Chaotic Turing structures
Ricard V. Sole
Complex Systems Research Group, Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya,
Pau Gargallo 5, 08028 Barcelona. Spain
and
Jordi Bascompte
Complex Systems Research Group, Departament d’Ecologia, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain
Turing-like structures are obtained in a two-dimensional coupled map lattice model. Such structures have a well-defined stable
pattern in space, but highly chaotic local fluctuations can also be observed. Two well-defined regions in phase space are separated,
defining the domains of emergence of phase separation. Chaotic behavior is widely present in both situations and the Allen—Cahn
theory of phase separation is applied. The transient behavior is also analysed and some implications for species competition in
ecology are outlined.
Starting from a small random initial perturbation
around a spatially homogeneous chemical concetration, an ordered macroscopic spatial pattern can
emerge from microscopic chaos. Such a surprising
result was obtained by Alan Turing in 1952 [1] in
relation with the chemical basis of morphogenesis.
Further extensive studies [2—5] revealed a rich variety of ordered patterns when spatial degrees of
freedom were introduced. Among other complex
phenomena, pattern formation in systems far from
equilibrium is certainly one of the most challenging
problems of nonlinear science. In this context, bifurcations of spatial and temporal structures have
been widely analysed in recent years [6] including
the conditions for soft- and hard-mode instabilities
and chemical turbulence.
Spatio-temporal chaos has been studied by means
of several mathematical and numerical tools. Coupled map lattices (CML) [7—91 (see also ref. [10]
for a recent review and applications) have been used
as the discrete alternative to partial differential
equations (PDE) which are usually difficult to manage when numerical integration needs to be per-
formed. The CML approach has been applied to several situations ranging from physical to biological
problems [11—13] and is actually among the most
popular objects of nonlinear dynamics.
A CML model is defined by a set of coupled maps
given by
(1)
Here 4(k) is the value at the time step n and at the
lattice node k of the ith variable involved, with i, j= 1,
N. Heref,, is assumed to be C(U) on a given
open set U c P and {,u, y} are some fixed parameters.
The space is defined as a discrete cl-dimensional lattice, i.e.
A(L)={k= (k1,
where k= 1,
...,
lcd): keZ}
(2)
L andf is defined by two terms,
in which F,,
represents the original rn-dimensional map and C,,
gives the specific kind of coupling.
Recent studies on lattice ecosystems have shown
that spatial order can coexist with highly chaotic local dynamics [14]. Usually, the chaotic temporal
...,
evolution is associated with propagating structures,
as chaotic spiral waves [151. As we will see, such a
situation can also be observed in a system in which
symmetry breaking leading to Turing-like patterns is
observed. Stable patches are formed, and the boundaries between these structures are stable in time even
in the chaotic domain of the parameters.
In this Letter we consider a 2D CML model based
on a two-dimensional map defined as
x,÷1
=F(U(x, y)=#1x( 1 —x,, —/31y)
,
,
(3a)
(3b)
which has been used as a model of competing populations [16,17]. Here x,, y,, are the population sizes
at a given discrete time step n, and (jig,
are the
growth and competition rates, respectively. Four
steady states are possible, i.e. P0 (0, 0), exclusion
points, P1= (1— l/M 0), P2= (0, 1— l/z2) and the
nontrivial coexistence point P, = (x*, y*) We will
here consider the completely symmetric case, i.e.
f3=fl and u=u for i= 1, 2. For this particular situationwe havex*=y*=(l_l/u)/(l+fl).
For this model, it is easily shown that competitive
exclusion may take place involving the extinction of
one of the competitors. For /1=0 we have two logistic
maps which show a period doubling route to chaos
for increasing t-va1ues. Chaos is observed for
The linear stability analysis leads
to the stable domains defined by the sets
S(P12)={(jt,fl)I /1>l;ue(l, 3)}andS(P)={(u,
/1)1 fle(O, 1); ue(1, 3)}. In ecological theory /1<1
is related with the so-called “coexistence” of species;
for/I> 1 coexistence is no longer possible and exclusion of one of the competitors takes place. In this
sense, fl =fl and jz =122 implies that, depending on
the initial conditions, one of the competitors will be
eliminated. It can easily be shown that for such a
symmetric case, the system undergoes a period-doubling scenario with identical bifurcation points as the
single logistic map. Now we will see that when the
spatial model is considered, competitors become
separated in well-defined and stable patches even
when strongly chaotic dynamics occurs. This result
is in agreement with previous studies on CML models
of interacting populations [10,121 where coexistence of temporal chaos and spatial order was shown
to exist. Such results have also been proved in other
/11)
326
similar discrete models of population dynamics [181.
The CML counterpart of eqs. (3a), (3b) is, for a
two-dimensional lattice A (L) a set of 2L2 equations,
x1(r)=ux(r)[1 —x(r)—fly(r)]
+D1V2x(r)
(4a)
,
y41(r)=uy(r) [1 —y(r) —/Ix(r)
+D2V2y(r),
(4b)
with r= (i, j) and where the coupling is defined as
V2x(r)= x(j)—qx(r)
(5)
Here q is the number of nearest neighbors (here we
take the Moore neighborhood). An additional rule
is used in order to prevent negative populations (and
numerical instabilities), i.e. x(r) =0 if x(r) <0 but
it only need to be applied for strongly chaotic states.
In this context, the continuous model has been recently analysed and the appearance of spatial heterogeneities was reported [191. It is important to
mention that this kind of formalism is able to describe several physical and chemical problems of interest. It is closely related with particle—antiparticle
annihilation in diffusive motion [20] and, in this
context, with cosmological problems as the fate of
singularities in the early universe [211. However, for
the continuous counterpart only steady state patterns are obtained. In our approach, because of the
discrete nature of the formalism, more complex patterns are expected to appear.
The stability of low-dimensional periodic orbits
can be analysed following previous studies [8,22].
In our case, denoting by (k) the transform of x(k’),
the corresponding Fourier transform of the linearized equations will be defined by
&(k, t)=L(k,P*)&(k, t)
.
(6)
The calculation of bifurcations leading to stable spatial patterns is performed by means of the characteristic equation [221
P()= ILp(oiF
U)
(P*))+ [De(k)—A)]oI
(7)
(‘ (P))
being the Jacobi matrix. The spatial dependence is introduced through the wavevector
LMP(ÔIF
8(k)
[(Jt(k+iv 1)) cos (7t(N
k_l)’
k=(k,1).
(8)
Following a previous analysis [22], we can find
the stability boundaries for our CML. Using fixed Dvalues, the domains in the phase space
F={(#,fl)I pE(O,4);/3>O}
are obtained from the critical condition A= ± 1. The
eigenvalues corresponding to the first two critical
points are given by
A1(k)=2—p+De(k)
(9a)
,
22(k)=u(l—fi)+fl+D8(k).
(9b)
And for the coexistence point we similarly obtain
21(k)=2—p+D8(k) and22(k)=fl[1—(1—1/U)/
(2 + 41?)] + D6’(k), but here we restrict out attention
to the first two. Using the first couple of eigenvalues
and the marginal stability conditions given by
)4f(k) = ± 1 we find the critical boundaries
(lOa)
(lOb)
[JL—l+D8(k)],
(lOc)
[u+l+D(k)].
(lOd)
For k= (0, 0) (i.e. spatially uniform states) and for
k= (iN, N) (i.e. checkerboard patterns), we have
(k) =0 and —8 respectively. Using these extremes,
the boundaries for symmetry-breaking bifurcations
leading to stable heterogeneous states are obtained
from
4=1, 4=3,
4/2=4
—
8D,
where
and 2 are the values
g, /i) for the
wavevectors k= (0, 0) and k= (iN, iN), respectively. As expected, the introduction of diffusion
terms generate a shift in the appearance of instabilities. For D=O.05, the phase space of our model is
shown in fig. I. This picture has been partially obtained from the previous stability analysis and also
from numerical studies, based on the computation of
the largest Lyapunov exponent [231. In all the studied (ii, fi)-combinations Turing patterns were
observed.
Some examples of our numerical simulations are
shown in fig. 2 for a 256x256 lattice. Forfi< 1, no
macroscopic spatial structures are observed (here
only the checkerboard pattern is stable as in the logistic CML). For /3> 1, and after some transients,
patches emerge and spatial symmetry breaking takes
place.
Patches can be defined as connected clusters of sites
on which the difference x—y has a definite sign. The
boundaries of these patches are stationary even if the
motion inside is chaotic. A careful study shows that
the distribution of x—y I inside any given patch is
very similar to the distribution of x for a logistic CML
with identical parameter ji [81.
In fig. 3 we also show the local dynamics of both
populations for chaotic parameter combinations in
a 20x20 lattice. Here /3=1.2, D=O.02 and u=3.6
2.0
(lla)
4/2=4
—
8D,
(I lb)
and
fl=l, fl=(p+l)/(#—l),
ON/2
p+
00
—p+—
/JI2fl_
(llc)
8D
/1—1
8D
(ile)
/2
Fig. 1. Phase space of models (4a) and (4b) for parameters involving the emergence of Turing patterns. Dotted domains indicate period-two oscillations (2P) and black regions corresponds
to period-four oscillations (4P), indicated by arrows. The vertical dashed line corresponds with the critical line ,t2 from eq.
(1 lb) and the left curve fiN/2 with eq. (lie).
327
b
d
Fig. 2. Snapshotsofourmodelfora256x256latticeforfl=1.2 and (a) D=O.O1,p=2.5, (b)D=O.O1,=3.6, (c) D=O.05,#=3.6 and
(d) D=O.05, jt=2.5. Here black points indicate x(i,j) >y(i,j). (a), (d) are stable patterns and (b), (c) are chaotic Turing structures.
(a) and 4.0 (b) after 100 transients were discarded.
Local extinction occurs from time to time in both
populations, including the dominant one. In fig. 3b
it can be observed how one of the species replaces
the other one: this situation is linked with the final
steps of local phase separation. At n 115, the dominance is yet established and the boundary is now
stable. The sharpness of the boundaries depends on
the specific value of fi. As /3 increases beyond /3 = 1,
the smooth boundaries are replaced by a more and
more sharp separation between the patches. In this
context, the transient time associated with the creation of boundaries between the patches will strongly
decrease with higher /3 values.
The way in which the two “phases” compete can
be analysed by studying how they shrink or grow in
time. Following the Allen—Cahn theory [241 we can
describe the dynamics of phase separation in systems with no conserved order parameters. Using this
approach, we can compare the late-stage behavior in
ordered and chaotic situations. One of the conse328
quences of this theory is that the domain size R(t)
scales as 1 Following Kapral [22] we consider here
the order parameter
‘•
w(r)=x(r)—y(r)
(12)
,
and the Allen—Cahn theory is applied by considering
the evolution of the following dynamic structure
factor,
If
Sw(On)çL2
kEA(L)
wn(r)),).
(13)
Here S,(O, n) is a particular case (when k= 0)
of the more general expression S(k,
R2(n)(kR(n)), with
the Fourier transform of
the order parameter. Here S(O, n) is expected to
scale as S(O, n) x n. In fig. 4 we show some of our
calculations for a 40 x 40 lattice. For fi= 1.2 and
D=0.05, we observe that for#< 2.8 the scaling islinear in n (as predicted) for n> 50. For periodic and
weakly chaotic states, when te (2.8, 3.6), the aver-
L=20x20
mu=3.6
beta=1.2 D=O.02
L=20x20
mu=4.O
beta=1.2 D=O.02
a
0.2
time
b
Fig. 3. Local dynamics of our CML model for both populations at r= (7, 7) after t= 100 transients were discarded. Parameter values are
indicated at the top in both figures. The arrow in (a) indicates an extinction event for both species. In (b) we can see how, at then 110th
step the dominant competitor is replaced by the second one. This situation is linked with the formation of a stable boundary in that
particular point.
D=0.05
D=O.05 beto=1.2 mu”3.6
beto=1.2
0.125
1.8[—003
a
b
1.5E—003
0.100
1.3E—003
‘0.075
1.0E—003
C
C
0.050
U)
7.5E—004
5.OE—004
0.025
2.5E—a04
0.000
6
time
0.OE+000
time
Fig. 4. Time evolution of the dynamic structure factor S(0, n) defined by eq. (13) for a 40X40 lattice with D=0.05 and fl= 1.2 using
several #-values. For t <2.8, linear growth is observed after n 50—60 time steps and this also applies for periodic and weakly chaotic
states, as for z= 3.2. For strongly chaotic motion, the linear scaling breaks down (b).
age behavior of the previous quantity also tends to
show linear scaling, even when an oscillation is observed (as for example in =3.2). For higher jtvalues (say for example u=3.6). in spite of the fact
that we certainly observe Turing patterns, the scaling
breaks down probably because of the existence of
strong local fluctuations.
Finally, the transient length T(L) has also been
329
state seems to be a key ingredient in order to support
high levels of species diversity in ecosystems. In our
model competition is strongly reduced when chaos
is present and coexistence is possible for a long time
in such a nonequilibrium state. Further studies will
be necessary in order to analyse the general validity
of these results.
480
-
440
ooona L=ixi
L=5x5
AAà
400-
36O-
NREP=100
D =002
fi = 1:20
320-
w
:
-
— 280-
240-
a
D
200E-1160120AO
2.00
2.50
3.00
I”,”
3.50
4.00
Fig. 5. Transient behavior of global exclusion for two small lattices. Here NREP = 100 initial conditions are used (see text).
Parameters are indicated. A maximum is obtained atp 3.84.
analysed for small lattices. We start with initial conditions x0(r), y0(r) =P+ with random e(—0.l,
0.1) and define T(L) as the time until all sites belong to one single patch. For L = 3, 5 we have estimated numerically the value of T(L) for fl= 1.2 and
D = 0.02 averaging over 100 different initial conditions. In fig. 5 we show how the transient length increases as we move inside the chaotic domain, with
a maximum at
3.84.
As we already said, the classical prediction of ecological theory [251 leads to the so-called competitive
exclusion principle: for fi> 1 two competing species
are unstable and unable to coexist when they are
similar. In our case, similarity (i.e. equal sets of parameters) leads to strongly stable spatially heterogeneous configurations. Here spatial degrees of freedom give a rich set of dynamical behavior together
with a stable coexistence. The underlying temporal
dynamics is not restricted to stable points; periodic
and chaotic attractors are also possible.
The existence of long transients is also relevant in
relation with the ecological organization. As recently
shown in several studies [12—141, a nonequilibrium
330
One of us (RYS) would like to thank A.S.
Mikhailov and H. Meinhardt for very interesting
comments about these structures at the Bielefeld
Workshop on Synergetics, as well as an anonymous
referee for several corrections to our first manuscript. We also thank Quim Valls, Jordi Cortadella
and Laurence Jacobs for his help with the Connection Machine algorithms as well as the European
Center for Parallelism of Barcelona. This work has
been partially supported by grants of CIRIT EE917
1 and UPC PR9119.
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