Spirals and targets in reaction-diffusion systems
A. Bhattacharyay
Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India
Received 21 November 2001; published 18 June 2001
The existence of spirals and targets is common in reaction diffusion systems of excitable dynamics. I present
a multiple scale perturbation analysis to show the existence of all these patterns near a Hopf bifurcation
boundary in a Turing type reaction-diffusion system.
DOI: 10.1103/PhysRevE.64.016113 PACS numbers: 82.40.Ck, 47.54.+r, 47.70.Fw
In two-dimensional reaction-diffusion systems, rotation-
ally symmetric patterns, known as targets or sinks, and a
generalization of them with broken circular symmetry, i.e.,
spirals are being investigated experimentally as well as theo-
retically 1 in many nonlinear systems. The Belousov-
Zabotinsky reaction is a well investigated excitable reaction-
diffusion system that shows all these patterns 2–4. Spirals
are characteristic patterns in slime mold aggregates 5–8
and are an important observation in cardiac arrythmias 9 as
well. Targets and spirals, which are generally found to form
around some defects, precede some defect mediated chaos,
commonly known as spiral defect chaos 10,11. All these
have made the study of the origin and stability 12 of these
patterns a subject of renewed interest. In this paper the exis-
tence of spirals and targets is reported in the Gierer-
Meinhardt GM model, representing a reaction-diffusion
system for biological pattern formation. Here I work near a
Hopf bifurcation boundary which separates the Turing state
and a homogeneous steady state from a homogeneous oscil-
latory state. The GM model represents a Turing type
reaction-diffusion system, which to my knowledge, has not
been investigated analytically for the existence of targets and
spirals.
In general, an investigation of targets in a system starts
with the inclusion of an additive inhomogeneity in the phase
equation followed by a Cole-Hopf transformation that gives
the equation the shape of a linear Schro
¨
dinger equation. The
target is a bound state of the Schro
¨
dinger equation; it then
remains to obtain the appropriate form of the inhomogeneity
from a suitable potential. This type of approach has made it
particularly controversial, whether intrinsic targets exist in
the vicinity of an oscillatory state or not. There is an explicit
solution for spirals in the - system due to Hagan 13. The
- system in simpler cases has the structure of a complex
Ginzberg-Landau equation without an imaginary part in the
coefficient of
2
. In his analysis, Hagan was of the opinion
that a spiral of a single branch will only persist if the higher
order spirals are unstable. It is also important to note that the
spirals obtained in these way are characterized by a quadratic
dispersion relation. Complex Ginzberg-Landau type ampli-
tude equations and eikonal equations 1, developed from the
curling up of a line defect, are standard approaches with
which to show spiral patterns. In what follows, we arrive at a
linear amplitude equation from the solvability criterion ap-
plied at first order in a perturbation expansion of the GM
model using multiple scales. It is shown subsequently that a
general solution of a spiral exists for that amplitude equation
in a region of phase space where a homogeneous oscillatory
state is stable. Targets and stars are shown to be special
cases. These things occur without any externally imposed
local inhomogeity. The other part of the result is obtained by
imposing an inhomogeneous distribution on the removal rate
of the interacting species; in the asymptotic region this gives
incoming spirals and targets with particular wave numbers.
It was Turing who first showed that the interaction of two
substances, say A and B, with differerent diffusion rates can
cause steady patterns to form. Two basic properties that ac-
count for the formation of pattern in a Turing system are
local self enhancement and long range inhibition. Local en-
hancement causes inhomogenieties to grow and long range
inhibition confines that effect if the antagonist is taken to be
fast diffusing. The two species A and B constitute a reaction-
diffusion system 14, known as the GM model:
A
t
=D
a
2
A +
a
A
2
1 +k
a
A
2
B
-
a
A +
a
, 1
B
t
=D
b
2
B +
b
A
2
-
b
B +
b
. 2
Here D
a
and D
b
are diffusion constants such that D
b
D
a
the condition for formation of a Turing pattern,
a
and
b
are the basic production terms, and
a
and
b
are
removal rates. All of these parameters are real and positive.
The natural pattern formation requires
a
b
to make an
autocatalytic local amplification of A effective to form steady
patterns 14. In the above mentioned model,
a
and
b
are
cross reaction coefficients; k
a
is a saturation constant, which
actually plays a role in determining the shape of the pattern,
and is also real and positive.
The model is simplified by setting k
a
=
B
=0, to make an
analytic treatment more easy without any loss of the qualita-
tive nature of the result. Then a change of scale as t
¯
=
a
t ,
l
¯
=
a
/ D
a
l , a =(
a
b
/
b
a
) A , and b =(
a
2
b
/
b
a
2
) B
allows us to reach the following set of equations 14:
a
t
¯
=
¯ 2
a +
a
2
b
-a + , 3
b
t
¯
=
¯ 2
b + a
2
-b . 4
Here D =D
a
/ D
b
, =
a
/
b
, =
b
a
/
b
a
, and
¯ 2
is the Laplacian operator in a changed length scale. The
PHYSICAL REVIEW E, VOLUME 64, 016113
1063-651X/2001/641/0161134/$20.00 ©2001 The American Physical Society 64 016113-1