Rethinking complexity: modelling spatiotemporal dynamics in ecology

chatventriloquistΤεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 3 χρόνια και 6 μήνες)

106 εμφανίσεις





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




References

We

would

like

to

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.

1

Moffat,

A.S.

(1994)

Science

263,

1090

1092

2

May,

R.M.

(1976)

Nature

261,

45
9

467

3

Schaffer,

W.M.

and

Kot,

M.
(1986)

Trends

Scot.

EvoL

1,

58

63

4

Hastings,

A., Horn,

CL.,

ElIner,

S.,
Turchin,

P.

and

Godfray,
H.C.J.
(1993)

Annu.
Rev.

Ecol.

Syst.

24,

1

33

5

Berryrnan,

A.A.

and

Millstein,

J.A.

(1989)

Trends

Ecol.

Evol.

4,

26

2
8












6

Nisbet,

R.,

Blythe,

S.,

Gurney,

B.,

Metz,

H.

and

Stokes,

K.

(1989)

Trends

Ecol.

Evol.

4,

238

239

7

Hastings,

A.

and

Harrison,

5.

(1994)

Annu.

Rev.

EcoL

Syst.

25,

167

188

8

Durrelt,

R.

and

Levin,

S.

(1994)

Theor.

Popul.

Biol,

46,

363

394

9

Czárán,

T.

and

Bartha,

S.

(1992)

Trends

Ecol.

Eva!.

7,

38

42

10

Taylor,

A.D.

(1990)

Ecology

71,

429

433

11

Hassell,

M.P.,

Comins,

N.H.

and

May,

R.M.

(1991)

Nature

353,

255

258

12

Sole,

R.V.

and

Valls,
J.

(1992).!

Theor.

Biol.

155,

87

102

13

Adler,

F.R.

(1993)

Am.

Nat.

141,

643

650

14

Allen,

J.C.,

Schaffer,

W.M.

and

Rosko,

D.

(1993)

Nature

364,

229

232

15

Ruxton,

GD.

(1994)

Proc.

R.

Soc.

London
Ser.

B

256,

189

193

16

Rasmussen,

D.R.

and

Bohr,

T.

(1987)

Phys.

Lett.

A

125,

107

110

17

S
ole,

R.V.,

Bascompte,

J.

and

Valls,
J.

(1992)

Chaos

2,

387

395

18

Ikegami,

T.

and

Kaneko,

K.

(1992)

Chaos

2,

397

407

19

Turing,

A.

(1952)

Phi/os.

Trans.

R.

Soc.

London

237,

37

72

20

Okubo,

A.

(1980)

Diffusion

and

Ecological

Problems:

Mathematical

Model
s
,

Springer

21

Segel,

L.A.

and

Jackson,

J.L.

(1972)

.1.

Theor,

Rio!.

37,

545

559

22

Winfree,

A.

(1987)

When

Times

Break

Down
,

Princeton

University

Press

23

Sole,

R.V.,

Valls,

J.

and

Bascompte,

J.

(1992)

Phys.

Lett A

166,

123

128

24

Sole,

R.V.,

Basc
ompte,

J.

and

Valls,

J. (1992)J.

Theor.

Biol.

159, 469

480

25

Goodwin,

B.C.

(1994)

I
-
low

the
Leopard

Changed

its

Spots,

Weinfeld
and

Nicholson

26

Kawata,

M.

and

Toquenaga,

Y.

(1994)

Trends

Ecol.

Evol
.

9,

417

421

27

Hassell,

M.P.,

Comins,

N.H.

and

May
,

R.M.

(1994)

Nature

370,

290

292

28

Kuramoto,

Y.

(1984)

Chemical

Oscillations,

Waves

and

Turbulence,

Springer

29

Pascual,

M.

(1993)

Proc.

R.

Soc.

London
Ser.

B

251,

1

7

30

Hastings,

A.

(1992)

Theor.

Popul.

Biol.

41,

388

400

31

Bascompte,

J.

and

Sole
,

R.V.

(1994)

J.

Anim.

Ecol.

63,

256

264

32

Kaneko,

K.

(1990)

Phys.

Lett

A

149,

105

112

33

Hastings,

A.

(1993)

Ecology

74,

1362

1372

34

Ruxton,

G.D.

(1994)J.

Anim.

Ecol.

63,

1002

35

Bascompte,

J.

and

Sole,

R.V.

(1994).!

Anim.

Ecol.
63,

1003

36

Ruxto
n,

GD.

(1995)

Trends

Ecol.

Evol.

10,

141

142

37

Hassell,

M.P.,

Miramontes,

0.,

Rohani,

P.

and

May,

R.M
.

.1

Anim.

Ecol.

(in

press
)

38

Bascompte,

J.

and

Sole,

R.V.

J.

Anim.

Ecol.

(in
press
)

39

Hanski,

I.

(1994)

Trends

Ecol.

Evol.

9,

131

135

40

Tilman,

D.
,

May,

R.M.,

Lehman,

CL.

and

Nowack,

M.A.

(1994)

Nature

371,

65

66

41

Connell,

J.H.

(1978)

Science

199,

1302

1
3
10

42

Bascompte,

J.,

Sole,

R.V.

and

Valls,

J.

(1993)

in

Proc.

1st

Copenhagen

Symp.

Comp.

Sim.

Biol.

Ecol.

and

Medic.

(Mosekilde,

E.,

ed.
),
p
p.

56

60,

The

Society

for

Computer

Simulation

International

43

Margalef,

R.

(1980)

La

Biosfera,

entre

Ia

Termodindmica

ye!

Juego,

Omega

44

Hastings,

A.

and

Higgins,

K.

(1994)

Science
263,

1133

1
136

45

Herrera, CM.

(1988)

Biol.

J

Linn. Soc.

35,

95

125

46

Sole,

R.V.

and

Bascompte,

J.

J.

Theor.

Biol,

(in
press
)