Download Extinctions in competitive communities forced by coloured

Oikos 116: 439 ! 448, 2007
doi: 10.1111/j.2006.0030-1299.15586.x,
Copyright # Oikos 2007, ISSN 0030-1299
Subject Editor: Owen Petchey, Accepted 30 October 2006
Extinctions in competitive communities forced by coloured
environmental variation
Lasse Ruokolainen, Mike S. Fowler and Esa Ranta
L. Ruokolainen ([email protected]), M. S. Fowler and E. Ranta, Integrative Ecology Unit, Dept of Biological and
Environmental Sciences, P.O. Box 65 (Viikinkaari 1), FI-00014 University of Helsinki, Helsinki, Finland. Present address for LR:
Dept of Ecology and Evolutionary Biology, Viikinkaar1, PO Box 65, FI-00014 University of Helsinki, Helsinki, Finland.
Understanding the relationships between environmental fluctuations, population dynamics and species
interactions in natural communities is of vital theoretical and practical importance. This knowledge is essential
in assessing extinction risks in communities that are, for example, pressed by changing environmental conditions
and increasing exploitation. We developed a model of density dependent population renewal, in a Lotka !
Volterra competitive community context, to explore the significance of interspecific interactions, demographic
stochasticity, population growth rate and species abundance on extinction risk in populations under various
autocorrelation (colour) regimes of environmental forcing. These factors were evaluated in two cases, where
either a single species or the whole community was affected by the external forcing. Species’ susceptibility to
environmental noise with different autocorrelation structure depended markedly on population dynamics,
species’ position in the abundance hierarchy and how similarly community members responded to external
forcing. We also found interactions between demographic stochasticity and environmental noise leading to a
reversal in extinction probabilities from under- to overcompensatory dynamics. We compare our results with
studies of single species populations and contrast possible mechanisms leading to extinctions. Our findings
indicate that abundance rank, the form of population dynamics, and the colour of environmental variation
interact in affecting species extinction risk. These interactions are further modified by interspecific interactions
within competitive communities as the interactions filter and modulate the environmental noise.
Climate and anthropogenic changes to the environment
pose a clear threat to biological diversity (Walther et al.
2002). Consequently, the relationship of population
extinctions to environmental variation has become an
elementary topic in ecological research in recent years.
The mechanisms behind, and effects related to, species
extinction have been studied from various angles.
Earlier studies focused on the effects of environmental
noise on extinction risk in single species populations
(Roughgarden 1975, Ripa and Lundberg 1996, 2000,
Kaitala et al. 1997a, 1997b, Petchey et al. 1997, Heino
1998, Ripa and Heino 1999, Heino et al. 2000, Laakso
et al. 2004, Pike et al. 2004, Schwager et al. 2006),
while it is now recognised that species interactions have
important consequences that cannot be ignored. Many
studies have focused on the cascading effects of species
removal events in ecological communities (Paine 1966,
Pimm 1980, Fowler 1990, Lundberg et al. 2000,
Fowler and Lindström 2002, Fowler 2005, Eklöf and
Ebenman 2006), while others have looked at how
extinctions are affected by different management
strategies in both single species microcosms (Fryxell
et al. 2005) and model multispecies communities
(Enberg et al. 2006).
Noise colour relates to the analogy between the
degree of autocorrelation of a time-series and the
frequency of visible light. ‘‘Red’’ noise, i.e. slowly
changing, low frequency noise is positively autocorrelated, whereas ‘‘blue’’ noise is fast changing, high
frequency noise, with a negative autocorrelation. White
noise has an equal proportion of high and low
frequencies and has no autocorrelation. Colour is an
important property of environmental fluctuation and
has been shown to translate to population dynamics
439
(Kaitala et al. 1997a, 1997b). Most theoretical and
experimental work with microcosms has concentrated
on the effects of red noise on population persistence
(Petchey et al. 1997, Heino 1998, Heino et al. 2000,
Petchey 2000, Pike et al. 2004), as blue noise is
considered unlikely in natural systems. The effect of
the full spectrum of noise colours on population
extinction risk has been considered in a few cases
(Ripa and Lundberg 1996, Ripa and Heino 1999,
Laakso et al. 2004). Here we chose to consider blue
noise for the sake of completeness. This choice can also
be justified in other ways, for example, considering
environmental variation over a different temporal scale
may change its spectrum (e.g. seasonal vs annual). In
addition, some climatic indices, such as El Niño
(Burgers 1999), may be negatively autocorrelated,
depending on the temporal scale.
While previous studies have considered the effects of
disturbance (Paine and Levin 1981), species removal
(Paine 1966, Pimm 1980), noise colour (Ripa and
Lundberg 1996, Petchey 2000, Pike et al. 2004),
species abundance rank (Fowler 2005) and population
dynamics (Roughgarden 1975, Ripa and Lundberg
2000) on species persistence, interactions between these
factors are still relatively poorly understood. We
approached these issues by modelling multi-species
communities using a deterministic skeleton for population renewal, and incorporating both demographic and
environmental stochasticity to the renewal function. We
examined the effect of environmental autocorrelation
on extinction probability of different community
members, under different dynamical scenarios and
when community members respond to the environmental variation in different ways. Our results indicated
that much information could be gleaned by ranking
species according to their abundance. Although there
were interactions between the different factors influencing extinction probability, much of the variation in the
results can be explained through knowledge of species
abundance ranks, the intrinsic growth rate of the
community members and their similarity in responding
to environmental variation.
Methods
The population model
We modelled multispecies population growth using
Ricker dynamics with Lotka !Volterra competitive
interactions as the kernel of the renewal process (Eq.
1). Behaviour of the Ricker model is well known in
single-species undisturbed situations (May 1974a, May
and Oster 1976). We incorporated demographic
stochasticity to the system by using a Poisson operator,
? ), where Nt"1
?
is the mean of the
Nt"1 #/Poisson(Nt"1
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Poisson distribution, taken from the population renewal function. This allowed us to address the
dynamics of small populations in a realistic manner.
Thus, the population renewal of each species i, in these
communities can be written as:
!
"#
Ni;t"1 #Poisson Ni;t exp r 1$
PS
j#1
aij Nj;t
Ki
$%&
;
(1)
where N is the population density, t is generation
number, r is intrinsic growth rate, S is the number of
species in the community, Ki is the species-specific
carrying capacity and aij indicates the per capita effect
of species j on the population renewal of species i.
Interspecific interactions, aij, among all S species in
Eq. 1 determine the off-diagonal elements in the
community matrix, A (the diagonal elements aii are
intraspecific interaction terms, standardized here to
unity). Equilibrium population densities N* (a column
vector of length S) for the S species (Eq. 1) were derived
as N* #/A$1K, from the equation K #/AN* (May
1974b), where K is the vector of species-specific
carrying capacities. Abundance ranking was determined
as follows: N*1 B/N*2 B/N*3 B/. . .B/N*S, where species S is
the most abundant. We drew aij values from a uniform
discrete random distribution, with limits [0.01, 0.8],
and selected only communities that were locally stable
for further investigation (for local stability conditions in
this model, see Ranta et al. 2006).
To find the baseline extinction probabilities in
locally stable communities with demographic stochasticity, we first explored the behaviour of the model in
the absence of environmental forcing. The intrinsic
growth rate (r) was varied between 0.1 (highly undercompensatory dynamics in a single species system) and
1.9 (close to the border of the 2-point bifurcation; May
1974a), and the minimum carrying capacity (K1) was
varied from 10 to 100, while the maximum was kept S
times higher. Values of Ki were drawn at random from
a discrete uniform distribution. This allowed us to
determine what the underlying probability of species
loss across a variety of population sizes was due to
demographic stochasticity (i.e. the Poisson process)
before communities were affected by environmental
noise.
Based on these results (Fig. 1), we used a minimum
equilibrium density for the least abundant species (N*1)
in the remaining analyses, which ensured that extinction events due solely to demographic stochasticity were
not too frequent under undercompensatory dynamics.
Following this, Ki values were assigned as above,
with limits [50, S %/50]. The minimum value was a
compromise between a tolerable level of background
extinctions and the disturbance level required to allow
Fig. 1. Mean extinction probability due to demographic
stochasticity alone, for different combinations of intrinsic
growth rate (r), and carrying capacity (K) of the least
abundant species in a multi-species community of competitors. Results based on 100 simulations for each parameter
combination.
environmentally driven extinctions. Community size
was set to S #/5 in all simulations (results were
qualitatively similar across different community sizes).
Populations were initiated at equilibrium, N*, to avoid
artefacts such as extinctions caused by low initial
densities. The inclusion of different forms of stochasticity ensured populations moved away from equilibrium densities immediately.
Environmental forcing
Environmental variation was introduced into the system
as a multiplicative process, through dynamic variation
of the carrying capacity of either a single community
member, or all community members. This was done
in order to investigate the effect of species position in
the abundance hierarchy where a focal species is
disturbed by a temporally autocorrelated environmental noise process. The noise was introduced as
follows:
(2)
Ki (t)#Ki g(t);
where Ki is the carrying capacity of the affected species
and g(t) is the environmental noise with given first-order
autoregressive structure (Ripa and Lundberg 1996)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g(t"1)#kg(t)"s 1$kv(t"1);
(3)
where k is the parameter characterising the autocorrelation in the noise signal, which can vary between $/1
(blue noise) and "/1 (red noise). When k #/0, the noise
is said to be white. The variable v is drawn randomly
from a normal distribution with standard deviation s.
Noise entered the model in a multiplicative way to
ensure that each species would be similarly affected,
irrespective of abundance. The ‘‘focal species’’ is the
species that is directly affected by the environmental
noise. Each community member was treated as a focal
species and tested in three different autocorrelation
regimes (k being either $/0.7, 0 or 0.7). This was
done to assess the effect of the rate of short-term
environmental change. In a second scenario, all community members were disturbed through their carrying
capacity simultaneously. The range of environmental
fluctuations was set such that the maximum disturbance
was 9/65% (s :/0.25) of the focal species current
density.
The model was simulated for 1000 time steps for
each combination of noise colour affecting each focal
species rank. Each parameter combination was iterated
1000 times, and for each iteration, a new A matrix was
constructed and a new noise time series was generated.
During the simulations all extinction events and the
abundance rank of the extinct species were recorded.
From this data extinction probability was scored for
each species.
Across simulations, several factors were varied: which
species experienced environmental forcing (a single, or
all species), the nature of the underlying population
dynamics (under-, exact-, or overcompensation), the
colour of environmental variation (blue, white, or red);
and the degree of correlation among species responses
to the environmental variation (uncorrelated or perfectly correlated) (summarized in Table 1). We also
tested the following cases of responses to environmental
variation ! (1) a single focal species, (2) all species
disturbed with uncorrelated noise terms (i.e. the
variance-covariance matrix of among species environmental variation was an S by S identity matrix) and
(3) all species with perfectly correlated noise terms (i.e.
the variance-covariance matrix of among species environmental variation was an S by S matrix of ones) !
under three different dynamical scenarios, within the
range of stable equilibrium dynamics in the absence of
noise. We examined undercompensatory (r #/0.25),
perfectly compensatory (r #/1) and overcompensatory
Table 1. Factors varied in this study and their respective levels.
Factor
Levels
Focal species abundance (species that is
1, 2, 3, 4, 5
directly affected by environmental forcing)
Abundance rank (increasing from least to
1, 2, 3, 4, 5
most abundant)
Growth rate (r)
0.25, 1, 1.75
Noise colour (environmental autocorrelation) $/0.7, 0, 0.7
(k)
Environmental correlation (correlation
0, 1
between noise terms for different species)
(r)
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(r #/1.75) equilibrium dynamics, in order to uncover
any relationships between intrinsic growth rate, noise
colour and extinction probability. As generation of
locally stable communities is not possible when r]/2,
we restricted our analyses to communities with r-values
less than 2.
Statistical analysis
The simulated data of extinction probabilities (based on
1000 trials) was analysed with ANOVA. In the single
focal species case the statistical model incorporated only
first-order interactions. This was due the fact that there
was no replication within factor levels, which made the
model run out of degrees of freedom. In the multi-focal
species case (whole community) no interactions could
be tested for the same reason. The analyses were carried
out separately for each r!value. As extinction probability was not normally distributed, the statistical
probability of the model parameters was estimated
with a permutation test. All statistical analysis where
carried out using R v2.2.0 (R Development Core Team
2005).
In order to uncover mechanisms behind the observed patterns, we analysed various aspects of the
population time series shown to be important in
predicting extinctions in single-species systems. For an
additional 100 simulations of the two scenarios where
the whole community was forced (with either uncorrelated or perfectly correlated environmental noise terms),
we recorded the autocorrelation, coefficient of variation
(CV), and standard error (SE) of extant time series of
the least abundant species. We also recorded population
densities one, and two time steps before extinction
(results not shown here).
Results
We began by exploring the impact of demographic
stochasticity alone on species extinctions (Fig. 1).
Increasing the intrinsic growth rate, r, or carrying
capacity of the least abundant species, K1, led to a
reduction in the probability of species loss in communities. These results were subsequently used to select
parameter values throughout the remainder of the
results, for which background extinctions would be
minor.
We next focused on assessing the significance of
environmental forcing, with varying autocorrelation
structure, on the extinction rate of each community
member. We selected each of the abundance-ranked
species in turn as the focal species, then scored
extinctions for the focal species and for the other
members of the community (influenced by the noise
442
only via the interaction terms aij). When this procedure
was repeated for all species in the community clear
patterns emerged (Fig. 2). Red (k #/0.7) noise always
yielded a higher probability of extinctions than did
white noise (k #/0) regardless the abundance rank of the
focal species (Fig. 2a !c). Extinction rates for the other
species in the community (non-focal) were clearly lower
than that of the focal species. These secondary extinctions were very rare for r]/1. In both cases (focal, nonfocal species) the risk of extinction correlated strongly
with abundance rank of the species (Fig. 2g !i). There is
a decrease in importance of the species rank associated
with an increase in r, in terms of extinction probabilities. With low r (Fig. 2g) most of the variation is
explained by species rank. When r#/1, species rank
declines in importance while the focal %/rank interaction increases (Fig. 2h). In this case, noise colour also
increases in relative importance, as compared with
abundance rank and focal %/rank interaction. Finally
with high r most of the variation in the results is due to
the different extinction probabilities of the different
focal species, reflected by the focal %/rank interaction
(Fig. 2i), while the noise colour has very little effect.
The effect of the modulating noise differed between
the different dynamical scenarios. With r#/ 0.25 and
r#/1 (Fig. 2a !b), extinction probability was higher
under red noise than under blue or white noise *this
being most pronounced with r #/1. There was no
marked difference in extinction probabilities between
blue and white noise. When r #/1.75, white noise
resulted in a lower extinction rate than either blue or
red noise when the focal species was relatively more
abundant (Fig. 2c). Noise colour only had minor effects
on (secondary) extinctions of the non-targeted species
(Fig. 2d!f).
We continued by exposing the whole community to
environmental forcing. First, with uncorrelated among
species noise terms, noise colour had a clear effect on
extinction risk (Fig. 3a !c). While this was true over all
r!values tested, the most pronounced differences were
observed with perfectly compensating dynamics (r #/1).
With r5/1 red noise resulted in a higher extinction
probability than did white and blue noise, whereas with
r#/1.75 extinction probability was highest under blue
noise and lowest under red noise. These patterns
changed somewhat when species responded identically
to the environmental variation (Fig. 3d!f). Here, noise
colour only had a significant effect on extinction risk
with r#/ 1.75. Most strikingly, red noise resulted in a
very low extinction probability, while under blue noise
extinction probability was close to one. This reduction
in extinction probability with red noise is a result of
increased synchrony in the response of the community
to the environmental variation.
Abundance rank played a major role in determining the extinction probability of individual species
r = 0.25
a
r=1
b
r = 1.75
c
κ = –0.7
κ=0
κ = 0.7
1
2
3
4
5
r = 0.25
d
1
g
2
3
r = 0.25
1
2
4
5
r=1
e
4
3
1
Abundance rank (focal)
5
1
2
3
4
5
r=1
1
i
3
4
5
r = 1.75
f
Abundance rank (non-focal)
h
2
2
3
4
5
r = 1.75
Noise
Focal
Rank
N F
N R
F R
Resid
Mean squares
Fig. 2. Mean extinction probability in the single-focal species scenario (mean9/95% CI for a binomial distribution), for three
different values of noise autocorrelations (k) and intrinsic growth rate (r). Mean extinction probability of the focal species (a !c)
and non-focal species (d !f). (g !i) Mean squares for the ANOVA design, p #/0.01 for all visible bars. Abundance rank is defined
as relative abundance within the community, where a higher rank corresponds to higher abundance. Results based on 1000
simulations for each scenario. Data points have been shifted horizontally to improve readability.
when there was no correlation between noise terms (Fig.
3g !i). For all r$/values, species with a relatively higher
abundance rank suffered fewer extinctions than those
with lower abundance ranks (Fig. 3a !c). However, with
perfect environmental correlation abundance rank was
an important factor only with undercompensatory
dynamics (r #/0.25, Fig. 3d and j). The very low
extinction rates when r #/1 (Fig. 3e) arise due to the
perfectly correlated response of the community to
environmental fluctuations and the perfect compensation of the dynamics. This can be contrasted with the
uncorrelated environmental signals, which produce
443
r = 0.25
a
r=1
b
r = 1.75
c
κ = –0.7
κ=0
κ = 0.7
1
2
3
4
5
r = 0.25
d
1
2
3
r = 0.25
g
1
2
4
5
r=1
e
4
3
1
Abundance rank
5
1
2
3
4
5
r=1
3
4
5
r = 1.75
f
1
Abundance rank
h
2
2
3
4
i
r = 1.75
l
r = 1.75
5
Noise
Rank
Resid
Mean squares
r = 0.25
j
k
r=1
Noise
Rank
Resid
Mean squares
Fig. 3. Mean extinction probability in the multi-focal species scenario (mean9/95% CI for a binomial distribution) (all species
affected by noise), for three different values of noise autocorrelations (k) and intrinsic growth rate (r). Correlation between
environmental noise terms (a !c) r#/0, (d !f) r#/1. (g $/l) Mean squares for the ANOVA design with either r #/0 (g$/i), or
r#/1 (j$/l), p#/0.01 for all visible bars. Abundance rank is defined as relative abundance within the community, where a higher
rank corresponds to higher abundance. Results based on 1000 simulations for each scenario. Data points have been shifted
horizontally to improve readability.
444
considerably higher extinction rates for all species
(Fig. 3b).
According to our analyses of the properties of extant
population time series, autocorrelation in the least
abundant species populations decreased with increasing
growth rate and decreasing autocorrelation in the
environmental signal (Fig. 4a !b). Population colour
tended also to be slightly less red when the community
a
b
c
d
was exposed to identical environmental fluctuation
(Fig. 4b). Nevertheless, the least abundant species
nearly always showed red dynamics, with only two
exceptions (considering blue and white noise, overcompensatory growth and correlated noise terms;
Fig. 4b). Fluctuation in population density (as measured by coefficient of variation, CV) of the least
abundant species was generally independent of environmental autocorrelation (Fig. 4c !d). In both cases
(environmental noise terms being either uncorrelated
or correlated between species), these relative fluctuations did not reflect the observed patterns in extinction
probability. When population fluctuations were measured as the standard error (SE) of extant time series,
fluctuation in population density of the least abundant
species tended to grow with increasing growth rate as
the community members reacted independently to
environmental variation (Fig. 4e). However, as with
CV, no reversal in fluctuation severity between blue and
red noise, with overcompensating dynamics, could be
observed. As the community members were forced by
identical environmental variation (Fig. 4f), the SE of
the least abundant species populations showed a similar
pattern to that of extinction probability. The least
abundant species clearly fluctuated more violently in a
blue environment as compared with the fluctuations
observed in a red environment, when the population
growth was overcompensatory.
Discussion
e
f
Fig. 4. Mean autocorrelation (a, b), mean coefficient of
variation (CV; c, d), and mean standard error (SE; e, f) of
extant population time series, calculated over the whole
community after 100 simulations (9/1 SD). The relatively
high autocorrelation values in all cases are due to the Poisson
operator, which is effectively a white noise process. Symbols
are as in Fig. 2 and 3.
This study has shown that environmental forcing with
different autocorrelation structure has a clear impact
upon extinction events in competitive communities that
are also affected by demographic stochasticity. Our
results indicate that the noise structure also interacts
strongly with the dynamical properties of the particular
species that are affected. A similar relationship arose
with species’ abundance ranking. Changes in the
amount of environmental correlation experienced across
the community also had an important effect on the
interaction between noise colour and population
growth rate. This means that while population dynamic
processes are of undoubted importance, they are not the
sole determinants of species response to different forms
of environmental variation. This response is further
modified by the similarity of different species’ reactions
to environmental variation in competitive communities.
In the single focal species case (1) red noise was
associated with the highest extinction probability (with
one exception, see Fig. 2c). This reflects results attained
with single species models (Johst and Wissel 1997,
Petchey et al. 1997). Extinctions are due to sufficiently
long runs of ‘‘bad’’ years, which are less likely to occur
under white or blue noise. Exact compensation, i.e.
445
when the environmental signal is tracked exactly, will
bring about the most pronounced differences in the
colour of the population time series in single species
systems (Kaitala et al. 1997b) and consequently also in
extinction probability. Our multi-species results are in
accordance with this. In a single species system blue
noise tends to amplify fluctuations in populations with
overcompensatory dynamics (Kaitala et al. 1997b). Red
noise should however reduce population fluctuations
when populations express overcompensatory growth
(Ripa and Heino 1999), but in this case very high
extinction rates were recorded. It would be tempting to
conclude that interspecific interactions in the community lead to this rather unexpected outcome. Further
investigation is required before reliable conclusions can
be made.
From our investigation of extinction risk due to
demographic stochasticity alone (Fig. 1) it may be
expected that extinction probability should also decrease with increasing growth rate in the presence of
environmental variation. However, an opposite trend
was observed here (Fig. 2 and 3). Extinction rates of
focal species’ tended to be highest in association with
overcompensatory dynamics. This pattern was evident
in all three cases analysed: (1) when noise only affected
a single focal species, (2) when species were affected by
uncorrelated noise terms and (3) when all species were
affected by correlated noise terms. The interaction
between demographic and environmental stochasticity
led to this surprising reversal in the direction of
extinction probabilities. Although both forms of stochasticity have previously been studied simultaneously
(Lande 1993, Halley and Iwasa 1998), the reversal in
extinction probabilities due to this interaction has not,
to our knowledge, been reported yet. The relatively
high extinction levels observed with undercompensatory
dynamics, both in the single species case (1) and
multispecies cases (2, 3), were mostly due to the
random demographic process (Fig. 1). Following the
loss of the focal species, in case (1), the remaining
community members are only affected by demographic
(but not environmental) stochasticity. Thus, when a
single species was forced by environmental noise,
secondary extinctions could arise through two mechanisms: 1) loss of species due to a change in the feasibility
or stability of the remaining community following
extinction of the focal species, i.e. an extinction cascade,
or 2) loss of species through demographic stochasticity.
In the second scenario examined here (2) extinction
rate was lowest under red noise in association with
overcompensatory dynamics (Fig. 3c). In the third case
(3) there was a considerable reduction in extinction
probability associated with the change from under- to
perfectly compensatory dynamics for all species. Contrary to the previous case, noise colour had no effect
within communities with under- or perfectly compen-
446
satory dynamics (Fig. 3d !e). However, overcompensatory dynamics led to a somewhat unexpected outcome,
with a clear difference in extinction probability under
different noise colours. Blue and white noise led to high
extinction rates for all species (similar to those with
uncorrelated noise terms), while red noise resulted in
almost negligable extinction rates (Fig. 3f). No such
pattern was observed for the single focal species case
(1), although it has been shown to occur in single
species systems (Petchey et al. 1997, Ripa and Heino
1999). These findings further indicate that while the
degree of correlation among species specific responses to
environmental variation is important (Ripa and Ives
2003), community interactions and the scale of
disturbance within a community play a considerable
role in affecting the extinction risk of community
members.
Inspection of both the coefficient of variation (CV)
and standard error (SE) of extant population time series
showed that the results can to some extent be understood by differences in population variability (Fig. 4), as
it has been demonstrated with single species models
(Roughgarden 1975, Ripa and Heino 1999, Laakso
et al. 2004). In turn, the magnitude of population
fluctuations is a product of the intrinsic population
dynamics, the degree of environmental autocorrelation
and demographic stochasticity. Population growth over
time translates into population colour (Kaitala et al.
1997b) and consequently it determines the speed at
which populations approach low densities. Undercompensating populations exhibit a high autocorrelation
coefficient irrespective of noise colour and approach
low densities gradually. Overcompensating populations
have a lower autocorrelation regardless of the colour of
external noise, and approach low densities rapidly. In
our study system, there was an overall decrease in
population autocorrelation associated with increasing
growth rate in all three cases (1, 2, 3; Fig. 4). Moreover,
the highest autocorrelations were recorded for populations experiencing red noise, and lowest for those
experiencing blue noise. In all cases, where the highest
extinction probability was associated with red noise, this
could be linked to increased fluctuations in population
densities. The extremely low extinction rates observed
in the case of perfectly correlated noise terms (2) and
perfect compensation was associated with relatively low
population variability for all noise colours. Furthermore, the increase in extinction probability associated
with increasing growth rate (Heino 1998) could partly
be explained by increasing variability.
In the case of perfectly correlated noise terms (3) and
overcompensatory dynamics, populations experiencing
red noise fluctuated relatively little, whereas those
forced by blue noise displayed more severe fluctuations.
However, the marked differences in extinction probability between different noise colours could not be
fully explained by differences in population variability.
Neither could they be accounted for by pre-extinction
densities (Ripa and Lundberg 1996, 2000) or the colour
of the population time series. This paper represents the
first attempt to quantify extinction probabilities in nonlinear competitive communities under the joint influence of demographic- and environmental stochasticity.
We hope we will stimulate further research to help
uncover the precise mechanisms behind the results
reported here.
The two multi-species scenarios we have considered,
i.e. either uncorrelated or perfectly correlated environmental variation across species, represent two extreme
cases. In real ecosystems it is likely that the degree of
correlation depends on a number of biogeographical
and ecological factors and neither extreme is likely to
occur. However, species in competitive communities
may be more likely to experience environmental
fluctuations at a similar scale. Hence, competing species
are likely to respond similarly in a shared habitat.
Moreover, it is also unlikely that any natural disturbances will affect only a single species in a community.
Anthropogenic disturbances, such as harvesting or
stocking, may however be very specific by their impact
on species assemblages (Enberg et al. 2006), and thus
our results considering single species disturbance are not
of academic interest only.
Although the actual abundance of the focal species
has been shown to be a relatively poor indicator of
community level effects, relative abundance rank has
been shown to play a major role in indicating extinction
probability when the community is disturbed through
species removal (Fowler 2005). This agrees with earlier
findings that the effect of species removal depends
considerably on the ecological role of the removed
species in the food web (Pimm 1980, Borrvall et al.
2000). For example, better competitors receive less
pressure from the rest of the community and are more
buffered against environmental changes (Ives 1995).
Our results confirm that abundance rank may be as
informative as the precise a !values when extinction risk
of a given species is evaluated (Fowler 2005). Measuring
a !values is likely to prove extremely tricky in real
population data (Wootton 1994, Laska and Wootton
1998, Berlow et al. 2004), so using a relevant proxy
such as abundance is likely to prove a much more
practical strategy in species management and conservation efforts.
According to Wichmann et al. (2003), the attributes
of autoregressive noise used in this and earlier studies
are insufficient to describe environmental variation.
While this might be the case, the autoregressive noise
model we have used is easy to apply, it gives a
reasonable approximation of short-term environmental
variation, and it allows a comparison between earlier
models. For long-term variation 1/fb -noise is suggested
to be more realistic (Halley 1996, Cuddington and
Yodzis 1999), while the importance of considering
the temporal scale of generation times should not be
neglected. Thus, our results may be specific to the noise
structure we have applied and they may best represent
short-term responses to environmental variation. As
concluded by Schwager et al. (2006), model structure
may also have profound effects on the relationship
between noise reddening and extinction risk. We used a
population model very similar to that of, e.g. Petchey
et al. (1997) and our results are in agreement with their
findings. It is thus unlikely that our results are due to
specific model assumptions.
In conclusion, population growth rate, environmental autocorrelation and abundance of constituent species
interact in subtle ways in influencing extinction risk in
competitive communities. The relative influence of
noise colour and species abundance on population
extinction probability depends critically on underlying
population dynamics. Moreover, the exact nature of
this relationship is not solely dependent on the
population growth rate, but also on the similarity of
species responses to the environmental fluctuation. The
abundance of an affected species is also likely to
influence the extent to which it responds to variation
in the pace of environmental fluctuation.
Acknowledgements ! We thank Veijo Kaitala, Jouni Laakso
and Antti Halkka for their illustrating comments on the
manuscript. We also thank Owen Petchey for helpful
comments on an earlier version of this manuscript.
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