Rethinking
complexity:
modelling
spatiotemporal
dynamics
in
ecology
Jordi
Bascompte
and
Ricard
Sole
logical
system
s
show
com
plex
patterns
in
both
time
and
space.
Consequently,
Fc
o
A

ecology
has
been
engaged
in
the
description
of
such
patterns.
A
ccordingly,
we
have
been
collect
in
g
information
on
how
rodent
popu
lation densities
change
year
after
year,
or
on
how
phytoplankton
cells
are
scattered
in
patches.
This
de
scriptive
approach,
which
ha
s
domi
nated
ecology
almost
throughout
its
history
1,
has
provided
us
with
an
image
of
our
complex
world.
As
a
n
additional
,
necessar
y
step,
we
must
understand
which
mech
anisms
are
producing
the
patterns
observed.
This
is
achieved
using
a
In
the
past
few
years,
part
of
theoretical
ecology
has
focused
on
the
spatiotemporal
dynamics
generated
by
simple
ecological
models.
To
a
large
extent,
the
results
obtained
have
changed
our
view
of
complexity.
Specifically,
simple
rules
are
able
to
produce
complex
spatiotemporal
patterns.
Consequently,
some
of
the
compl
exity
underlying
nature
does
not
necessarily
have
complex
causes.
The
emerging
framework
has
far

reaching
implications
in
ecology
and
evolution.
Thi
s
is
improving
our
understanding
of
topics
such
as
the
problem
of
scales,
the
response
to
habitat
fragmentat
ion,
the
relationship
between
chaos
and
extinction,
and
how high
diversity
levels
are
supported
in
nature.
the
most
direct
way
to
introduce
space is
by
coupling
a
set
of
N
such
maps
by
diffusion.
Now,
we
can
establish
different
spatial
scales;
local
scales
—
that
is,
the
dynamics
shown
in
a
local
point
or
patch;
and
global
scales
—
that
is,
the
glo
bal
or
metapopulation
dynamics.
Metapopulation
extinction
There
is
a
wide
body
of
litera
ture
regarding
the
stabilizing
role
of
space,
focusing
on
how
disper

sal
between
local
populations
can
enhance
the
persistence
of
meta
populations
despite
local
popula
tion
extinctions
(see,
for
example,
Refs
9
and
10).
According
to
this
parallel
theoretical
development’,
in
which
mathematical
modelling
general
fra
mework,
the
argument
against
chaos
could
be
relaxed
and
computer
simulation
play
an
important
role
.
Temporal
processes,
for
example,
can
be
described
by
means
of
deterministic
mathemati
cal
models
such
as
partial
differ
ential
equations
or discrete
maps.
What
are
the
lessons?
Jordi
Bascompte
is
at
the
Dept
d’Ecologia,
Universitat
de
Barcelona,
Diagonal
645,
08028
Barcelona,
Spain;
Ricard
Sole
is
at
the
Dept
de
Fisica
i
Enginyeria
Nuclear,
Universitat
Politêcnica
de
Catalunya,
Sor
Eulalia
d’Anzizu
s
/n,
Campus
Nord,
Môdul
B5,
08034
Barcelona,
Spain.
when
space
was
introduced;
de
spite
unstable
chaotic
dynamics,
the
global
metapopulation
is
per
sistent
when
the
coupling
is
local,
that
is,
when
dispersal
is
reduced
to
nearest

neighbour
patches”’2.
Ho
wever,
the
evolutionary
ar
gument
against
chaos
in
popula
Complexity
in
time
and space
The
surprising
discovery
that
simple
ecological
models
could
display
complicated
dynamics
was
first
reported
by
May2.
Non

linear
terms
are
the
seed
for
a
random

li
ke,
aperiodic
behaviour
known
as
deterministic
chaos
(see
Box
1).
Although
some
evidence
for
the
presence
of
this
new
kind
of
order
in
the
fluctuations
of
natural
populations
was
soon
reported3,
there
is
no
unambiguous
proof
of
its
occur
rence
in
nature4
.
Without
any
definitive
conclusion
from
the
real
world,
some
ecologists
have
been
sceptical
about
the
role
of
chaos
in
ecological
systems.
Chaos
has
been
thought
to
be
mal
adaptive
because,
as
populations
fluctuate
violently,
it
would
increase
the
probab
ility
of
extinction5,
as
shown
in
Box
1
(although
this
view
is
controversial6).
However,
there
is
still
one
ingredient
missing;
space.
Spatiotemporal
model
s
are
mathematical
descriptions
of
dynamical
systems
in
which
space
is
explicitly
introduced.
There
are
different
approaches
describin
g
spatiall
y
extended
models,
reviewed
recently
in
order
to
illustrate
their
simi
larities
and
differences7’8.
Many
of their
predictions
seem
robust
in
the
face
of
changes
in
the
modelling
approach,
although
classical
mod
els
emphasize
stable
equilibrium
so
lutions,
while
more
recent
approaches
show
new
complex
spatial
and
temporal
dynamics7
.
We
focus
here
on
the
latter,
stressing
mainly
the
emergence
of
compleiiity
and
its
impli
cations.
There
are
different
‘recipes’
for
explicitly
intro
ducing
space
in
such
a
way9
(briefly
explained
in
Box
2).
If
we
are
dealing
with
discrete
temporal
models
as
in
Box
1,
tion
dynamics
is
not
all
that
is
relaxed.
At
higher
spatial
scales,
the
relationship
between
chaos
and
metapopulation
extinction
can
be
the
opposite;
chaos
can
enhance
persis
tence.
Spatially
induced
stability
needs
asynchronous
dynamics
among
patches
to compensate
for
buffer
crashes.
One
source
of
such
an
asynchrony
is
heterogeneity
among
patches.
But
even
if
all
p
atches
are
identical,
it
is
possible
to
induce
asynchrony
by
means
of
different
initial
con
ditio
n
s’
As
one
important
property
of
deterministic
chaos
is
its
high
dependence
on
initial
conditions,
very
close
trajec
tories
in
phase
space
will
diverge
exp
onentially
with
time.
Although
the
probabilit
y
of
local
extinction
increases
in
the
chaotic
domain,
the
probability
of
global
(
metapopulation
)
extinction
decreases
because
the
higher
the
dependence
on
initial
conditions,
the
faster
the
desynchronizing
rate
’4’5.
As
can
be
proved16,
chaos
leads
to
short
spatial
correlation
lengths,
which
is
the
origin
of
such
stability
at
global
scales
(more
rigorously,
coherence
length
decreases
as
the
in
verse
of
the largest
Lyapunov
exponent
at
the
onset
of
chaos’6).
When
the
coupling
is
limited
to
the
nearest

neigh
bour
patches,
global
population
dynamics
looks
like
a
steady
state
with
added
noise,
its
variance
being
reduced
by
increasing
the
lattice
size.
This
global
stability,
as
induced
by
the
desynchronizing
effect
of
chaos,
has
been
termed
‘chaotic
stability”
7
or,
in
a
different
context,
The emergence
of
spatial
patterns
Biological
systems
are
open,
far

from

equilibrium
sys
tems.
Under
such
conditions,
self

organization
can
operate
via
symmetry

brea
king
instabilities.
Man
Turing’s
seminal
paper’9
showed
how
the
coupling
of
reaction
and
diffusion
can
induce
pattern
formation,
stationary
inhomogen
eities
now
called
Turing
structures.
These
patterns
were
reported
in
continuous
spatially
extend
ed
predator
—
prey
models20’21.
Similar
structures
such
as
target
patterns
and
spiral
waves
are
observed
in
a
wide
set
of
chemical
and
biological
systems,
ranging
from
oscillating
reactions
to
the
aggregation
of the
slime
mold
Dic!yostelium
discoideum22.
Mother
kind
of
spatial
pattern
is
spatial
chaos,
in
which
concentration
waves
change
aperiodically
through
space,
as
in
turbulence.
All
the
above
spatiotemporal
patterns
are
shown
in
Fig.
1.
The
y
can
b
e
modelled
usin
g
continuou
s
and
discrete
models.
Couple
d
map
lattices
can
easily
reproduce
all
the
spatial
patterns
observed”23’24,
as
well
as
more
exotic
ones,
for
ex
ample,
chaotic
Turing
structures24
(a
generalization
of
the
classical
Turing
patterns
in
which
stable
spatial
patterns
coexist
with
chaotic
fl
uctuations).
In
all
these
situations,
order
and
chaos
can
coexist,
arising
from
the
same
source
—
non

linearity.
Three
main
lessons
should
be
pointed
out.
First,
the
same
restricted
patterns
arise
in
a
wide
range of
physical
Box
1.
Detenninistlc
chaos
and
population
dynamics
Simple
mathematical
models
can
display
complicated
dynamics2.
This
fact
was
discovered
when
studying
deterministic
difference
equations
in
order
to
model
population
s
wit
h
discrete,
non

overlapping
generations.
These
maps
can
be
wri
tten
as
follows;
N1÷1=F(N)
(1)
that
is,
the
population
value
(N)
at
generation
f+
1
is
a
given
function
of
the
den
sity
a
t
the
previous
generation.
For
a
given
example
of
F,
considerthe
logistic
map;
N11=
N,(1
—
N)
(2)
where
is
the
population
growth
rat
e,
the
population
value
is
normalized,
and
(1
—
Ne)
represents
the
density

dependent
term.
The
dynamics
generated
by
map
(2)
depends
on
the
value
of
the
bifurcation
parameter,
As
p.
is
increased
the
map
is
led
through
different
dynamical
behav.
iours
(stati
onary,
periodic
and chaotic)
in
a
well

defined
period

doubling
scenario
depicted
in
the
bifurcation
diagram
below.
As
usual,
for
each
1s

value,
the
corre
sponding
long

term
density
values
are
plotted.
If,
as
proved,
complex
dynamics
can
be gener
ated
by
simple
causes,
the
fluc
tuations
recorded
in
natural
systems,
traditionally interpreted
as
a
consequence
of
environmental
stochasticity,
might
have
a
deterministic
origin.
However,
two
crit
cisms
to
chaos
arise
from
an
inspection
of
the
bifurcati
on
diagram.
First,
chaotic
dynamics
imply
large
unstable
fluctuations,
with
some
population
values
very
near
the
extinction
threshold5
(see
arrow).
Second,
periodic
windows
appear
inside
the
chaoti
c
domain
at
all
scales.
Thus, a
small
change
in
the
bifurca
tio
n
parameter
(
p.
)
can
change
the
dynamics
from
chaotic
to
periodic.
In other
words,
chaos
is
struc
turall
y
unstable.
Both
criticisms
are
affected
by
the
introduction
of
space.
systems
and
mathematical
models.
This
means
that
the
de
tail
is
not
so
rele
vant,
that
is,
the
wa
y
in
which
entities
inter
act
is
more
important
than
their
particular
nature
(see Ref.
25
for
a
development
of
this
idea
and
further
biological
examples).
Second,
these
structures
are
emergent,
in
the
sense
that
they
are
macroscopic
patterns
emerging
from
simple
and
local
diffusion
rules.
In
other
words,
they
are
not
programmed
in the
equations’
structure.
There
is
a
re
cursive
‘dialogue’
between
the
parts
and
the
whole:
the
parts
create
a
macroscopic
pattern
that
in
turn
modifies
t
he
boundary
conditions
in
which
the
parts
interact26
.
Third,
all
these
spatiotemporal
phenomena
arise
from
very
simple
causes,
which
has
huge
implications
for
our
understanding
of
complexity.
Finally,
it
should
be
noted
that
all
these
patterns
have
a
larg
e
influence
on
the
persistence
of
populations9.”24’27.
For
example,
the
spatial
extension
of
discrete
models
for
two

species
competition
has
shown
that
even
for
high
inter
specific
competition
rates,
global
persistence
is
possible
despite
local
exclusi
on,
a
phenomenon
closely
related
to
pattern
formation
and
spatial
segregation24.
Effects
of
space
in
local
dynamics
It
has
been
proved
that
the
couplin
g
of
otherwise
periodic
oscillators
can
induce
chaotic
dynamics28.
This
property
is
referred
t
o
as
diffu
sion

induce
d
chaos,
and
has
been
found
in
chemical
systems2
8
as
well
as
in
continuous2
9
and
discrete’23°
models.
One
related
phenomenon
is
that
of
spatially
induced
bifurcations3t.
This
term
refers
to
the fact
that,
as
the
spatial
domain
is
increased,
the
local
dynamics
undergo
succes
sive
bifurcations.
So,
two
populations
of
the
same
species
may
have
distinct
dynamical
patterns.
This
result
can
help
to
explain
the
observed
geographical
trend
in rodents
and
lagomorphs,
with
populations
cycling
in
the
north
ern
part
of
their
range, where
there
are
larger
forested
areas,
while
populations
in
the
south,
where
habitat
is
fragmented
,
show
stationary
dynamics3’.
On
the
other
hand,
as
noted
in
Box
1,
chaos
is
struc
turally
unstable
because
of
the
dense
nesting
of
periodic
windows
inside the
chaotic
domain.
A
small
change
in
the
bifurcation
parameter
may
collapse
the
aperiodic
motion
into
a
periodic
one.
The
same
is
true
for
a
spatially
homo
geneous
counterpart,
but
now
the
basin
of
attraction
toward
such
a
solut
ion
is
small,
that
is,
the
solution
is
unstable
in
the
presence
of
small
perturbations32.
Consequently,
spatiotemporal
chaos
(i.e.
the
local
chaotic
motion
in
a
spa
tiall
y
extended
system)
is
structurally
stable
because
of
the
destruction
of
periodic
w
indows
via
two
processes
called
spatiotemporal
intermittency
and
supertransients32.
The
existence
of
supertransients
is
an
important property
of
spatiotemporal
systems.
The term
‘transient’
refers
to
the
initial
number
of
iterations
required
to
reach
t
he
long

term
dynamics,
that
is,
in
order
to
be
captured
by
the
attractor.
When
space
is
introduced,
the
transient time
is
much
higher
than
in
uncoupled
maps,
and
it
is
called
supertransient.
There
are
two
kinds
of
supertransients
depending
on
the
coupling
rate.
In
type

I
supertransient,
the
length
of
the
transient
time
diverges
slower
than
exponentially
with
the
lattice
size,
and
the
macroscopic
measures
decrease
with
time,
which
is
an
indication
of
the
fact
that
we
have
not
reached
the
steady
state.
On
the
other
hand,
in
type

Il
super

transients,
the
length
of the
transient
time
diverges
expo
nentially
or
faster
with
the
systems
size,
and
there
is
a kind
of
quasistationarity
(we
cannot
detect
whether
the
system
is
in
its
attractor)32.
An
additional
sou
rce
of
complexity
in
spatiall
y
extended
systems
is
multiple
attractors2333,
which
introduces
new
levels
of
uncertainty.
For
the
same
parameter
values,
a
system with
different
initial
conditions
will
evolve
toward
qualitativel
y
different
dynam
ic
s.
Furthermore,
the
bound
ary
between
different
attrac
tors
may be
fractal,
which
make
s
it
impossible
to
predict
future
dynamical
behaviour33.
These
result
s
suggest
that
complex
motion
is
enhanced
by
dispersal.
However,
some
recent
studies
have
pointed
i
n the
opposite
direction.
Some
consider immigration
to
b
e
a
time

independent
con
stant,
others
use
coupled
populations
in
which
the
link
age
is
a global
mixing
rather
than
being
restricted
to
the
nearest
neighbours
,
and
others
suggest
that
the
timing o
f
different
biological
processes
ha
s
a
great
influence
in
chang
in
g
the
dynamic
s
observed
(for
discussion,
see
Refs
33
—
38;
see
also
Box
2).
Ecological
and
evolutionary
implications
All
the
results
outlined
here
represent
a
change
in
the
way
we
look
at
co
mplexity
in
ecological
systems,
and
they
are
leading
to
major
rethink
ing
in
ecology.
Although
heterogeneity
has
long
been
recorded
in
ecology,
it
has
traditionally
been
interpreted
as
a
con
sequence
of
environmental
Box
2.
Recipes
fo
r
modelling
s
patiotemporal
dynamics
There
are
different
ways
to
make
a
mathematical
description
of
spatiotemporal
dynamics.
They
can
be classified
in
two
groups.
(1)
Continuous
space
and
time
models
(reaction
—
diffusion
mathematical
models)
These
are dynamical
systems
that
are
continuous
in
time,
space
and
state2021.29:
=
F(N,t)
+
DV2N
St
D
being
the
diffusion
rate
and
V’N
being
the
Laplacian
operator.
Reaction
—
diffusion
mathematical
models
represent
the spatial
extension
of
partial

differential
equations
and,
in fac
t,
they
are
simple
analogues
of
physicochemical
models
of
reaction
—
diffusion
kinetics.
(2)
Discrete
time
and
space
models
Coupled
map
laffice
s
—
these
are
dynamical
systems
with
discrete
time,
discret
e
space
and
continuous
state”32,
and
can
be
expressed
a
s
follows:
=
(1

D)
r{No
,
Jfl
+
D[F(N,o
+
1,j)}
+
F(Nr(l
1
,
)}+
F{NQ,j
1)
+
F{N,Q,j
+
1)
}l
where
(ij)
determines
the
spatial
coordinates
of
a
given
patch
in
the
discrete,
two

dimensional
lattice
of
points
in
which
the
dynamics
is
defined.
Now
the
co
upling
is
the
discretized
version
of
the
Laplacian
operator,
that
is,
a
passive
diffusion.
Normally,
the
coupling
is
reduced
to
the
four
or
eight
nearest
patches.
However,
global
mixing
is
also possible.
Reaction
and
diffusion
can
act
at
different
stages,
as
state
d
in
the
formula,
or
simultaneously,
as
a
discretize
d
version
of
th
e
reaction
—
diffusion
models.
Some
controversy
has
recently
arisen
about
how this
choice
modifies
the
obsemed
dynamics34

’8.
In
particular,
while
diffusion

induce
d
chaos
has
been
sh
own
fo
r
a
particula
r
map
when
reaction
and diffusion
act
simultaneously31,
i
t
is
not
observed
when
th
e
two
processes
act
at
different
stages,
in
a
more
realistic
approach3’
that
has
been
adduced
to
mean
that
such
complex
dynamics
arise
solely
from
the
unre
alistic
dispersal
rule
(see discussion
in
Refs
3T,38
)
.
For
other
kinds
of
system,
diffusion

induced
chaos
has
been
reported
in
continuous
models,
experimental
systems25,
and
other
discrete
models
even
when the more
realistic
dispersal
rule
is
used’°.
Apar
t
from
this
point,
the
particular
description
of
the
coupled
map
lattice
used
seems
not
to
be
so
important
in
producingthe
othe
r
patterns
described
here.
As
an
example,
the
emergence
of
spatial
structures
is
the
same
with
both
dispersal
rules”27,
and
it is
observed
independently
of
the
particular
recipe
used
for
modelling
spatiotemporal
systems.
Multiple
attractors
and
supertransient
s
have
also
been
found
using
both
dispersal
rules23’8.
Coupled
map
lattices
are
the
simplest
way
to
introduce
space
in
discr
ete
maps
like
those
of
Box
1.
Cellular
aufomaf
a
models
—
these
are
dynamical
systems
that
are
discrete
in
time,
space
and
state.
There
is
a
set
of
rules
that
determine
the
state
of
an
automaton.
There
is
often
a
very
small
number
of
possible
states
suth
as
‘0’
and
‘1’.
Each
automaton
state
evolves
in time
according
to
such
rules,
which
determine
the
state
of
an
automaton
at
the
next
time
step
as
a function
of
the
state
of
the
neighbouring
automata
(located
at
a
radius
r)
at
the
previous
time
step91’.
This
can
be written
as
follows:
a
1
=
F[aI_r
a
.
.
I+rl
There
are
also
stochastic
cellular
automata,
in
which
some
rules
are
not
deterministic.
variability.
The
results
summarized
here
suggest
that
the
distribution
of
populations
in
space
may
thus
result
fro
m
intrinsic
mechanisms,
that
is,
the
coupling
hetween
interac
tion
and
diffusion27.
Some
similarities
between
ecology
and
scale
coexistence
depends
on
the
size
of
the
spatial
domain.
If
space
is
reduced
beyond
a
minimal
threshold
in
which
there
cannot
be
any
bifurcation
from
the
homogeneous
state,
there
is
an
abrupt
reduction
in the
number
of
co
development
arise
as
a
result
of
this
structuralist
view,
2
m
p
4
eti
t
o
7
r
s
.
These
lessons
are
very
important
for
the
man
weakening
a
reductionist
approach
.
In
a
broad
sense,
lower

level
entities
(genes
or
individuals)
are
necessary,
but
not
sufficient
to
understand
higher

level
patterns
(morpholo
gical
or
ecological
patterns)
that
are
just
emergent
agement
of
natural
resources.
Habitat
destructio
n
is
one
of
the
most
serious
problems
of
the
biosphere.
Recent
work
on
the
dynamics
of
fragmented
landscapes
has
provided
new
information
on
which
to
make
quantitative
predi
and
subjected
to
dynamical
constraints.
Furthermore,
a
s
3
c
t
i
o
n
s
.
Amon
g
the
resu
lts
obtained,
there
is
the
unexpected
mentioned
above,
these
macroscopic
patterns
modify
the
boundary
conditions
under
which
such
lower

level
entities
interact2526.
This
point
is
of
paramount
importance
in
understanding
the
difference
between
a
structural
ist
approach
and
a
re
ductionist
one.
Although
many
reductionists
do not
believe
that
the
parts
are
enough
to
understand
the
whole,
they
do
believe
in
unidirectional
causality
from
micro

to
macro
level
behaviours,
as
opposed
to
the
bidirectional
causal
ity
stressed
by
structuralists.
The
coexistence
of
competing
species,
as
noted
above,
is
enhanced
when
space
is
considered,
which
suggests
that
some
ideas
on
niche
theory
and
the
competitive
exclusion
principle
should
be
reviewed24.
The
number
of
patches
r
e
sulting
from
self

organization
and
enhancing
such
large

evidence
that
habitat
destruction
can
cause
a
selective
ex
tinction
of
most
successful
competitors40,
or that
extinction
events
occur
generations
after
perturbation,
creating
a
‘debt’,
wh
ich
may
explain
the
equivocal
small
number
of
extinctions
that
take
place
instantaneously
after
a
pe
rtu
r
b
ati
o
n
It
is
well known
that
high
diversity
levels
can
be
sup
ported
only
in
nonequilibrium
conditions41.
The
existence
of
supertransients
maintain
s
the
system
far
from
asymp
totic
behaviour.
As
transients
show
an
abrupt
transition
between
order
and
chaos
(at
the
edge
of
chaos,
see
Ref.
26),
it
has
been
suggested
than
the
highest
diversity
levels
are
maintained
in
this
state’842.
This
framework
is
close
to
the
‘baroque
of
the
natural
world’
as
termed
by
Margalef42.43.
This
concept
reflects
the
existence
of many
more
species
than
would
be
necessary
if
biological
efficiency
alone
were
ai
organizing
principle.
Supertransients
an
d
multiple
attractors
have
caused
important
theoretical
breakdowns.
Ecologists
have
tra
(litionally
looked
at
long

term
behaviour,
but
as
a
conse
quence
of
the
extremely
long
duration
of
supertransients,
transient
dynamic
s
of
models
may
b
e
a
bette
r
des
cription
of
ecosystems
than
asymptotic
dynamics
of
models44.
This
theoretical
prediction
can
explain
some
population
erup
tions
i
n
which
ecologists
hav
e
failed
t
o
find
an
underlying
cause44.
On
the
other
hand,
because
of
the
presence
of
inuhipl
e
attracto
rs
,
different
dynamical
patterns,
which
have
traditionally
been
interpreted
as
the
reflection
of
large
dif
ferences
in
th
e
operating
processes,
can
arise
In
the
same
system
from
such
extreme
sensitivit
y
on
initial
conditions1’.
Finally,
the
richness
of
sp
atiotemporal
dynamics
sug
gested
by
this
theoretical
framework
has
larg
e
consequences
for
the
evolution
of
mutualistic
systems.
For
example,
the
variation
in
time
and
space,
at
different
scales
,
of
the
com
position
and
abundance
of
pollinator
s
create
s
a
s
patiotem
poral
mosaic
of
selective
pressures
.
This
sort
of
unpredllc
tability
In
adaptive
traits
preclude
s
fine

tuning
specialization
o
l
the
plant
to
particular
polIinators4.
Detecting
spatiotemporal
chaos
Whether
chaotic
behaviour
is
common
in
real
eco
sys
tem
s
is
still
controversial
because
of
th
e
problems
in
apply
ing
dynamical
systems
techniques
to
the
short

terni
and
noisy ecological
time
series.
Although
new
and
ingenious
methods
hav
e
recently
bee
n
developed
,
the
problem
remains
open4.
A
recent
appro
ach
i
s
particularl
y
linked
wit
h
th
e
clos
e
re
lationshi
p
between
temporal
and
spatial
processes
reported
here.
The
Idea
Is
t
o
detect
spatiotemporal
chaos
by
using
temporal
as
well
as
spatial
in1ormation’.
It
is
based
on
a
spatiall
y
define
d
average
of
local
divergenc
e
of
trajectories.
that
is
,
on
the
compariso
n
of
different
spatial
points
with
an
initially
similar
population
level,
in
order
to
measure
the
rate
at
which
such
dynamically
close
local
states
will
separate
over
time
(see
Box
3.
The
main a
dvantage
of
this
approach
is
that
under
a
common
deterministic
mech
anism
operating
at
each
lat
tice
point,
we
can
character
ize
chaos
from
very
short
time
series,
which in
turn
re
duces
the
probability
of
tran
sitions
between
different
dy
namical
behaviours46.
An
additional
field
in
which
such
a
measure
would
be
use
ful
is
neurodynamics
,
another
spatiotemporal
system
dis
Box
3.
Characterizin
g
spatiotemporal
chaos
Chaotic
dynamics
is
characterized
by
its
high
dependence
on
initial
conditions,
a
property
whereby
two
very
close
initial
states
will
diverge
exponentiall
y
wit
h
time,
thus
preventin
g
all
long

term
forecasting.
One
dynamical
measure
quantifies
this
degree
of
stretching:
the
Lyapunov
exponent.
However,
it
s
computation
requires
very
lar
ge
and
noiseless
temporal
series,
a
serious
problem
for
ecological
data.
A
new
class
of
Lyapunov
exponent
has
recently
been
proposed
to
characterize
spatiotemporal
chaos46.
It
uses
spatial
as
well
as
temporal
information,
and
its
main
advantage
is
that
i
t
can
work
well
even
forvery
short
temporal
series,
provided
that
there
are
enough
spatial
repliques.
In
a
spatially
extended
system
there
is
a
local
temporal
series
in
each
lattice
point
or
patch
K
=
(i,j),
which
we
can
write
as:
f(K)
=
{x1(K)
x1(K),...,
Xm(K)},VK
=
(
i,j
)
where
m
is
the
length
of the
temporal
series.
We
now
construct
d

dimensional
vectors,
for
a
given
embedding
dimension
d,
using
the
lagging
method,
that
is:
playing
phase
transitions
and
self

organizing
processes.
F4(K)
=
{X(K)
[x,
(K)
,
x+1
(K)
,...,
forf=1
m
—
d+1
X.s
1(K)]}
Prospects
In
the
near future
we
can
expect
the
following
devel
opments:
•
A
more

detailed
understand
ing
of
the
relationship
be
tween dispersal
and
chaos,
with
emphasis
on the
kind
of
dispersal,
the
timing
o
f
differ
ent
biological
processes,
and
the
radius
of
coupling.
•
Further
investigation
of
the
dynamics
of
fragmented
land
scapes,
in
order
to
improve
our
knowledge
of
the
re
sponse
of
ecosystems
to
hu
man
perturbation.
A
deep
search
for
theoreti
cal
implications
of
the
idea
of
emergent phenomena
in
multiscale systems,
as,
for
example,
in
the
relationship
between
micro

and
macro

evolution.
Improvement
and
develop
ment
of
current
and
future
methods to detect
chaos
in
ecological
data
as
well
as
to
distinguish
between
self

gen
erated
spatial patterns
and
spatial
randomness27.
As
all
these
questions
will
b
e
answere
d
and
incorporated
For
a
given
time
ste
p
we
search
for
all
the
pairs
o
f
lattic
e
points
<RH>
whose
populatio
n
value
is
very
similar,
tha
t
is
,
thos
e
pairs
for
which
the
following
inequality
holds:
—
Xt(H)
€
€
being
a
predefined
initial
difference
(very
small).
The
next
step
is
to
calculate
the
distance
between
the
two
points
at
the
next
time
step,
that
is:
IIxr+i
(o)
—
X+1
(H)N
The
reference
system
is
exemplified
in
the
following
figure:
The
aver
age
behaviour
of
the
spatiotemporal
system
can
now
be
characterized
by
estimating
whether
this
initial
difference
between
spatial
points
increases,
decreases
or
remains
constant.
Thus,
we
define
the
spatiotemporal
Lyapunov
exponent
in
the
following
way:
—
X+(H)N
into
a
common
body
of
work,
a better
understanding
on
spatiotemporal
dynamics
will
be
reached.
Complexity
will
still
be
as
fascinating
but
per
haps
a
little
less
unexpected.
Acknowledgements
X
(d)
=
s’
Nfrçj

>
IIX
(
1O

X
(H)U
where
N
is
th
e
total
number
of
spatial
points
<K,H>
used
in
the
calculation,
that
is
,
those whose
initial
difference
is
less
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thank
Robert
M.
May,
Michael
P.
Hassell,
Graeme
0.
Ruxton,
Jordi
Flos,
Pere
Alberch
and
Ramon
Margalef
for
m
any
helpful
comments,
criticisms
and
suggestions.
We
are
also
grateful
to
Quim
Garrabou
and
to
the
computer
service
of
the
Universitat
de
Barcelona
for
graphic
facilities.
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