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E¡ects on population persistence: the interaction

between environmental noise colour, intrasp eci¢c

competition and space

OWEN L. PETCHEY

1

, ANDREW GONZALEZ

1

A N D

HOWARD B. WILSON

2

1

NERC Centre for Population Biology, Imperial College at Silwood Park, Ascot, Berkshire, SL5 7PY, UK

(o.petchey@ic.ac.uk; a.gonzalez@ic.ac.uk)

2

Department of Biology, Imperi al College at Silwood Park, Ascot, Berkshire SL5 7PY, UK

(h.b.wilson@ic.ac.uk)

SU M M A RY

It is accepted that accurate estimation of risk of population extinction, or persistence time, requires predic-

tion of the e¡ect of £uctuations in the environment on population dynamics. Generally, the greater the

magnitude, or variance, of environmental stochasticity, the greater the risk of population extinction.

Another characteristic of environmental stochasticity, its colour, has been found to a¡ect p opulation persis-

tence. This is important because real environmental variables, such as temperature, are reddened or

positively temporally autocorrelated. However, recent work has di sagreed about the e¡ect of reddening

environmental stochasticity. Ripa & Lundberg (1996) found increasing temporal autocorrelation

(reddening) decreased the risk of extinction, whereas a simple and p owerful intuitive argument (Lawton

1988) predicts increased risk of extinction with reddening. This study resolves the apparent contradiction,

in two ways, ¢ rst, by altering the dynamic behaviour of the population models. Overcompensatory

dynamics result in persistence times increasing with increased temporal autocorrelation; undercompensa-

tory dynamics result in persistence times decreasing with increased temporal autocorrelation. Secondly, in

a spatially subdivided population, with a reasonable degree of spatial heterogeneity in patch quality,

increasing temporal autocorrelation in the environment results in decreasing per sistence time for both

types of competition. Thus, the inclusion of coloured noi se into ecological models can have subtle interac-

tions with population dynamics.

1. I N T RO DUC T ION

Physical environments are rarely static, and the varia-

bility in important environmental parameters such as

temperature and rainfall, have widely recognized

impacts on natural populations of plants and animals.

It is generally acknowledged that an important charac-

teristic of environmental variability, or noise, is its

temporal variance (Leigh 1981; Goodman 1987; Wissel

& StÎ cker 1991; Lande 1993). Recently, the potential

importance of another characteristic of environmental

noise has been recognized: the variance spectra, or

colour, of the noise (Goodman 1987; Halley 1996).

White noise (or noise with a white variance spectrum)

contain s no temporal autocorrelation and is essentially

a series of independent random numbers. For example,

if a series of daily temperatures were white, tomorrow's

temperature would be independent of today's tempera-

ture. Red noise, however, contains positive temporal

autocorrelation: tomorrow's temperature is likely to be

similar to today's. The colour of environmental noise is

potentially important because (i) the variance spectra

of real environmental variables are reddened

(Mandlebrot & Wallis 196 9; Monin et al. 1977; Steele

1985; Williamson 1987; Halley 1996); (ii) the variance

spectra of natural population sizes t hrough time are

reddened (Pimm & Redfearn 1988; Ari ·o & Pimm

1995; Sugihara 1995); and (iii) recent theoretical inves-

tigations suggest that population dynamics are sens itive

to noise colour (Roughgarden 1975; Foley 1994; Caswell

& Cohen 1995). A seemingly strong intuitive argument

for how noise colour might a¡ect population dynamics

is that reddened noise contains runs of bad conditions,

and whereas a population may survive one setback, a

series of poor conditions will lead to a higher risk of

extinction (Lawton 1988; see also Halley 1996). Thus,

deciding whether environmental £uctuations are c orre-

lated or not is essential when attempting to estimate a

population's probability of extinction.

This paper investigates the problem of how environ-

mental autocorrelation a¡ects time to extinction and,

speci¢cally, why in some cases it increases estimated

per sistence time, while in others it decreases estimated

per sistence ti me (Ripa & Lundberg 1996; Foley 1994).

In an interesting recent paper, Ripa & Lundberg

(1996) (hereafter referred to as R & L) showed that

Proc. R. Soc. Lond. B (1997) 264, 1841 ^18 47 1841 & 1997 The Royal Society

Printed in Great Britain

increasing temporal autocorrelation in the noise

process (reddening the noise) caused a decrease in

extinction risk. This is in direct contradiction to

Lawton's (1988) intuitive argument, which predicts

reddened noise to increase (relative to white) the risk of

population extinction. Both overestimation of time to

extinction, and underestimation are potentially very

dangerous; while species become extinct at unprece-

dented rates (Lawton & May 1995), and more become

endangered, accurate predictions about extinctions are

paramount. The apparent contradiction in the e¡ects of

environmental autocorrelation on extinction probabil-

ities raises serious doubts about the accuracy of

predictions of population persistence: not even the

expected direction of the e¡ect of reddened spectra

appears to be consistent, let alone the magnitude. R &

L suggest that model choice may severely in£uence esti-

mates of extinction times. This paper extends that of R

& L by including di¡erent forms of intraspeci¢c

competition, thereby changing the dynamic behaviour

of the model and identifying the di¡erence between R

& L's study and the intuitive argument. The signi¢-

cance of spatial subdivision of populations on the risk

of extinction under di¡erent types of noise is also inves-

tigated, further demonstrating that the inclusion of

coloured noise into extinction models can have subtle

interactions with population dynamics.

2 . IM P OR TA NC E OF C OM PET I T I ON

(a) Model formulation

R & L used the discrete-time Ricker growth model

to investigate the e¡ect of the colour of environmental

noise on the probability of extinction of a single-species

population. Density dependence in the Ricker model

results in overcompensatory dynamic behaviour. In

order to investigate the e¡ects of a range of dynamic

behaviour, the model used by R & L was generalized to

N

t1

N

t

exp(r(1 ÿ n

t

=K

t

b

)), (1)

where the parameter b c ontrols the dynamic behaviour,

from overcompensatory dynamics when b 1.0 (the ca se

of R & L) with a continuous change to undercompensa-

tory dynamics when b&1 (with perfect compensation

lying somewhere in-between) (¢gure 1). The form of

mortality for equation (1) is unusual because mortalities

are not equal at low densities for di¡erent values of b

(¢gure 1a). Whereas in the model of Hassell (1975),

given by

N

t1

N

t

=(1 N

t

=K

t

)

, (2)

mortalities are equal at low densities (¢gure 1b). In

equation (2), K

t

is the reciprocal of the a usually used

in the Hassell model. We concentrate on equation (1)

to allow comparison with R & L, but also present

results using Hassell's model (in both cases the qualita-

tive results are the same). Demographic stochasticity

was included in both models by taking population size

as an integer. The population size in the next genera-

tion was then a random number taken from a Poisson

distribution with the deterministic expectation of the

population size as its mean. Hence, equation (1)

becomes

N

t1

Z(N

t

exp(r(1 ÿ N

t

=K

t

b

))), (3)

where Z*Poisson (N

t

exp( r (1 ÿ N

t

=K

t

b

))). The

distribution of the sum of N

t

indep endent Poisson

distributions each with t he same mean

(exp(r(1 ÿ (N

t

=K

t

b

))), is the same as a Poisson

distribution with mean (N

t

exp(r(1 ÿ N

t

=k

t

b

))).

Therefore, equation (3) can be interpreted as repre-

senting N

t

individuals each having a random (Poisson-

distributed) number of o¡ spring with mean

exp(r(1 ÿ N

t

=K

t

b

)), so that competition between

individuals decreases with per capita fecundity. By

formulating the model in a stochastic framework,

extinctions occur in a biologically plausible manner,

rather than by setting a minimum population size.

Environmental noise entered the model in exactly

the same manner as in the model studied by R & L.

Noise had an additive e¡ect on the mean carrying

capacity K

0

; K

t1

K

0

t1

. Autocorrelation in the

noise proces s,

t

, was determined by a ¢rst order auto-

regressive process described by

t1

t

t1

, (4)

where the parameter controls the degree of auto-

correlation,

t

is a normally distributed random

deviate with zero mean and unit variance, and

1842 O. L. Petchey and others Noise, competition, space and persistence

Proc. R. Soc. Lond. B (1997)

(a)

(b)

Figure 1. Mortality as K values (log(N

t

=N

t1

)) against log

population size for (a) equation (1) and (b) equatio n (2)

showing the e¡ect of altering (a) b and (b) on the type of

intraspeci¢c competition. A high value of b or results in

overcompensatory dynamics (or scramble competition); a

low value of b or results in undercompensatory dynamics

(more similar to contest competition).

determines the magnitude of the environmental £uc-

tuations. When C(1 ÿ

2

)

1=2

, the variance in K

t

is

constant for any value of . C is therefore used to alter

the magnitude of environmental £uctuations, and is set

here at 70. (R & L used a C value of 20, but this results

in very long persistence times when the parameter b is

low, hence C is set at 70 instead. The exact value of C

used does not in£uence the qualitative results presented

later.) As in R & L, our objective i s simply to study the

e¡ects of the degree of autocorrelation in the noise

process, not to study the full range of possible noise

signals. When 0, the noise is white; for 0 < < 1,

the noise is positively autocorrelated, i.e. reddened. If

ÿ1 < < 0, the noise is negatively correlated (blue

noise); however, blue noise i s not generally considered

to be a relevant model of environmental variation and

is not included here.

Although threshold e¡ects (such as the probability of

extinction within 1000 model generations) are a

per fectly reasonable measure for comparing di¡erent

e¡ects, where exactly the threshold is drawn can be

arbitrary. In stead, therefore, we use d the mean persi s-

tence time, t(e), over 1000 independent s imulations. The

maximum length of one simulation t

max

was set very

large (200 000 generations); if extinction did not occur

by t

max

, t(e) was set to t

max

.

(b) Results of model simulation

For model (3), when b 1 (overcompensatory

dynamics), our results concurred with the study of R

& L: increasing autocorrelation in the noi se process

increased mean persistence time from about eight

generations at 0 to about 100 generations at

0.95 (in both cases all 1000 simulations went

extinct by t

max

) (¢gure 2a). However, when b 0.1

(undercompensation), mean persistence time decreased

from 150 000 generations (572 extinctions by t

max

) when

0 to about 700 generations (1000 extinctions by

t

max

) when 0.95. The e¡ect of increasing auto-

correlation in the noise process was, therefore,

reversed by decreasing b, and the e¡ect was continuous

from b 1.0 to b 0.1. Hassell's model returns qualita-

tively similar results: overcompensatory dynamics

result in an increase in persistence ti me with increased

autocorrelation and a decrease in persistence time with

undercompensatory dynamics (¢gure 2b). In Hassell's

model, only e¡ects whether the model is overcompen-

satory ( 41), undercompensatory (51) or has perfect

compensation ( 1). The value of has no e¡ect on

this (although it can change the stability of the equili-

brium). Therefore, the change in the two types of

pattern seen in ¢gure 2b occurs at 1. In equation

(1), the change from overcompensation to undercom-

pensation is a complex function of both r and b, with

an increase in either leading to more overcompensatory

dynamics. However, the change in the types of pattern

seen in ¢gure 2a occur s, as in Hassell's model, at the

point where the dynamics change from over- to under-

compensation. It should be stressed that we are

interested in making comparisons of how pers istence

time changes with ; rather than comparing persis-

tence times for di¡erent values of b (a s, for example,

changing b in Hassell's model changes the equilibrium

population size).

By examining the distribution of population sizes just

before extinction (N

t(eÿ1),

), R & L identi¢ed the

mechanism causing extinction as overcompensatory

crashes (all population sizes prior to extinction were

greater than K). Figure 3a shows distributions of N

teÿ1

for the four combinations of and b representing the

four extremes of the simulations. The dist ribution of

N

teÿ1

when 0 and b 1 was spread widely, but the

high p opulation density prior to extinction suggests

that demographic stochasticity was not the mechanism

responsible for extinction. When 0.95 and b 0.1,

all values of N

teÿ1

were less than 10. This strongly

suggests that demographic stochasticity was the

mechanism responsible for extinction; an overcompen-

satory crash would be very unlikely from such a low

density. Due to the £uctuating carrying capacit y,

however, distributions of N

teÿ1

give limited informa-

tion when attempting to determine the extinction

mechanism.

To elucidate the mechanisms causing extinctions,

therefore, the distributions of log

10

(N

t(eÿ1)

=K

t(eÿ1)

)

Noise, competition, space and persistence O. L. Petchey and others 1843

Proc. R. Soc. Lond. B (1997)

(a)

(b)

1000000

100000

10000

1000

100

10

1

mean persistence time

0

0.2

0.4

0.6

0.8

0.9

0.7

0.1

0.3

0.5

competitio

n

parameter b

autocorrelation

parameter

1000000

100

1000

10000

100000

mean persistence time

0 0.2 10.80.60.4

autocorrelation parameter

Figure 2. Mean persi stence t ime (of 1000 simulations) as a

function of t emporal autoc orrelation in the environment, ,

and the dynami c be haviour of the model, controlled with b,

for the non-spatial model ((a) equation (1), and (b) equa-

tion (2)). The carrying capacity K

t

is given by the

stochastic process K

t1

K

0

t1

;

t1

t

t1

,

where

t

is a normally distributed d eviate with mean zero

and uni t varianc e. The amplitude of the noise, , is given

by 70(1 ÿ

2

)

1=2

, hence the variance (K

t

) 70

2

for all

. Other paramete rs are r 1:5, K

0

100 and

t

max

200 000.

were plotted for each case (¢gure 3b).

Log

10

(N

t(eÿ1)

=K

t(e ÿ 1)) > 0, was used to deter mine

whether population size should have increased or

decreased from t(e71) to t(e), and then to infer the

mechanism responsible for extinction. If

Log

10

(N

t(eÿ1)

=K

t(eÿ1)

) > 0, population si ze should have

decreased from t(e71) to t(e) and if, in addition,

population density was high, an overcompensatory

crash alone is inferred to have caused extinction.

Conversely, if Log

10

(N

t(eÿ1)

=K

teÿ1

) < 0, population

size shou ld have increased from t(e71) to t(e) and if,

in addition, populat ion densit y was small, demo-

graphic stochasticity alone is inferred to have caused

extinction. Simulations w ith b 1 resulted in distribu-

tions with Log

10

(N

t(eÿ1)

=K

t(eÿ1)

) being generally

greater than zero (see ¢gure 3b) and population densi-

ties were generally high before extinction (see ¢gure

3a). Thus, overcompensatory crashes were inferred as

the extinction mechanism. Distributions from simula-

tions with b 0.1 were generally less than zero,

indicating that population growth was expected,

when actually the population density decreased to

zero (see ¢gure 3b). In addition, population sizes were

generally small (see ¢gure 3a), so that demographic

stochasticity was therefore the mechanism responsible

for extinction.

The patterns in persistence times described above

were, for both models, with parameters that resulted

in single point equilibria. The models can display non-

equilibrium dynamics when density dependence is

high ly overcompensatory. However, the stability of

the local equilibrium has no e¡ect on the general

pattern of increasing or decreasing persistence time

with changes in the autocorrelation parameter

(although it may have an e¡ect on the absolute value

of the persistence time). This is because £uctuations in

K

t

result in £uctuations in the population size far from

the local e quilibrium point. The e¡ects of the type of

competition (under- or overcompensation) are then

the determi ning factor, rather than the dynamics in

one local area of the phas e space (around the equili-

brium). To con¢rm this, we investigated t he e¡ects of

noise colour on persistence t imes for parameter combi-

nations that resulted in non-equilibrium dynamics. We

found that persistence time increased with autocorrela-

tion. This is consistent with the dynamic behaviour of

the models being overc ompensatory, the dominant

cause of extinction being overcompensatory crashes

and white noise therefore resulting in lower persistence

times.

The conclusions about the mechanisms responsible

for extinct ion in di ¡erent simulations were reinforced

when the strength of demographic stochasticity was

increased (by increasing the variability around the

mean population size). For example, a negat ive bino-

mial distribution, with the variance greater than the

mean, could be used in equation (3), instead of a

Poisson distribution. Since demographic stochasticity

does not cause extinction in the overcomp ensatory

model, changing its strength should not change the

mean persistence t ime. Acc ordingly, changing the

strength of demographic stochasticity was found to

decrease mean per sistence time in the model with

undercompensation, but to have no e¡ect in the model

with overcompensat ion.

The opposite e¡ects of noise colour seen with

di¡erent types of competition, and therefore dynamic

behaviour, represent the intuitive argument of the

introduction in this paper and the theoretical study of

R & L. The intuitive argument assumes demographic

stochasticity as the extinction mechanism. R & L

studied the e¡ects of noise colour in a model with over-

compensatory dynamics, and thereby revealed a

fascinating counter-intuitive result caused when over-

compensatory crashes are the mechanism responsible

for extinction.

3. T H E IM P ORTA NC E OF SPAC E

(a) Model formulation

Consider a spatially subdivided population. Chan-

ging habitat quality across a landscape leads to spatial

variation in the chances of individual survival and

reproduction, resulting in heterogeneity in the local

number of individuals. This spatial variability, when

coupled to temporal environmental variability, leads to

an ever-changing spatial pattern in t he chances of

survival and reproduction for an individual, and conse-

quently local variability in population size. Such

circumstances, it has been suggested, result in a

regional metapopulation structure in which persistence

is attained via constant di spersal between transient

local sub-populations (Gilpin & Hanski 1991).

1844 O. L. Petchey and others Noise, competition, space and persistence

Proc. R. Soc. Lond. B (1997)

(a)

(b)

1

0

0.2

0.4

0.6

0.8

20 40 60 80 100 120 140 200+200160 180

N

t(e-1)

proportion

1

0

0.2

0.4

0.6

0.8

proportion

–2.5 –2 21.510.5–0.5–1.5 –1 0

log

10

[N

t(e-1)

/K

t(e-1)

]

Figure 3. Frequency distri butions of (a) population size

N

teÿ1

, and (b) log

10

(N

t(eÿ1)

=K

t(eÿ1)

) o ne time-step prior to

extinction, t(e71), for equation (1). Four distributions are

shown, for each combination of low (0) and high (0.95)

with undercompensatory (b 0.1) and overcompe nsat ory

(b 1.0) dynamics. All other model details are as in ¢gure 2.

The consequences of changing the temporal correla-

tion in environmental noise for metapopulation

per sistence in a spatially heterogeneous environment

were explored by employing a spatial extension of

R&L's model. The metapopulation structure was

modelled as 100 patches linked by dispersal. Local,

within-patch dynamics were modelled using equation

(1) with b 1.0 (overcompensatory dynamics for all

simulations) and were therefore identical to those used

in R & L, except for the addition of demographic

stochasticity.

After the local dynamics a dispersal phase follows,

with the probability of any given individual dispersing

being ¢ xed and density independent. The dispersal

process is global rather than local, so that the prob-

ability of a dispersing individual reaching any patch in

the landscape is identical.

Metapopulation extinction occurs when N

t

0 for

all patches. Mean persistence time of t he metapopula-

tion over 100 simulations was used as the response

variable. The maximum length of one simulation t

max

was large (100 000 generations); if extinction did not

occur by t

max

, t(e) was set to t

max

.

Environmental variability is included in two ways:

(i) temporal variability due to £uctuations in t he mean

carrying capacity across the landscape, K

t

, determined

in the same way as in the previous section (the value of

therefore determines the level of autocorrelation in

the variability in K

t

); and (ii) spatial heterogeneity in

patch quality across the landscape, represented by

di¡erent carrying capacities of the individual patches.

At any time the carrying capacities of individual

patches are normally distributed with mean K

t

and

standard dev iat ion, . Therefore, is a parameter that

controls the heterogeneity in patch quality across the

landscape.

(b) Results of model simulation

When the model is run with a single patch, or when

0, the results are qualitatively the same as those of

the previous section and of those of R & L. However,

when the population is subdivided between a number

of patches (100 in this case), and is large enough, the

results are reversed (¢gure 4). An interaction between

the temporal autocorrelation in K

t

and the level of

spatial heterogeneity in patch carrying capacity (deter-

mined by ) in determining mean metapopulation

per sistence time was found. With 920, increasing

leads to increasing t(e); but with 020, increasing

leads to decreased t(e).

The e¡ect of reveals the underlying mechanism

behind this result. Starting with a low value of (i.e.

white noise), increasing increases the spatial hetero-

geneity in the carrying capacity of the environment.

Above 20, the carr ying capacity of the patches at

any given t now di¡ers su¤ciently such that when an

unfavourable environmental event (for population

growth) occurs, although some patches su¡er local

extinction through over compensatory crashes, others

per sist. Extinct patches are then rescued by dispersal

from those that survived. It is the spatial heterogeneit y

of the landscape which desynchronizes population

dynamics (Heino et al. 1997), and then the rescue e¡ect

that promotes persistence. Increasing the level of

temporal autocorrelation causes t his mechanism to

break down. The temporal smoothing of the environ-

mental noise that results from an increase in

autocorrelation now allows the populations to track

their environment (¢gure 5) and synchronize through

space, increasing extinction risk. Thus, during periods

of poor environmental quality, because one bad event

is more likely to be followed by another and more and

more patches su¡er sequential extinction, there is a

decline in the number of rescuing patches. If the ser ies

of unfavourable events is long enough, the number of

rescuing patches becomes too small to keep pace with

the extinction of existing ones.

The results of the spatial model were found to be

sensitive to the number of patches in the landscape,

more patches increased persistence times. Land scape

size did not alter t he interaction of spatial heterogeneity

() with temporal autocorrelat ion, only the level of

heterogeneity needed to cause reversal in the e¡ect of

environmental autocorrelation. The larger the land-

scape, the greater the probability that the dynamics in

some patches are not in synchrony with all the other

patches, and the lower the level of spatial heterogeneity

therefore needed.

When the as sumption of global dispersal was relaxed

to local dispersal (dispersal only to eight nearest neigh-

bours) persistence times were not a¡ected signi¢cantly,

and the reversal in the e¡ect of autocorrelation still

occurred when environmental heterogeneity was

increased. In this case, assuming local dispersal has

little e¡ect since an individual's probability of dispersal

is low (0.05), neither a decrease in the rescue e¡ect nor

a decrease in synchrony occurs as a result of assuming

local dispersal (although at higher dispersal rates local

dispersal can have an e¡ect). The synchrony in popula-

tion dynamics over the landscape, and therefore

population extinction risk, is overwhelmingly

Noise, competition, space and persistence O. L. Petchey and others 1845

Proc. R. Soc. Lond. B (1997)

100000

10000

1000

1

10

100

0.0

0.2

0.8

0.6

0.4

autocorrelation

parameter

landscape or

spatial

heterogeneity

50

40

30

20

10

0

mean persistence time

Figure 4. Mean persistence time (1 00 simulations) as a

function of temporal autocorrelation in the environment

and the degree of environmental heterogeneity, , among

the 100 patches . K

t

is the mean patch carrying capacity at

time t and is given by the same stochastic process as in the

non-spatial model. is the standard deviation of patch

carryin g capacities at any one time. t

max

100 000, in divi-

duals' dispe rsal probability 0.05.

determined by landscape heterogeneity. However, when

there are non-equilibrium population dynamics (e.g.

chaos), changes in the persistence time e¡ected by auto-

correlation and environmental heterogeneit y are

complex. Asynchrony between patches (which

promotes persistence) is now a¡ected by the local

dynamics, the landscape heterogeneity and the degree

of dispersal b etween patches. In th is case the outcome

can be complex: for example, with local dispersal and

chaotic dynamics, the relationship between autocorre-

lation and persistence can b e humped with greatest

per sistence at intermediate autocorrelation. Lack of

space here prevents exhaustive reporting of these

results.

Summing up, in our simple, spatially heterogeneous

model, R & L's result is now seen to be a non-general

case, in which populations exhibit overcompensatory

dynamic behaviour, and where patches are relatively

uniform in quality across the landscape (520). Where

habitat quality varies from patch to patch (420), even

with overcompensatory dynamics, the assertion that

white noise leads to an optimistic extinction probability

(Lawton198 8; Halley 1996) remains valid.

4 . DI SCUS SIO N

R & L showed that the probability of extinction in a

single-species, single-patch model decreased as noise

colour was reddened. As R & L suggested, t his result

is contingent on overcompensatory population

dynamics. When dynamic behaviour is undercompen-

satory, reddening environmental noi se increases the

probability of extinction. It is the dynamic behaviour

of the models that determines the e¡ect of environ-

mental noise colour. The dynamic b ehaviour of the

models was controlled by changing the t ype of

competition (¢gure 1) from scramble competition,

where high population densities result in high

mortality, through pure contest competition, where a

constant number of individuals survive competition, to

undercompensatory competition. A new result has been

found: the colour of environmental noise and the

dynamic behaviour of the population interact. Further-

more, whether a population is spatially subdivided also

has important in£uences on population persistence.

Here it is shown that, when coupled with overcompen-

satory intraspeci¢c c ompetition, white noise results in a

highly per sistent metapopulation. Persistence of the

same metapopulation, however, is signi¢cantly

reduced as the environment is increasingly reddened, a

process shown to be due to the sens itivity of population

dynamics to noise colour. Harrison & Quinn (1989)

showed a similar e¡ect of decreased metapopulation

per sistence by reddening the temporal sequence of

extinction probabilities. However, populat ion dynamics

were not explicitly modelled in this study, thereby

precluding more detailed analysis of possible

extinction-causing mechanisms.

An immediate question raised by our results is the

relationship between environmental autocorrelation

and persistence time that would we predict in real

populations. We would argue that populations gener-

ally display undercompensatory dynamic behaviour

and, therefore, that their persistence time will be over-

estimated (rather than underesti mated as predicted by

R & L) by the assumption of white noise. Although the

evidence is limited for this assertion, Has sell et al.

(1976), and Bellows (1981), estimated of Hassell's

(1975) model for a limited number of (insect) taxa,

and found values that were typically less than 1 (i.e.

the predicted dynamic behaviour of the populations is

undercompensatory).

1846 O. L. Petchey and others Noise, competition, space and persistence

Proc. R. Soc. Lond. B (1997)

(a)

(b)

(c)

(d)

Figure 5. Time-series of metapopulation s ize and patch carryin g capacity K

t

for the spatial model with (a) no autocorrela-

tion ( 0) in K

t

, and (c) positive temporal autocorrelation ( 0.95) in K

t

. Scatter plots of K

t

against metapopulation size

for (b) 0 and (d) 0.95 indicate how closely metapopulation size tracks K

t

.

Moreover, most natural populations are spatially

subdivided, increasingly so due to habitat de str uction.

The results of the previous sect ion suggest that, once

again, use of white noise will overestimate population

per sistence times. The spatial model investigated above

could also be extended to explore e¡ects of spatial auto-

correlation in patch quality: patches close to each other

would be more similar than distant patches. This is

obviously a more realistic assumption than assuming

no spatial correlation, as in this study. The likely

outcome is that since spatial autocorrelation will lead

to less environmental heterogeneity, its e¡ect should be

the same as decreasing , it should lead to a greater risk

of metapopulation extinction.

In summary, the demonstrated importance of intra-

speci¢c competition, in terms of how it alters dynamic

behaviour, and space, strongly suggests that, for the

majority of real populations, assuming white environ-

mental noise in population models w ill bias estimates

of risk of extinction and persistence time, and that the

direction of this bias will be non-conservative.

These results need placing in context in the way in

which environmental noise was simulated: an auto-

regressive process was used to produce reddened noise

(as did R & L). Other processes can also produce

reddened noise, such as 1/f processes (Halley 1996). An

important di¡erence between a ¢rst order autoregres-

sive process and a 1/f process is that where a ¢rst order

process converges with time to have a constant

variance, the variance of a 1/f process increases

continually with time, without limit. The likely impor-

tance of these two di¡erent ways of generating

reddened environmental noise on the persistence of

populations requires further investigation (Halley &

Iwasa 1997; Halley & Kunin 1998). An experimental

system designed to investigate the e¡ects of 1/f noise

on real populations has been designed (Cohen et al.

1998) and should provide insight into the sensitivity of

real organisms to coloured noi se.

With growing evidence that environmental variables

really are reddened, the need to understand how this

might a¡ect population dynamics also grows. Theo-

retical studies have already shown environmental

colour to in£uence populations (Rougharden 1975;

Foley 1994) and communities (Caswell & Cohen

1995). Here the form of competition (under- or over-

compensation) has been shown to have a large e¡ect

on the in£uence of environmental colour, as has the

e¡ect of spatial subdivision of a population on the in£u-

ence of noise colour. These interactions between noise

colour and competition, and noise colour and spatial

subdivision, lead us to echo R & L's closing statement.

If ecological theory is to give accurate, non-biased

information on how best to direct conservation and

restoration e¡orts, it must realize the importance of an

accurate model of environmental variability, accurate

with respect to mean, variance and colour.

Great thanks to Professor John Lawton for many discussions

and comments on the manuscript. O.L.P. and A.G. are

supported by NERC studentships.

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Received 4 August 1997; accepted 28 August 1997

Noise, competition, space and persistence O. L. Petchey and others 1847

Proc. R. Soc. Lond. B (1997)