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SUMMARY 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, hasbeen found to a¡ect population persis- tence. This is important because real environmental variables, such as temperature, are reddened or positively temporally autocorrelated. However, recent work has disagreed 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 powerful 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 persistence time for both types of competition.Thus, the inclusion of coloured noise into ecological models can have subtle interac- tions with population dynamics.
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ects on population persistence: the interaction
between environmental noise colour, intrasp eci¢c
competition and space
NERC Centre for Population Biology, Imperial College at Silwood Park, Ascot, Berkshire, SL5 7PY, UK
Department of Biology, Imperi al College at Silwood Park, Ascot, Berkshire SL5 7PY, UK
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.
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.
(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
exp(r(1 ÿ n
)), (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
=(1 N
, (2)
mortalities are equal at low densities (¢gure 1b). In
equation (2), K
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)
exp(r(1 ÿ N
))), (3)
where Z*Poisson (N
exp( r (1 ÿ N
))). The
distribution of the sum of N
indep endent Poisson
distributions each with t he same mean
(exp(r(1 ÿ (N
))), is the same as a Poisson
distribution with mean (N
exp(r(1 ÿ N
Therefore, equation (3) can be interpreted as repre-
senting N
individuals each having a random (Poisson-
distributed) number of o¡ spring with mean
exp(r(1 ÿ N
)), 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
; K
. Autocorrelation in the
noise proces s,
, was determined by a ¢rst order auto-
regressive process described by
, (4)
where the parameter controls the degree of auto-
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)
Figure 1. Mortality as K values (log(N
)) 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
, the variance in K
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
was set very
large (200 000 generations); if extinction did not occur
by t
, t(e) was set to t
(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
) (¢gure 2a). However, when b 0.1
(undercompensation), mean persistence time decreased
from 150 000 generations (572 extinctions by t
) when
0 to about 700 generations (1000 extinctions by
) 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
), 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
for the four combinations of and b representing the
four extremes of the simulations. The dist ribution of
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
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
give limited informa-
tion when attempting to determine the extinction
To elucidate the mechanisms causing extinctions,
therefore, the distributions of log
Noise, competition, space and persistence O. L. Petchey and others 1843
Proc. R. Soc. Lond. B (1997)
mean persistence time
parameter b
mean persistence time
0 0.2
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
is given by the
stochastic process K
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
, hence the variance (K
) 70
for all
. Other paramete rs are r 1:5, K
100 and
200 000.
were plotted for each case (¢gure 3b).
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
) > 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
) < 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
) 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
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
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.
(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)
20 40 60 80 100 120 140 200+200160 180
–2.5 –2 21.510.5–0.5–1.5 –1 0
Figure 3. Frequency distri butions of (a) population size
, and (b) log
) 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
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
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
was large (100 000 generations); if extinction did not
occur by t
, t(e) was set to t
Environmental variability is included in two ways:
(i) temporal variability due to £uctuations in t he mean
carrying capacity across the landscape, K
, determined
in the same way as in the previous section (the value of
therefore determines the level of autocorrelation in
the variability in K
); 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
standard dev iat ion, . Therefore, is a parameter that
controls the heterogeneity in patch quality across the
(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
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)
landscape or
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
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
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
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.
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
1846 O. L. Petchey and others Noise, competition, space and persistence
Proc. R. Soc. Lond. B (1997)
Figure 5. Time-series of metapopulation s ize and patch carryin g capacity K
for the spatial model with (a) no autocorrela-
tion ( 0) in K
, and (c) positive temporal autocorrelation ( 0.95) in K
. Scatter plots of K
against metapopulation size
for (b) 0 and (d) 0.95 indicate how closely metapopulation size tracks K
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.
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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-
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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)
... Here we modeled climatic conditions via changes in prey productivity rate, r(t), but there are other parameters that could also be linked to climatic conditions (Marangon et al., 2020;Peers et al., 2020). Furthermore, population dispersal could connect neighbouring regions that are suffering very different changes in climate variability (Petchey et al., 1997), leading to questions about the effect of spatial coupling on the likelihood of P-tipping. ...
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Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator-prey ecosystems to extinction via so-called phase tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator-prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator-prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.
... Seasonal changes in resources, disturbance, succession, and climate change can all add temporal variation to the resource supply in a given environment. Simultaneous heterogeneity in space and time is likely to produce complex effects, depending on the extent, frequency, and intensity of the spatial and temporal changes (Petchey et al., 1997;Bissonette and Storch, 2007;Yang et al., 2010). Accordingly, there is a rich history of spatial and temporal models in ecology that attempt to understand these effects (Huffaker, 1958;Hutchinson and MacArthur, 1959;Connell and Orias, 1964;MacArthur and Pianka, 1966;Wright, 1983;Holling, 1992). ...
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Canola (Brassica napus L.) is a valuable crop that occupies a large part of the Canadian prairies, and is visited by wild and managed bees. However, the distribution of foraging bees in mass-flowering crops (MFCs), the value of bee visitation, and how MFCs and semi-natural land (SNL) affect wild bee populations in the context of canola agro-ecosystems is unclear. Using three separate studies, I related pollination and yield to bee visitation, and landscape composition and canola abundance to wild bee abundance. The first study examines honey bee (Apis mellifera L.) visitation in commodity canola fields using a simulation model, and reveals that honey bees most closely followed predictions for solitary efficiency-maximizers, valuing nearby flower sources much more than distant ones. The second study relates visita-tion rates of honey bees and alfalfa leafcutting bees (Megachile rotundata (Fabr.)) to pollen deposition and seed production using a structural equation model. Leafcutter bee visitation strongly increased pollen deposiion, but not honey bee visitation, and extra pollen deposition increased seed production in seed canola, but not in commodity canola. The final study uses solitary bee abundance data from a set of landscapes across southern Alberta, and relates bee abundance to landscape composition from the current and previous year. While the overall response of bee abundance to SNL was positive, individual species' response to SNL ranged from positive to negative, and canola had little effect on any species' abundances. These results reveal that insect visitation in flowering crops decay rapidly with distance, that the plant growth context of canola is equally important as the pollination context, and that while the effect of SNL on wild bee abundance is generally positive, it likely varies depending on the traits of individual bee species. This work provides mechanistic insight into the foraging behaviour and contextual value of pollination by managed bees, and sheds light on how agro-ecological landscapes shape wild bee communities.
... Deterministic models with stable equilibria can exhibit undercompensatory dynamics, where the population approaches equilibrium monotonically, or overcompensatory dynamics, where the approach to equilibrium exhibits damped oscillations. Many studies have shown that reddened environmental spectra increase extinction risk for undercompensatory populations and blue spectra increase extinction risk for overcompensatory populations (Danielian, 2016; García-Carreras & Reuman, 2011; Mustin et al., 2013;Petchey et al., 1997;Ripa & Heino, 1999;Ripa & Lundberg, 1996;Ruokolainen et al., 2009;Schwager et al., 2006). While some studies reached different conclusions (Heino, 1998;Heino et al., 2000), Schwager et al. (2006 showed that these contrasting results depend on the modeling details and a consideration of the likelihood of catastrophic events. ...
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Competition from invasive species is an increasing threat to biodiversity. In Southern California, the western gray squirrel (Sciurus griseus, WGS) is facing competition from the fox squirrel (Sciurus niger, FS), an invasive congener.We used spectral methods to analyze 140 consecutive monthly censuses of WGS and FS within a 11.3 ha section of the California Botanic Garden. Variation in the numbers for both species and their synchrony was distributed across long timescales (>15 months).After filtering out annual changes, concurrent mean monthly temperatures from nearby Ontario Airport yielded a spectrum with a large semi-annual peak and significant spectral power at long timescales (>28 months). The cospectrum between WGS numbers and temperature revealed a significant negative correlation at long timescales (>35 months). Cospectra also revealed significant negative correlations with temperature at a six-month timescale for both WGS and FS.Simulations from a model of two competing species indicate that the risk of extinction for the weaker competitor increases quickly as environmental noise shifts from short to long timescales.We analyzed the timescales of fluctuations in detrended mean annual temperatures for the time period 1915-2014 from 1218 locations across the continental USA. In the last two decades, significant shifts from short to long timescales have occurred, from <3 years to 4-6 years.Our results indicate that (i) population fluctuations in co-occurring native and invasive tree squirrels are synchronous, occur over long timescales, and may be driven by fluctuations in environmental conditions; (ii) long timescale population fluctuations increase the risk of extinction in competing species, especially for the inferior competitor; and (iii) the timescales of interannual environmental fluctuations may be increasing from recent historical values. These results have broad implications for the impact of climate change on the maintenance of biodiversity.
... Ripa and Lundberg (1996) applied the Ricker model of population growth (Ricker, 1954) to show that redder noise lessens the risk of extinction, concluding that autocorrelation was critical in determining whether a population grew (under red noise) or declined, ultimately to extinction (under blue noise, with β < 0). Similar results were obtained by Cuddington and Yodzis (1999), using another variant of the Ricker model that explicitly included coloured noise (see also Petchey et al., 1997), extending the noise colours to encompass black noise with β → 3. Extremely high values were found to be associated with increasingly long population persistence times compared to less reddened noise (0.5 ≤ β ≤ 1), suggesting that evolution under highly autocorrelated noise produces robust populations that can adapt effectively to rare but potentially highly significant environmental events. The conclusions of Cuddington and Yodzis (1999) are confirmed by Halley and Kunin (1999) and Inchausti and Halley (2001, 2002, all of whom note that the effect of reddening is to increase the variance observed in longer time series but that, contrary to the expectation that populations are more likely to become extinct in more extreme environments, evolution in such conditions is more robust to environmental variance. ...
Simulations of evolutionary dynamics often employ white noise as a model of stochastic environmental variation. Whilst white noise has the advantages of being simply generated and analytically tractable, empirical analyses demonstrate that most real environmental time series have power spectral densities consistent with pink or red noise, in which lower frequencies contribute proportionally greater amplitudes than higher frequencies. Simulated white noise environments may therefore fail to capture key components of real environmental time series, leading to erroneous results. To explore the effects of different noise colours on evolving populations, a simple evolutionary model of the interaction between life-history and the specialism-generalism axis was developed. Simulations were conducted using a range of noise colours as the environments to which agents adapted. Results demonstrate complex interactions between noise colour, reproductive rate, and the degree of evolved generalism; importantly, contradictory conclusions arise from simulations using white as opposed to red noise, suggesting that noise colour plays a fundamental role in generating adaptive responses. These results are discussed in the context of previous research on evolutionary responses to fluctuating environments, and it is suggested that Artificial Life as a field should embrace a wider spectrum of coloured noise models to ensure that results are truly representative of environmental and evolutionary dynamics.
... Empirical studies reported that natural populations often exhibit positively autocorrelated temporal dynamics (Pimm and Redfearn 1988, Halley 1996, Inchausti and Halley 2001. Theoretical models showed that population growth rate and the timescale structure of environmental fluctuations have significant influences on the timescale-specific patterns of population dynamics (Ripa and Lundberg 1996, Kaitala et al. 1997, Petchey et al. 1997. Specifically, populations with low growth rates converge gradually to its equilibrium when perturbed (referred to as 'under-compensatory growth'; Ruokolainen et al. 2009), resulting in population dynamics with positive temporal autocorrelation. ...
Understanding the processes that stabilize species populations is a fundamental question in ecology and central to conservation biology. In metapopulations, dispersal can act as a ‘double edged' sword for species stability by simultaneously decreasing local population variability (thereby decreasing local extinction risk) while increasing spatial synchrony (thereby increasing landscape-level extinction risk). These dynamics may operate at different timescales, complicating efforts to assess their relative importance for long-term stability. Here, we use a simple metapopulation model to understand how dispersal affects population variability and spatial synchrony across timescales. Our model shows that dispersal has contrasting effects at short versus long timescales on the variability and synchrony of populations. For populations that exhibit slow recovery when perturbed (i.e. under-compensatory growth), dispersal decreases local population variability while increasing spatial synchrony at long timescales. In contrast, at short timescales dispersal increases local population variability while decreasing spatial synchrony. For populations that recover via damped oscillation when perturbed (i.e. over-compensatory growth), the effects of dispersal are all opposite to those for populations with under-compensatory growth, at both short and long timescales. The timescale-dependent effect of dispersal has important implications for empirical studies. Specifically, studies conducted over short periods may only observe population variability increasing and spatial synchrony decreasing with dispersal, whereas the opposite patterns may predominate over longer periods. Our results provide novel insights on the dynamics underlying the role of dispersal and highlight the importance of time series length in empirical studies of metapopulations.
... The principal determinant of the direction of the effect is 114 the curvature of the growth rate's response to the relevant climate variable. However, higher-order 115 properties such as temporal autocorrelation can also play a role (Petchey et al. 1997;Heino et al. 116 2000). Many populations near the edge of their range show greater population variability and 117 climate sensitivity (Myers-Smith et al. 2015;Mills et al. 2017) and so may be expected to be 118 particularly responsive to these effects. ...
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Recent developments in understanding and predicting species responses to climate change have emphasised the importance of both environmental variability and consideration of the wider biotic community. To date, the interaction between the two has received less attention. However, considerable bodies of theory and empirical results suggest that multi-species consequences of variability can have strong impacts on range limits and the speed of range shifts. Here we demonstrate how biotic interactions and temporal variability can act together to influence range shift dynamics and highlight the need to understand these interactions in order to predict how species will respond to global change. We emphasise the value and utility of partitioning approaches applied to parameterised models to determine the direction and relative importance and direct of these forces in empirical systems. Authorship JCDT wrote the manuscript and built the models. All authors contributed significantly to the editing and manuscript development. Funding The work was supported by NERC grant NE/T003510/1 Data Sharing and Data Accessibility Code to generate all results is publicly available at and should the manuscript be accepted will be permanently archived. The paper contains no new datasets.
... Les effets qui entrent en synergie sont particulièrement dangereux pour l'équilibre des écosystèmes qui sont soumis à de fortes pressions et ainsi rendus encore plus vulnérables (Folke et al., 2004). Petchey et al., 1997;Begg et al., 1999;Lundberg et al., 2000). ...
Dans le contexte actuel de surpêche et de réchauffement climatique, il est nécessaire d’adopter une gestion écosystémique des ressources halieutiques. Ce travail de thèse apporte des éléments de réponse à cette problématique en se penchant sur le cas de la pêcherie crevettière guyanaise et de son impact sur les communautés de poissons associées. Comme dans la plupart des régions tropicales, les eaux côtières de Guyane se sont réchauffées d’environ 1°C entre 1990 et 2017. En revanche, comme dans peu d’endroits au monde, la pression de pêche s’est quasiment effondrée sur cette même période. Ce cas d’étude est donc une bonne opportunité pour comprendre la réaction des communautés de poissons de cette région tropicale après une diminution progressive et à grande échelle de la pression de pêche dans un contexte de réchauffement climatique. Les résultats de ce travail montrent que les facteurs environnementaux jouent un rôle dominant dans la structuration des communautés de poissons marins en Guyane. Les analyses montrent notamment des augmentations de certains indicateurs de diversité, une augmentation du spectre des tailles maximales théoriques des espèces et une augmentation de la redondance fonctionnelle qui est un facteur clé pour la stabilité des peuplements. Ces éléments témoignent d’une restructuration des communautés et mettent en avant leur capacité à retrouver un état riche et stable en une période relativement courte d’une douzaine d’années voire moins. On observe néanmoins une tropicalisation des communautés qui pose des questions sur l’évolution future de la diversité des écosystèmes tropicaux au regard des différents scénarios climatiques.
One of the biggest threats to the survival of many plant and animal species is the destruction or fragmentation of their natural habitats. The conservation of landscape connections, where animals, plants, and ecological processes can move freely from one habitat to another, is therefore an essential part of any new conservation or environmental protection plan. In practice, however, maintaining, creating, and protecting connectivity in our increasingly dissected world is a daunting challenge. This fascinating volume provides a synthesis on the current status and literature of connectivity conservation research and implementation. It shows the challenges involved in applying existing knowledge to real-world examples and highlights areas in need of further study. Containing contributions from leading scientists and practitioners, this topical and thought-provoking volume will be essential reading for graduate students, researchers, and practitioners working in conservation biology and natural resource management.
It is already well known that environmental variation has a big effect on real evolution, and similar effects have been found in evolutionary artificial life simulations. In particular, a lot of research has been carried out on how the various evolutionary outcomes depend on the noise distributions representing the environmental changes, and how important it is for models to use inverse power-law distributions with the right noise colour. However, there are two distinct factors of relevance-the average total magnitude of change per unit time and the distribution of individual change magnitudes-and misleading results may emerge if those factors are not properly separated. This article makes use of an existing agent-based artificial life modeling framework to explore this issue using models previously tried and tested for other purposes. It begins by demonstrating how the total magnitude and distribution effects can easily be confused, and goes on to show how it is possible to untangle the influence of these interacting factors by using correlation-based normalization. It then presents a series of simulation results demonstrating that interesting dependencies on the noise distribution remain after separating those factors, but many effects involving the noise colour of inverse power-law distributions disappear, and very similar results arise across restricted-range white-noise distributions. The average total magnitude of change per unit time is found to have a substantial effect on the simulation outcomes, but the distribution of individual changes has very little effect. A robust counterexample is thereby provided to the idea that it is always important to use accurate environmental change distributions in artificial life models.
Recent developments in understanding and predicting species responses to climate change have emphasised the importance of both environmental variability and consideration of the wider biotic community. However, to date, the interaction between the two has received less attention. Both theoretical and empirical results suggest that the combined effect of environmental variability and interspecific interactions can have strong impacts on existing range limits. Here we explore how competitive interactions and temporal variability can interact with the potential to strongly influence range shift dynamics. We highlight the need to understand these between‐process interactions in order to predict how species will respond to global change. In particular, future research will need to move from evaluating possibilities to quantifying their impact. We emphasise the value and utility of empirically parameterised models to determine the direction and relative importance of these forces in natural systems.
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A recurrent problem in ecology and conservation biology is to estimate the risk of population extinctions. Extinction probabilities are not only imperative for conservation and management, but may also elucidate basic mechanisms of the regulation of natural populations (Burgman et al. 1993; Pimm 1994). The usual way of modelling stochastic influence on population dynamics has been to assume that the external noise is uncorrelated. This means that each and every randomly drawn noise value is totally independent on previous ones. This is what is generally called `white' noise. However, the noise itself can be temporally autocorrelated. That is, the values of the random numbers used in the noise process will depend on previous ones. Here we show that the autocorrelation, or colour, of the external noise assumed to influence population dynamics strongly modifies estimated extinction probabilities. For positively autocorrelated (`red') noise, the risk of extinction clearly decreases the stronger the autocorrelation is. Negatively autocorrelated (`blue') noise is more ambiguously related to extinction probability. Thus, the commonly assumed white noise in population modelling will severely bias population extinction risk estimates. Moreover, the extinction probability estimates are also significantly dependent on model structure which calls for a cautious use of traditional discrete-time models.
What is the minimum viable population (MVP) of a particular species? Besides the obvious implications for conservation, especially of endangered species, this question raises important issues in population biology. MVP obviously varies with demographic, life history and environmental factors, but also depends upon genetic load and genetic variability. This book addresses the most recent research in the rapidly developing integration of conservation biology with population biology. Chapters consider the roles of demographic and environmental variability; the effects of latitude, body size, patchiness and metapopulation structure; the implications of catastrophes; and the relevance of effective population size on inbreeding and natural selection. Other topics addressed include the role of decision theory in clarifying management alternatives for endangered species, and the opportunities for improved co-operation between agencies responsible for management. The book concludes with a forward-looking and plain-speaking summary on future research and its application for conservation practice.
We explore extinction rates using a spatially arranged set of subpopulations obeying Ricker dynamics. The population system is subjected to dispersal of individuals among the subpopulations as well as to local and global disturbances. We observe a tight positive correlation between global extinction rate and the level of synchrony in dynamics among the subpopulations. Global disturbances and to a lesser extent, migration, are capable of synchronizing the temporal dynamics of the subpopulations over a rather wide span of the population growth rate r. Local noise decreases synchrony, as does increasing distance among the subpopulations. Synchrony also levels off with increasing r: in the chaotic region, subpopulations almost invariably behave asynchronously. We conclude that it is asynchrony that reduces the probability of global extinctions, not chaos as such: chaos is a special case only. The relationship between global extinction rate, synchronous dynamics and population growth rate is robust to changes in dispersal rates and ranges.
Illustrates points about communities, stability and time-scales with examples from bird and plankton communities, and bring in points about the nature of environmental variability and its important consequences for the practical decision about whether a particular community can be said to be stable. -from Author
1. A linear model for population dynamics in stochastic environments is introduced by linearizing the logistic equation in discrete time for small stochastic variations around a large average carrying capacity. The analysis of this model involves the relatively simple mathematics of time-series analysis and is not restricted to white noise (i.e., totally unpredictable) environments. 2. In this linear model the population size at any time is a reflection of many historical effects. The intrinsic rate of increase, r, is a measure of the responsiveness of the population to changes in k. A responsive population (r ≈ 1) does not retain historical effects for long, whereas a sluggish population (r ≈ 0) does. 3. The variance of the population size through time always increases with r. But the presence of much predictability in the environment, as measured by the serial correlation between consecutive values of K, reduces the influence of r. High serial correlation allows time for even a sluggish population to ac...
Stochastic factors affecting the demography of a single population are analyzed to determine the relative risks of extinction from demographic stochasticity, environmental stochasticity, and random catastrophes. Relative risks are assessed by comparing asymptotic scaling relationships describing how the average time to extinction, T, increases with the carrying capacity of a population, K, under each stochastic factor alone. Stochastic factors are added to a simple model of exponential growth up to K. A critical parameter affecting the extinction dynamics is r, the long-run growth rate of a population below K, including stochastic factors. If r is positive, with demographic stochasticity T increases asymptotically as a nearly exponential function of K, and with either environmental stochasticity or random catastrophes T increases asymptotically as a power of K. If r is negative, under any stochastic demographic factor, T increases asymptotically with the logarithm of K. Thus, for sufficiently large populations, the risk of extinction from demographic stochasticity is less important than that from either environmental stochasticity or random catastrophes. The relative risks of extinction from environmental stochasticity and random catastrophes depend on the mean and environmental variance of population growth rate, and the magnitude and frequency of catastrophes. Contrary to previous assertions in the literature, a population of modest size subject to environmental stochasticity or random catastrophes can persist for a long time, if r is substantially positive.
A unifying approach described by a random birth and death process which includes both environmental and demographic noise is introduced. It is shown that both of these noise sources play an essential role in extinction processes in general. The probability distribution of the lifetime of a population is determined and its dependence on the parameters of the model is discussed. Finally a population divided into subpopulations is modeled. The lifetime of this ensemble of subpopulations is compared to the lifetime of one large population.