Spiral waves, chaos and multiple attractors in lattice models
of interacting populations
Ricard V. Sole, Joaquim Valls
Complex Systems Research Group, Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica 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
Stable spatial patterns are shown to appear in a 2D coupled map lattice (CML) approach. The model is constructed from a
two-dimensional nonlinear map based on the dynamics of interacting populations. Such structures are mainly spiral waves. Chaotic and periodic dynamics are present for different parameter combinations, as shown by the largest Lyapunov exponent. Multiple attractors are present for some range of initial conditions and spiral waves are also present at the chaotic domain of parameters. Implications for ecosystem dynamics are also discussed.
In recent years, increasing attention has been given
to nonlinear phenomena and chaos in natural systems [11. As is now well known, low-dimensional
dynamical systems can display complex behaviour,
often related to the existence of a fractal attracting
set of trajectories. The dynamics of biological populations is in fact one of the first fields in which chaotic dynamics was succesfully applied through the
use of discrete map models [2]. On the other hand,
the introduction of spatial degrees of freedom involves the emergence of new, and sometimes unexpected, properties [3]. Spatial structures are actually known as the spontaneous behaviour of many
physical systems when driven far from equilibrium.
The appearance of such an organization can take
place in several ways, involving Turing structures,
excitable waves and solitons [4] and even more exotic phenomena like chaotic Turing structures [51.
Recently, it has been shown that complex spatial patterns are present in simple two-dimensional models
of interacting populations. Spatial patterning is in
fact a key point in the understanding of ecosystem
.
dynamics. In this sense, models of nonlinear dissipative dynamical systems with spatial degrees of
freedom can provide a framework for a theory of
ecosystem organization [61 or even in chemical evolution [7]. In such systems, recent studies showed
that space and time cannot be separated: they are two
faces of the same coin [6,8,9].
The appearance of some particular types of spatial
structures like spiral waves has been particularly well
studied. Target patterns and spiral structures have in
fact been observed not only in chemical systems (like
the Belousov—Zhabotinskii reaction [101) but also
in some well known biological systems like the slime
mold Dictyostelium or in calcium waves [111.
For those dynamical systems defined by discrete
maps, the spatial counterpart has been described by
the so-called coupled map lattice (CML) formalism
[12]. A CML is a dynamical system with a discrete
time, discrete space and continuous state. Several
studies on reaction—diffusion models have used CML
approaches in order to simulate dissipative structures [13]. In this paper we will see that spiral wave
patterns can be modeled using a CML approach. Interestingly, such waves are present even in the chaotic domain of parameters. A two-dimensional map
with biological relevance will be used in our study.
In this Letter we consider a 2D CML model based
on a nonlinear map:
x,,÷
=
ix,( I —x,,)F(x,,, y,,)
periodic windows are also observable. In fact, it can
be easily shown that such ‘host—parasitoid” map
models are unlikely to have stable periodic orbits of
order below six [15].
We can construct a CML counterpart in a straightforward way,
(la)
(Ib)
which has been used, for a suitable choice of F, in
population dynamics [14]. Here x,, and y,, are prey
and predator populations at a given generation n
(technically, a so-called “host—parasitoid” model).
We will assume that FECr(U) (r>l), Ucil2
being an open set. We consider two additional conditionsonF(a)F(x, oo)=O and (b)Fp(x, O)1.
:
In this sense, if y= 0, eq. (la) is a logistic map with
a period doubling route to chaos. Chaotic dynamics
appears for y=3.5699... [2]. In this Letter we use
F(x, y) =exp( —fly). A rich set of bifurcations takes
place for this map. Here we use y=4 and several
combinations of fi. The corresponding bifurcation
scenario is shown in fig. 1 for fin (1, 7). We can see
that for fl<fi= 1.75 chaotic dynamics is present as
a consequence of the dominant logistic part, i.e. of
px(l —x). Forfie(fi, 3.4), the chaotic attractor collapses to a stable state, followed by quasiperiodic attractors, six-periodic orbits and again chaos. Some
+l)1V2x(r),
(2a)
y±1(k)=x(k){l —exp[—fly(k)]}
(2b)
in which the coupling is given by a discrete diffusion
operator, i.e.
where jeA(L) are the q nearest neighbours (here we
take q = 4). An additional rule is used in order to
avoid negative populations (and numerical instabilities) i.e. x(k)=0 ifx(k)<O.
The dynamics is then defined on a two-dimensional lattice
A(L)={k_—(i,j)eZ2 Il i,jL}.
An interesting situation is an initial small number
of predators appearing at random positions into
A(L). In this sense, we consider an initial condition
for which both populations are randomly generated
as
öl <x0(k)<5 VkEA(L),
and
ö <yo(k)<c52 VkeK,
Fig. I. Bifurcation diagram for model (1). Here i=4. Arrows
show (from left to right) the three studied situations, i.e. quasiperiodic, periodic and chaotic motions (see the text and the following figures).
124
K={k1, k2,.., k} being a finite number (s<L2) of
randomly selected lattice positions (seeds) keA (L).
The ô, are such that 0< ó1 <J2 < 1; here we take
5=0.3 and 52=0.4. Using some fixed number of
initial seeds, we can study the effect of initial conditions on the dynamics of our system.
The time evolution of populations for different
parameter combinations has been studied together
with the corresponding spatial patterns. Lattice sizes
up to 512 x 512 have been used with both periodic
and zero-flux boundary conditions. Some examples
of the obtained patterns are shown in figs. 2a—2d; in
B
D
Fig. 2. Spiral waves obtained from system (2) for 256 x 256 lattices. Here i=4 and (a) fl=4.25 (periodic), D1 =0.001, D2=0.20 with
50 initial seeds, (b) same as (a) but /3=4.0 (quasiperiodic map), (c) same as before but /3=5.0 (chaotic map), (d) here /3=4.0 but
= D2 = 0.11. The last situation is linked with the transition from spirals to chaotic patterns.
figs. 2a, 2b spiral waves are generated at the periodic
(46=4.25) and quasiperiodic (46=4.0) states. Such
spirals are stable structures and as we will see a long
transient is necessary in order to reach the steady
state. The unexpected result of our study comes from
the chaotic domain, i.e. flCh> 4.64. In such a domain,
no well-defined structures are expected to appear.
When chaos is present, the coherence length decays
as Aj (a 1), )L being the largest Lyapunov exponent. In our model, robust spiral waves have been
observed (fig. 2c) for up to fl6.2 and no pattern is
observed for larger values. These chaotic spirals are
not locally stable: they move through the lattice and
are created and destroyed; the current position of a
given spiral will change with time in an apparently
random way.
In order to characterize the previous results, a
quantitative measure can be obtained through the
largest Lyapunov exponent Am. We compute Am from
the global x-population,
125
X(n)=
x(k).
D={(D1,D2)ID2k—D1}
keA(L)
As is well known, the Lyapunov exponent can be estimated from
Amz
following the Wolf algorithm [161, r being the number of sampled points and
6(r,
IIX(t±e)—X’U+z)II
t)=
IILI’ &)II
An extensive numerical study of Am was performed for several CML configurations using lattice
sizes up to 100 x 100. The number of seeds was s= 50
and calculations were made using 2500 time steps
after 1500 transients. By means of Am we can characterize the phase space of our system. The qualitative behaviour of our model is shown in fig. 3.
For /ie (3.5, Pch), i.e. in the quasiperiodic and periodic domain, a zone of diffusion induced chaos is
observed around the critical condition
Fig. 3. Qualitative behaviour of the phase space of model (2) for
several diffusion rates, obtained from the computation of the
largest Lyapunov exponent. Three main zones are observed. Below the dashed line (Do, see text) spiral waves are observed
Vfle(3.5, 6.2). For fl.cflh the Lyapunov exponents are zero. A
chaotic domain is observed over 13, in which no coherent spatial
patterns are observed. The size of this chaotic region depends
upon fi. A frozen lattice behaviour is obtained over such a chaotic
zone, with period-three oscillations.
126
Here the line given by D defines two separated regions, namely D (for D2<—D1) and D (otherwise). For the previously mentioned fi values, the
Lyapunov exponent is zero and spiral waves appear
for all (D1, D2)eD,j. As mentioned before, even in
the chaotic domain spirals will appear but also restricted to the subset D Over D, diffusion induced chaos is then present with no well-defined
spatial patterns, although spatial correlations are still
present (see fig. 2d). In this region Lyapunov exponents are Am 0.4—0.5. Small stable period-three
(3P) domains are now present. As we move far from
D, frozen 3P domains grow and percolation of such
domains leads to chaos suppression. In the frozen
pattern zone, period three is observed and a “crystal
lattice” is spatially present. This phenomenon is in
fact well known for the logistic lattice in relation with
pattern competition and checkerboard patterns [12].
The CML model is a dynamical system of a large
number of equations (we have in our two-dimensional lattice 2L2 equations). In such a system, it can
be expected to have a large number of steady states.
For small numbers of seeds, macroscopic structures
are present as we have seen before, and the underlying dynamics shows several behaviours, characterized by Am. However, as the number of initial seeds
is increased, the relevance of initial conditions becomes less important. After a given critical threshold
s is reached, there is no observable important change
in the largest Lyapunov exponent Am. In figs. 4a, 4b,
we can see how initial conditions modify the qualitative dynamical behaviour, for (a) periodic
boundary and (b) zero-flux conditions. In this context, we can imagine the observed result as the existence of multiple attractors in the underlying 2L2dimensional phase space. For some initial conditions, the system is attracted to a basin linked with
macroscopic coherent structures. An explanation of
the observed variability in Lyapunov exponents
comes from two origins. First, the chaotic behaviour
of the logistic CML part introduces a high instability
in the lattice if S<<Sc and a long time is necessary in
order to stabilize the patterns and to eventually fall
into a low-dimensional attractor. On the other hand,
a long pattern selection process takes place between
spirals. Even for small lattices, r 3x lO time steps
.
00000
a
beta,4.OO
b
)
A
beta.5.OO
.
ci)
A
A
0
.
Q)
a)
A
0
-
A
A
.
A
•o
0.1
A
00.2-
A
0
A
A
0
0
0
ci
ct1
0
00
. 0
0.
o.
0
0
0
0.
0
0.0
•
0
0
•
0
Jr
11
I
Iii
•
•J
120
0
Number of
-
seeds
0
00
O
••
0
40
80
120
Number of seeds
Fig. 4. Dependence on the initial conditions. The existence of multiple attractors for our CML is shown. Calculations were performed
using 2500 time steps after 1500 transients. The largest Lyapunov exponent for a given set of parameter values is shown to be very
sensitive on the initial number of seeds. Here a 50x 50 lattice is used, D1 =0.001, D2=0.20, j=4 and several fl-values. Both periodic (a)
and zero-flux (b) boundaries are considered. The arrows show the critical value s, (see text).
A
B
C
Fig. 5. Transient behaviour for a quasiperiodic map, with fl= 4.0 and D1 =0.001,
= 0.20. Competition and interaction between spirals
take place. Here we show three snapshots at (a) t= 25 x I 0, (b) 5 x 1o and (c) lO time steps.
are necessary in order to decrease a positive Lyapunov exponent towards zero in the periodic and
quasiperiodic domains. An example of this transient
behaviour is shown in fig. 5 for a 512 x 512 lattice.
After r= lO time steps, a unique spiral is going to
dominate, but other initial conditions (with only a
small initial difference) lead to a much longer transient. So, sensitivity to the initial number of seeds is
then very high.
For s> s, the Lyapunov exponents for the periodic and quasiperiodic parameters decay to zero, because of the number of initial seeds, which is large
127
enough to generate some kind of “percolation”. In
such cases, the chaotic logistic transient is suppressed and periodic or quasiperiodic temporal patterns are well defined. Such a situation is not present, as expected, for the chaotic state.
The present results illustrate the importance of the
initial conditions in these lattice models of interacting populations. The existence of such a dependence
on the number of initial seeds and the appearance of
coherent structures in the chaos regime can be particularly important in theoretical ecology. Our example shows clearly that interacting populations can
show self-organization in such nonequilibrium conditions. The relevance of such a behaviour has been
considered in experimental and theoretical studies
[17] of particular importance in our understanding
of ecosystems. Further studies will be necessary in
order to characterize the general properties of these
systems and the existence of scaling laws.
The authors would like to thank Xavier Cañete and
Joan Antoni Blanco for their help at different stages
of our studies. We also thank Jordi Cortadella and
Laurence Jacobs for his help with connection machine algorithms as well as the European Center for
Parallelism of Barcelona. This work has been supported by grants of CIRIT ref EE91 / 1.
References
[1] L. Glass and M.C. Mackey, From clocks to chaos (Princeton
Univ. Press, Princeton, 1988).
128
[2] R.M. May, Nature 261 (1976) 459;
M.J. Feigenbaum, J. Stat. Phys. 21(1979) 669.
[31 J.D. Murray, Mathematical biology (Springer, Berlin,
1989).
[4] T.S. Akhromeyeva, S.P. “i,rdyumov, G.G. Malinetskii and
A.A. Samarskii, Phys. Rep. 176 (1989) 189;
0. Erti, Science 264 (1991)1750.
[5] R.V. Sole, J. Bascompte and J. Valls, submitted to J. Theor.
Biol.
[6] R.V. Sole and J. Valls, J. Theor. Biol., to be published.
[7] M.C. Boerlijst and P. Hogeweg, Physica D 48 (1991) 17;
R.M. May, Nature 353 (1991) 607.
[8] R.V. Sole and J. Valls, Phys. Lett. A 153 (1991) 330.
[9] M.P. Hassell, H.N. Comins and R.M. May, Nature 353
(1991) 255.
[101 A. Winfree,Science 175(1972)634.
[111 J.J. Tyson, K.A. Alexander, V.S. Manoranjan and J.D.
Murray, Physica D 34(1989)193;
J. Lechleiter, S. Girard, E. Peralta and D. Clapham, Science
252(1991) 123.
[12]K. Kaneko, Physica D 34 (1989) 1; 37 (1989)60; Phys.
Lett.A 149(1990) 105.
[13]M.Chee, R. Kapral and S. Whittington,J. Chem.Phys.92
(1990)7315;
D. Barkley, M. Kress and S. Tucherman, Phys. Rev. A 42
(1990) 2489.
[141 M.P. Hassell and R.M. May, J. Anim. Ecol. 42 (1973) 693.
[15]R.May,Proc.R.Soc.A413(1987) 27;
H. Lauwerier, in: Chaos (Manchester Univ Press,
Manchester, 1986).
[16] J.B. Wolf, H.L. Swift, H.L. Swinney and J.A. Vaslano,
PhysicaD 16(1985)285;
R.V. Sole andJ. Valls, Phys. Lett. A 161 (1991)241.
[171 K. Kaneko and T. Ikegami, Physica D 56 (1992) 406.