigated in X - w systems (see [24-27]

0895-7177/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(03)00084-O

Typeset

by AM-w

768

D. SCHLEY

and references

therein),

class of models

which are more amenable

can be represented

to analytical

treatment.

In complex

form, this

by

wt = (X(lwl) -t iw(]w])) w + V2w? with suitable

constraints

on X so that

limit cycle oscillations

where u and v are two species or chemicals. and/or

spiral solutions

in reaction

Rigorous

diffusion

may be sustained.

results

systems,

Here, w = u + iv:

exist for the bifurcation

including

that

to rotating

of [28] for the N-dimensional

system u, = v2u

+ F(X, U),

V(z,t)

E RN,

z E R2.

Studies of rotating and spiral waves have been mostly confined to multispecies systems, rather than scalar equations. The primary reason for this is that, in nondelay models at least, two coupled

equations

are necessary

is a prerequisite. conditions.

The main

for the kinetic

exceptions

By [29], rotating

wave solutions

periodic

boundary

waves arise in models

conditions

But, chemical

out

experiments, of special

be appropriate, example,

=

v2u + x u

+ f(u),

boundary

in [31], the most Neumann restricting

boundary

but may often

they can introduce

f E0

conditions

Since these waves often though

not

spatially

suitable

bounda.ry

spiral,

rotating

and spiral

(1)

7

This requires

the solution

has exactly

ourselves

boundary

conditions.

conditions

for chemical

u = mu0

on

the form of

to a finite is not.

be viewed

as more mathematically

unnatural

Dirichlet

inhomogeneity,

arise through

solutions

may

equilibrium)

boundary

come

through

about

in

may sometimes

interesting

than

uniform

oscillatory

realistic

(for

solutions

address the problem of whether rotating with homogeneous Neumann boundary it is natural

such bifurcations.

(through

in scalar equations

conditions

and prevent

a Hopf bifurcation,

solutions

species

When modelling populations and/or domain is usually appropriate, while the

conditions

which can destabilise

uniform

that

(u2>

are imposed.

which are commonly observed in reality). We, therefore, and/or spiral solutions are possible in scalar systems conditions. of time delays,

equation,

spiral.

a dish are homogeneous imposition

which

boundary

x E R/21rz,

for some circle (i.e., the boundary),

Archimedean

as pointed

oscillations (rotating)

diffusion

for z E [0,27r]. In [30], 1 ‘t is shown

disc BR when spiral

dBR (m E Z+), so that an m-armed

temporal

with specialised

of the form Ut

on a circular

to exhibit

exist for the reaction

%x + f(u,u,),

ut = with

equations

to this are models

a Hopf

with Neumann

to consider

bifurcation boundary

the effects

We find that from

rotating,

a nontrivial

conditions.

Time delays arise naturally in many biological and ecological systems, since responses may not always be instantaneous. For example, delays may represent resource regeneration times; maturation periods, feeding times, or be included to take into account the age structure of a population.

Delay equations

include

u(t-cr),

terms

a>0

such as

or

t .I’

k(t - s)u(s)

ds,

-cX

for some kernel lc. The former is discrete delay, and implies that one particular time in the past, is the most important; the latter gives a distributed delay (see [32] for a comparison). Good reviews of delay equations are given in [33,34]; more recent work and reviews of the analytical methods

available

are given in [35-371.

769

Rotating Waves To show how such waves can be brought equation

with discrete

the equations

about,

we consider

a general

delay on the unit disc with homogeneous

having

been centred

about

some positive

scalar

Neumann

equilibrium.

reaction-diffusion

boundary

Our model

conditions,

equations

are

V%(T, 0, t) - Ut(T, 8, t) = o! U(T, 8, t - T) + PU(T, 8, t) + f (u(r, 8, t - T), U(T. 8, t)) , T (5 10,11, 0 E iO.27r). VU(l, 8, t).t where f is the outward all the higher-order

= 0,

pointing

large class of models, delay logistic equation

7 > 0,

normal

(nonlinear)

to the boundary (i.e., the radial vector) and f represents a in U(T, 8, t) and U(T, 8, t - T). This equation represents

terms

once recentred with diffusion

about each of their equilibria (i.e., the Fisher equation)

in turn,

such as the classic

u(x, t - T)

(

V%(x, t) + pu(x, t) 1 -

Ut(X, t) =

(2)

K

.

>

A more plausible formulation of this model is given by

Ut(X, t) =

t) V2u(x,t) + pe-%(X, t - T) - p-U2(X! K

’

since it is difficult to find a meaningful biological interpretation for a delayed logistic equation, although its analysis is instructive for comparison with previous results. A meaningful food limited

model

has been proposed

by [38] ( see d iscussion

Ut(X, t) = An example diffusive

of a scalar

Nicholson’s

with specific

Once written

about

motivation

and derivation

is the

models

(one of) the uniform

t - 7-)e+++‘).

the population

data of [41,42] (see [43]).

equilibrium(ia),

each of these

equations

may be

in the form (2).

It is easier to consider the delay strength waves)

t) - 6u(x, t) + Pu(x,

by [40], and successfully

recentred

biological

equation

ut(x, t) = V2u(x, It was proposed

-

u(x, t - T) K + cu(x, t - T) K

V2u(x,t) + pu(x, t)

delay equation

blowflies

in [39]), given by

only

occur

bifurcations

7, even though in the presence

with respect

to the linear parameter

some of these bifurcations of the delay.

(including

Interesting

cy than with respect those leading

bifurcations

with

to

to rotating

respect

to the

parameter cy occur for both positive and negative cr, although for many biological systems, only one sign will be relevant. The paper [30] considers the same linear equation with spiral boundary conditions but without delay (see equation (1) above, where (Y = 0, /?I = -X), and found rotating solutions

only for positive

(instantaneous)

feedback

2. LINEAR Since we are assuming of interest,

the linearised

that

equation

equations

ANALYSIS

(2) is centred

about

ZLt(T,e,r) = UTT(T, 8, t) + +?+,

(p < 0).

around

this equilibrium, e,t) + &9(0,

some positive on the circular

uniform domain,

equilibrium are

t) - au(r, e,t - 7) - Pu(r, e,tj, (3)

Vu(r,

8, t)f = 0,

when r = 1.

We look for solutions of the linearised problem of the form U(T, 8, t) = eat eine 21,(r),

n E 2,

(4)

770

D. SCHLEY

so that (3) becomes

where the prime denotes differentiation /I

and letting z = fir,

w.r.t. r. Setting =

--cyemu7 - p - g

(5)

W,(Z) = U,(T), our equation becomes

wf+ LJ; + r

(

1-

$

>

21, = 0,

which is Bessel’s differential equation of order n. Solutions are, therefore,

the Bessel functions,

but since we seek solutions which are well defined everywhere on the unit disc, Bessel functions of the second kind Y,(Z)

are not admissible,

as they possess a singularity

at the origin.

The

solution of equation (3), in terms of the original variables, therefore, takes the form U(T, 8, t) = cot eine J, (4

7-).

(6)

Applying the homogeneous Neumann boundary conditions at r = 1, we require X6=0 or

J:,(fi) = 0, which has only real solution Jii

(7)

(see [44, p. 719]), i.e.,

The possible values of /.Lfor 0 < p < 100 are shown in Table 1. We know that these are the only such solutions for p in this range, since the smallest positive zero of .7;(+)

(Vn E Z) is always

strictly

greater than n. For each n E Z+, we shall refer to the ith positive solutions of (7) by n*i, and note that the smallest such value is /.L ‘11 = 3.390; /J = 0 is obviously a solution for all n. P Table

1. The solutions

of (7), and for which

7~ they

occur,

for 0 < b < 100

We now note that if cr = 0, then U=-$LL--, and only bifurcations to steady solutions (a E R) will be possible. In what follows, we show that the delay term alone (CX# 0, p = 0) may generate interesting behaviour by itself, such as periodic and rotating solutions (see below). From this point on, we therefore neglect the undelayed linear term and consider p = 0. The analysis is possible for general Q: and ,D, but does not reveal any new behaviour which is not present in the delay-only linearisation, and only complicates interpretation unnecessarily. Such a generalisation requires various additional criteria on our parameters, as a result of bifurcations only being possible in certain regions of the (a,@) parameter plane. What we are interested in here is showing that novel solutions exist, in all models of form (2). From what follows, the method for analysis of any specific equation (where restrictions of the (a, p) parameter plane being relevant) is clear.

usually result in only a subspace

Rotating

2.1. Bifurcation Such

to Inhomogeneous

bifurcations

increases.

will occur

We, therefore,

771

Waves

Steady Solutions

if an eigenvalue

look for solutions

crosses

the imaginary

axis through

of form (4) with 0 = 0, which

implies

zero as u

that

a = -p

by (5). Bifurcation will occur for each value LY= -/_L~I~ given by (7), and at cy = 0, since it can be shown that the roots always cross the imaginary axis (w.r.t. cy) with nonzero speed e -rr

da

-

=&#O.

da o=o = crre-o’ - 1 o=o Furthermore,

we see that

half of the complex bifurcation

for n = 0 (since

homogeneous

For future (see Section 2.2.

only

at cy = 0 will result

be nontrivial spatially

each of these

plane)

in a linear un(0)

we note that bifurcations

to Spatially

If p = 0, then

cross into the right

Q 5 0.

of the form eat einevn(0), so that

through

Homogeneous

c = iw (w > 0 without

either

Note

that

which

the bifurcating

real eigenvalues

at cy = o?(7),

bifurcation

Periodic

the

will only

solution

here is

will never occur if Q > 0

and

-,LL

which requires,

crsinwr

Using

+ l)$

to show that,

we need

2.1), so

for all k E Zf .

where kg

w’“(7) = (2k + l)$,

occurs,

(8)

in w = 0 (see Section

(8), 1 ‘t is simple w = We,

by (5), that

= w.

= 0. The first case results

with frequency

actually

Solutions

loss of generality),

+ 1) (k E Z+).

CY~(T) = (-l)k(2k that

=

a = 0 or coswr

wr = (7r/2)(2k

bifurcate

‘To confirm

solution

= OVn E Z/(O)),

crcoswr

namely

(roots

/_A2 0, we require

and of the form cut.

reference,

We now consider

solutions

is destabilising

Since

2.4).

Bifurcation

we consider

bifurcations

if ~7 > 1.

Z+.

to check the nondegeneracy

condition.

that a==a~(T)

This is simple

# 0.

to verify, since

so that 4 (-l)“+’ a=&(T) Once again, bifurcating

since p = 0, the only nontrivial

solutions

= 4 - (2k + 1)2n2’ solution

of form (4) is when

n = 0, so that

the

have the form u(r, 8, t) = eiwkcT) t.

The smallest such solution (w.r.t. CX)occurs when k = 0, so that the uniform to periodic solutions when cy = a?(~) = 7r/27. 2.3.

Bifurcation

to Rotating

We now show that solutions

when

equilibrium

bifurcates

Waves

,LL# 0 (Le., for each /.L”>~)we have bifurcation

for all T > 0, and furthermore,

that

a delay is required

w.r.t.

for this to occur.

cr to rotating

772

D. SCHLEY

tanm

Figure 1. For each p > 0 and T > 0, there exists a unique solution w of tanwT -w/p in each interval (10).

When

I_L# 0, we require

T, w, and o to satisfy tanwr

For each I_L= pnlZ, there in Figure

1. We denote

exist infinitely the solution $

when p = p,n,i by wnvzlj (r). is not included

cr > 0. Together

Bifurcation

many

a2 =ws+p2.

w satisfying

which occurs (2j + 1,zj

Note that

that (91

the above,

for every T, as can be seen

in the j th interval j E z+,

+ 2))

w = 0 has already

(10)

been considered

previously,

and thus,

here.

If w is in one of the intervals that

= -W P’

(8), implying

=

(lo), then cos wr < 0. If in addition

with the second

of (9), this implies

will take place if the nondegeneracy

p > 0, then it follows from (8)

that

requirement

is satisfied.

This is straightforward

to verify since Re

2 (

= >

coswr

- CYr

1 - c&r2 - 2ar cos WI-

# 0.

If this were not so, we would require p = -_(y27 by (8), but since 7, p > 0 and cy E R, we have a contradiction. We, therefore, have Hopf bifurcation whenever (1~= cr(n, i, j, 7) (n E Z+), where cr(n, i,j, T) = to solutions

+ (w”>“J(T))~, \/ (P”>“)~

of the form U(T, 0, r) = eiw’L,2x3 (7) t ,in 0 J,, (@r-j

+ C.C.

(11)

The smallest such value is o( 1, 1, 1,~). This follows immediately from the fact that cy is monotonically increasing in w and p. Since w is also monotonic increasing in ~_l.~,‘,the smallest o is given by the smallest p, i.e., pull1 (see Table 1). If n = 0, then we have no angular dependence.

Rotating

Waves

‘1 LAS 0 1

z

-2’

/ ~~

__

__

__

__

__

__

~_

__

__

__

__

__

__.__

__

~_

__

__

__

~_

_._

-.

~~

_.~

.~

-4

t -6

1 Figure 2. The bifurcation values of o w.r.t. the delay 7, to solutions of form (4). Bifurcating solutions may be steady homogeneous (a = 0 axis), steady inhomogeneous (dashed line), periodic homogeneous (dotted line) or periodic inhomogeneous (solid line) solutions, the latter giving rotating waves. For the range of a shown, only the bifurcation value a(l, 1, 1, r) gives such waves; it should however be remembered that there exist infinitely many bifurcation curves for each type of solution (except for bifurcations to homogeneous steady solutions, which occur only when o = 0) for T > 0, (II E R. The uniform equilibrium (zero solution) is linearly asymptotically stable in the parameter region bounded by Q = 0, T = 0, and a = a’(~).

resulting in target-patterns.

Solutions with radial symmetry obviously cannot be seen to rotate,

and we instead have bifurcation to oscillatory solutions. For all n, however, bifurcations are to inhomogeneous periodic solutions. Note that if r = 0, we see immediately from (5) that (T = -cr - p, and since p E R from (6) only real solutions for 0 are possible, so that Hopf bifurcation does not occur. Since we have Hopf bifurcation with S0(2)-equivariance,

the existence of nontrivial rotating

waves follows (see [45, p. 3591). We see from the form of the solutions (11) that because all solutions ~1of (7) are real, our solutions (rotating with frequency wnli”(r))

do not form spiral

waves. This is in contrast to [30], where rotating boundary conditions (‘spirals at infinity’) were considered, giving rise to solutions of form (11) but with complex CL,thus, leading to the evolution of near stable spiral waves. Figure 2 shows the bifurcation curves in the (7, T-)parameter plane for all the above-mentioned solution types. 2.4.

Steady-State

Stability

Having considered all possible bifurcations in previous sections, we may now give necessary and sufficient conditions for the spatially uniform linear equilibrium, solutions from which the above-mentioned

solutions bifurcate.

1. When p = 0, the zero solution of equation (2) is linearly asymptotically general perturbations of form (4)) if and only if THEOREM

o
stable

(to

774

D. SCHLEY

For the proof of this, we shall use the method characteristic equation is of the form

fiJ:,(Jis;) For stability,

we require

Re(a)

now consider becomes

the introduction

described

= 0,

Since this equation state

< 0 for all roots 0, which, since p > 0, simply of an infinitesimally

bifurcations

stable

have already

the first such solutions.

is ‘retarded’

remains

Note that

such time

been considered region

as some root

in the previous

(i.e., the ODE model),

occurs

is shown in Figure

this is also the stability so that spatial

3.1.

Explicit

we give examples

Our bifurcating The smallest

when

0 crosses sections,

appear

solutions

LYT = 7r/2, to periodic

properties

do not affect the local stability

uniform

delayed

the Such that

homogeneous linear

equation

of the equilibrium.

WAVES for rotating

waves.

will take form (6), and we are of course particularly

the frequency

of rotation

waves occur is cy = Q( 1, 1 , 1,T) , giving

+

c,c,

is not. against

T and 0 is possible

p).

n), or by changing

Examples

of such solutions

a linear

solution (12)

and the point from which it bifurcates The eigenfunction T in Figure in solutions

the ‘scaling’-the

is shown in Figure

of form

(ll),

but

only

3, and because

waves on the unit disc the order of rotational

range of the Bessel function

are given in Figures

are both

4.

Jn(JTir) is real for all p 2 0, n E Z. The complexity of any rotating may, therefore, be increased in two distinct ways: either by changing (increasing

in the

- to) + Q) J1 (1.841 r)

M cos (&+)(t

of the solution

L&~~~(T) plotted

of the variables

interested

of form (11) occur.

on the delay T, the eigenfunction

Separation

(increasing

Thus, axis.

and it has been shown

for the spatially

of the eigenfunctions

cx for which rotating

although

the frequency

symmetry

at --co.

the imaginary

condition

U(T,19,t) = e iJ31,1(T)t eieJ1 (&Jir)

dependent

equation

Solutions

values of cx and 7 when solutions

Note that

Q > 0. We

2.

3. ROTATING In this section,

requires

small delay, so that the characteristic

(in the sense of [47]), a 11 new roots

until

the delay, the

u-p--_e-(rT=O.

(in the sense of cv increasing)

The stability

without

p=---(Y-p.

dGJ&‘P) = 0, steady

in [46]. First,

5 and 6. The second

Jn(Ji;ir) of these is

equivalent to considering the problem on a disc with increased radius; indeed, Figure 3 may be viewed as a close up of the centre of Figure 5. When these two ‘effects’ are combined (i.e., bifurcations for higher and higher values of a), the resulting solutions may become quite complex. Figure 7 show the rotating wave solution bifurcating from Q = a(5,4, j, T), that is when n = 5 and ,u = p5,4 M 299.735. As an example! when 7 = 1 (quite a weak delay), the solution will bifurcate when a x 299.751, rotating with frequency 3.2.

3.131, whereas

when

T

=

10, cr M 299.735 but w z 0.314.

Stability

It is difficult to predict analytically whether these rotating wave solutions are stable, although it is reasonable to assume that most of them are not. The spiral waves found by [30] were computed to be unstable, but only just (in that only one Floquet multiplier was, slightly, greater than 1).

Rotating

\

x

.0.6

-0

Waves

775

A

I

-0.4

6

0.4

1

-1

Figure 3. The eigenfunction of solution (12), which bifurcates when a! = a(l,l, 1,~) (p z 3.390, w = w’~‘~‘(~)).

1

2

from the equilibrium

3

4

5

z

Figure 4. The frequency with which bifurcating solutions rotate is dependent delay 7, plotted here for the example (12) shown in Figure 3.

on the

776

D. SCHLEY

0.6

K

0

2

i

L

-0.6 -1

-0.8

0

-0.6

-0.4

o.4oGos6 1

-“.4 -1

Figure 5. The eigenfunction of the solution bifurcating from CY= ~(1, 7, j, r), given rotational frequency by (11) when n = ~11,7 z 447.931, n = 1 (note that the solution’s wiv7,j(~) will depend on which a the solution bifurcates from, both of which are functions of the delay T).

0.4 7

3 5

0

3

-0.4 -1

: -0.6

\ /

.6

.0.4

1 Figure 6. The eigenfunction of the solution by (11) when /J = p 531 z 41.160, n = 5.

-1 bifurcating

from cx = a(5,

l,j,

r),

given

.0.6 .6

Rotating

Waves

777

0.4

= s 3

0

-0.4 -1

/

A

-0.8

\

-0 .6

z -0.4

Figure 7. The eigenfunction

of the solution bifurcating

from a! = c~(5,4, j, T)

4. CONCLUSIONS We have shown the existence homogeneous

Neumann

Such solutions

may bifurcate

not form in this manner. system

provided

spiral

of rotating

boundary

wave solutions

conditions,

provided

from a nonnegative

This is in contrast boundary

conditions through

positive

through

negative

waves brought about

about

by rotating

by delay (with Neumann

clear extension numerically

boundary boundary

simulation

although

feedback

It is perhaps

conditions)

are for fundamentally

the stability

worth

with

spiral waves do in a scalar

mentioning

(Q > 0, ,B = 0). Thus,

(in the undelayed

on a circular

systems

in the model.

that

(p < 0 with no delay, (Y = 0), whereas

feedback

conditions

of this work is to investigate

through

equilibrium,

were imposed.

our rotating

bifurcate

uniform

diffusion

delay is present

to [30], where such waves were possible

the spiral waves in [30] bifurcate solutions

in scalar reaction a suitable

model)

of such solutions,

different both

the rotating

and those brought systems.

analytically

A and

domain.

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