Spectral color, synchrony and extinction risk
1.
Introduction
In conservation biology, and ecology in general, it is important to correctly estimate and understand the causes of
increased risk of extinction risk, and fluctuation over time is an essential factor for determining the persistence of a
population (Inchausti and Halley 2003). The mix between variance and mean has marked impact on the extinction
risk and is often measured as the coefficient of variation. Still, fluctuations over time have other properties that may
also influence the probability of extinction, and an example of this is autocorrelation, which is often called colored
noise. Several investigations of natural, abiotic, and biotic time series have shown positive autocorrelated variation
referred to as red noise (Steele 1985; Pimm and Redfearn 1988; Halley 1996; Inchausti and Halley 2002; Vasseur
and Yodzis 2004). The impact of red noise on population dynamics and extinction risk has been a matter of debate
for several years. Over the same period, theoretical ecology has also been concerned with the spatial dimension and
its effect on population dynamics, an interest that originated with introduction of the metapopulation concept
(Hanski 1998). The aspects of noise and spatial dimension have been combined in some studies of extinction risk,
but those efforts have not been able to provide general and conclusive results. In the present study, we combined the
two in a set of models of single species dynamics in a subdivided population. To reduce the risk of
misinterpretations and erroneous results, we used a novel method to jointly handle variance, noise color, and the
spatial entities as patches.
The autocorrelation in time series is usually measured using a spectral representation obtained by applying Fourier
transform. The spectral representation of a random time series has an equal mix of all inherent frequencies and
therefore, analogous to white light, it is termed white noise. A dominance of low frequencies is denoted red noise,
which has positive autocorrelation. A dominance of high frequencies, called blue noise, is thus negatively
autocorrelated. According to Halley (1996), red noise is generally the better null model for ecological time series,
because such natural series generally show positive autocorrelation (Steele 1985; Pimm and Redfearn 1988; Pimm
1991; Halley 1996; Inchausti and Halley 2002; Vasseur and Yodzis, 2004). In theoretical studies one has either used
autoregressive methods (AR) to generate colored time series of environmental noise or spectral methods for strictly
1/fγ environmental noise. The methodology introduced in this study generates time series by such spectral methods.
The γ value measures the color by being the slope of log(amplitude) versus log(frequency).
Studies have shown that red noise has both positive (Johst and Wissel 1997; Petchey et al. 1997) and negative
(Roughgarden 1975; Ripa and Lundberg 1996) effects on persistence time or extinction risk, and inconclusive
results have been reported by Mode and Jacobson (1987), Petchey et al. (1997), Heino (1998), and Cuddington and
Yodzis (1999). One explanation for this discrepancy concerns how variance is scaled to make white and red noise
comparable. Heino et al. (2000) observed opposite effects simply by changing the scaling procedure. It should be
mentioned that this scaling problem occurs primarily when using autoregressive (AR) noise methods. The model
used to represent population dynamics can also alter the effect of environmental noise (Ruokolainen et al. 2009).
Heino et al. (2000) and Petchey et al. (1997) have shown that the representation of density-dependent dynamics (i.e.,
over- versus under-compensation) can have marked effects on the extinction rates, which in turn, can be explained
by the extinction path. Schwager et al. (2006) concluded that the extinction risk decreases with increasing
autocorrelation, if the population is sensitive, which can be interpreted as being over-compensative; in other cases
the extinction risk should increase. In general, noise color should have the same effect at the extreme end of high γ
(i.e., there should be extremely high autocorrelation) regardless of the inherent dynamics. A time series with very
high γ can be considered to be constant over a finite time interval, and hence any stable dynamics (over or under
compensatory) will eventually become the same, exemplified by the high autocorrelation observed by Cuddington
and Yodzis (1999). Yet the paths to this extreme end may differ, and some other behavior might be expected at the
1
other end, namely, white noise, since such noise may interact with over- and under-compensatory dynamics in fairly
different manners. Also, the way noise is incorporated into the model can have significant effects on the population
dynamics, as recently reported by Mutshinda and O’Hara (2010). Whether environmental noise should be
introduced to affect the carrying capacity or the growth rate, or both, depends to some extent on the type of
ecological system in focus (Roughgarden 1979). In their review Ruokolainen et al. (2009) conclude that the
population dynamics is important and that reddening of noise decreases (increases) extinction risk for overcompensatory (under-compensatory) dynamics for the one-patch single-species population. For the undercompensatory dynamics there is actually a humped shaped response in the case noise is affecting carrying capacity
K, which is not discussed by Ruokolainen et al. (2009) but can be seen in for example Cuddington and Yodzis
(1999) and Schwager et al. (2006). In present study, we investigated both over- and under-compensatory models,
and we introduced the noise as a resource fluctuation by varying the carrying capacity or as a noise affecting the
growth rates.
When adding space and patches to a system with temporal fluctuations, one is directly faced with the problem of
correlation (synchrony) between patches. The environmental variation within a patch can be completely
uncorrelated or completely correlated with the variation in other patches, or somewhere in between. Here, we use
the term synchrony to describe this correlation, and our study was focused on synchronized environmental noise and
how this affect the extinction risk of populations. Compare with the Moran effect (Moran 1953; Bjørnstad et al.
1999) on how environmental noise synchronizes population dynamics and for other mechanisms and topics related
to synchrony, see Greenman and Benton (2001); Engen et al. (2002); Liebhold et al. (2004). Adding space is not a
simple task, because it is not easy to assess the degree of synchrony, which should nonetheless be chosen and tested
if space is introduced to a stochastic spatial system. Surprisingly, this main feature of subdivided populations has not
been a key issue in theoretical ecology. Complete synchrony of patches will have an effect that is quite straightforward, because it will behave as a single population system. On the other hand, a completely unsynchronized
system will have dynamics involving a high probability of rescue and will therefore reduce the extinction risks.
Between these two extremes, a gradual change in the extinction risk should be expected (Heino et al. 1997;
Palmqvist and Lundberg 1998; Amritkar and Rangarajan 2006; Ruokolainen and Fowler 2008), although it is
unclear whether this gradual transformation will interact with other features of the system, such as population
dynamics and noise color.
Since synchrony and noise color are coupled to each other, there is a risk of introducing large variability in noise
color and, for example, unintentionally changing the color when setting the synchrony and vice versa (Vasseur
2007). Ruokolainen and Fowler (2008) have demonstrated a solution for AR methods that offered good control over
generated values of both dimensions, but it was applied to a set of species (a food web), not a set of patches. Vasseur
has demonstrated a solution also for 1/f methods, but it was applied to a two-patch system and is hard to expand to a
larger system of more than two patches. Consequently, studies such as ours are needed that take these features into
account in a one-species setting, as well as in space, to enable in-depth investigation of the mechanisms and also to
clarify the effects of over- and under-compensatory dynamics. Most of the research in this area has used methods
that are extensions of the AR model first described by Ripa and Lundberg (1996). Considering the non-obvious
errors such as rescaling variance or whitening red series that have occurred in some of the studies following Ripa
and Lundberg (1996), we have included a 1/f analysis of the generated time series to confirm that our study do not
repeat previous mistakes.
Our aim was to present a novel method of generating 1/f noise with specified values of variance and correlations
over both time (noise color) and space (synchrony). Previously, this has only been satisfactorily achieved for AR(1)
noise in a multi-species setting. The prime goal, achieved by the novel method, was to conduct a more complete and
correct analysis of the effects exerted on extinction risk by red-shifted environmental noise in spatially structured
populations with local dynamics. Thus we performed a more correct analysis of the same dynamics used in earlier
2
studier (over-compensatory Ricker dynamics and noise entering K) and filled in some missing gaps for the spatially
structured population case (under-compensatory dynamics and noise affecting growth rate).
2.
Methods
2.1.
A novel method for generating noise in two dimensions
With our method, time series of patches env(i,t), are generated as 1/|f|γ -noise using a two-dimensional spectral
synthesis approximation obtained by fast Fourier transform (FFT) and applied according to a technique similar to
that employed by Halley et al. (2004). One of the dimensions corresponds to time and the other to space. For more
information about how FFT is used to generate two dimensional “noise-landscapes” in general, see the appendix in a
paper published by Keitt (2000). The 1/|f|γ -noise obeys a power law where the power of the amplitudes A(f)
increases with decreasing frequency f at a rate determined by the spectral exponent γ, i.e., the noise color. Thus the
spectral exponent γ is the slope of the curve in the log(amplitudes) versus log(frequencies) plot, but not including
any zero frequency.
A( f )  1 / f 
f 0
(Eq. 1)
In the two-dimensional version, this curve/slope has the shape of a cone (Fig. 1). It should be noted that when time
series and surfaces are represented by sine functions, it is only the amplitudes of the sine functions at non-zero
frequencies that determine the variance. Hence, the variance of the time series can be assessed by multiplying all
these frequencies by a parameter β, as shown in Eq. 2. The Fourier transform represents the time series env(i,t) by
the amplitudes A(fi , ft), where fi and ft are the frequencies of sine functions in patch and time dimensions,
respectively.
 1
var env(i, t )     2
M

 A( f
ft
i
, f t )2 
fi

1
1
1
A(0,0) 2   2 A(0, f i ) 2   2 A( f t ,0) 2 
2

M
fi M
ft M

(Eq. 2)
M2 is the number of grid points in the two-dimensional space. The overall mean is represented by the amplitude in
origin A(0,0). Consequently, the mean of the time series of each of the patches,
i
env t (i ) , is then represented by the
i
amplitudes along the axis of f , A(0, f ), and the variance of these means is the sum of the amplitudes along this axis.


var env t (i)  
fi
1
A(0, f i ) 2
2
M
The mean over patches at any specific time,
(Eq. 3)
env i (t ) , is represented along the axis of ft , A(ft, 0), and the variance
of these means is the sum of the amplitudes along this axis.


var env i (t )  
ft
1
A( f t ,0) 2
2
M
(Eq. 4)
Adjusting the means along any of these axes by multiplying all amplitudes by a constant α will not alter the overall
variance, the mean, or the slope of the 1/f cone.
3


var env i (t )   
ft
1
A( f t ,0) 2
2
M
(Eq. 5)
These three representations (Eq. 2-4) can then be used to adjust means and variances, and this can be done
independently of the spectral exponent, which is the slope of the cone. Furthermore, we can specifically adjust the
means along specific dimensions (Eq. 3 and 4) to generate specific synchrony between patches. Considering the
variance of means along the time axis (generated by a large α in Eq. 5), it becomes evident that a variance as large as
that of a single time series implies a perfectly correlated set of time series (Fig. 1, right panels), whereas a small
variance of means (generated by a small α in Eq. 5) indicates an almost uncorrelated set of time series (Fig. 1, left
panels).
In runs of our method, we used pair-wise cross-correlation (Bjørnstad et al. 1999) as the measure of synchrony, ρ,
between patches. There are no simple relationships between the spectral representation and the synchrony measured
as pair-wise cross-correlation, and since no analytical relationships could be used, we applied numerical method by
first generating a large dataset from which we could determine what values of β were needed for different noise
colors γ to achieve a specific synchrony ρ at a given variance. The dataset D then consisted of 30 replicates of
env(i,t) generated for γ ranging from 0 to 2.2 and α ranging from 1 to 96. The environmental noise color (γ) and
synchrony (ρ) were measured for each replicate, and this dataset was used for interpolation when setting the
parameter value α for actual runs to achieve the expected ρ. We also measured the color of environmental noise in
all runs to ensure correct values. To reduce the variance in noise color between replicates, we used the method of
random phase-shifts between replicates (Vasseur [2007] and the general transform theory) instead of adjusting an
initial white noise dataset. We also checked for linearity in the power spectrum to ensure a 1/|f|γ relationships
between amplitude and frequency.
Given the dataset, the complete procedure we used can be summarized as follows:
a)
Data for amplitudes of the frequencies A(fx,fy) are generated for the spectral representation of the time
series, and the “cone” is formed (see above). The slope of the cone is 2γ (γ is the spectral exponent).
b) The values (of amplitudes) along one axis of frequencies A(0,fy) are multiplied by a factor α, which
determines the synchrony. This value is set by interpolation from a given dataset D.
c) All values (of amplitudes), except along the axis A(0,fy) and A(fx,0), are multiplied by a factor β, which
determines the variance of all local time series.
d) The value in the origin is set to adjust for the mean of all local time series.
Now the cone of amplitudes is adjusted in “height/intercept” position with β (variance over time), the slope is 2γ
(color), and the cone has a valley or a peak along one axis according to α (synchrony).
Random phase shifts (uniform over the interval [0,2π]) are applied to the frequencies. The phase shift is the
random component which generates replicates.
f) The data set with uniform random phase shift is inversely Fourier transformed to the set of real time series
(see the general transform theory presented by Cuddington and Yodzis 2004 and Vasseur 2007).
g) A number of time series (n) are randomly chosen.
h) Each time series is analyzed for noise color, variance, and mean to ensure correct methodology.
i) The synchrony between time series is measured.
e)
2.2.
Applying the novel noise method in a spatially implicit model
4
We used the well-known Ricker model to describe population dynamics. In line with Petchey et al. (1997), we
changed from over- to under-compensatory (i.e., from oscillatory to monotone) dynamics by the parameter b:
N i ,t 1  N i ,t e
r (1( Ni ,t / Ki )b )
(Eq. 6)
where Ni,t is population density i at time t, r is per capita rate of increase, and K is the carrying capacity. The range
of b is from 0 to 1, and b = 1 and b = 0.1 indicate over- and under-compensation respectively. The model described a
spatially subdivided population: a metapopulation consisting of a number of local subpopulations. We modeled
landscape implicitly and dispersal as simply as possible. Dispersal was a density-independent mass-action mixing
process (i.e., regional dispersal), and all subpopulations were therefore connected equally. Dispersal occurred first,
then reproduction, and finally census:
n


  n

N i ,t 1   N i ,t 1   d ji     d ij N j ,t  e


j i

  j i


 

 
r  1  N i , t  1


 

 


d ji  






d ij N j , t  / K t 




b





d ji  d ij  d i /( n  1)
(Eq. 7)
(Eq. 8)
where di is the overall per capita dispersal rate, and n is the number of local subpopulations.
Environmental variation was a 1/|f|γ -noise with spectral exponents ranging from 0.2 to 1.2 (i.e., so-called ‘pink’
noise; Halley 1996). From a two-dimensional noise “landscape” (Section 2.1), we picked time series for each local
(sub-) population. Therefore, environmental noise (ε) entered locally, either in carrying capacity K (as indicated by
Ruokolainen and Fowler 2008) or in growth rate r (as described by Kaitala et al. 1997):
N t 1  N t e r (1( Nt / K (1 t ))
b
)
(Eq. 9)
N t 1  N t e r (1( Nt / K ) ) (1   t )
b
(Eq. 10)
In total, we simulated three model cases: (i) noise entering K (Eq. 4) and over-compensatory dynamics, (ii) noise
entering K (Eq. 4) and under-compensatory dynamics, and (iii) noise affecting growth rate (Eq. 5) and overcompensatory dynamics. The magnitude (or variance) of the noise was scaled to achieve the same population
density variance in the three model cases (i-iii), although the magnitude of the noise was kept constant for each case.
The population equation (Eq. 1) was formulated in such a way that the impact of K on density regulation decreased
when parameter b was decreased to obtain under-compensatory dynamics. Consequently, the variance of the
environmental variation had to be larger in that case to acquire the same impact of environmental variation as in the
over-compensatory dynamics. Also, in model case iii, the noise affecting the growth rate has to be larger to obtain
the same population variance.
We used a local perspective when setting parameter values, and the mean carrying capacity of local habitats was not
changed according to number of subpopulations. Consequently, the total sum of individuals on a regional scale
increased with increasing numbers of subpopulations. On the other hand, conditions at a local scale were kept
constant and independent of the number of subpopulations. This agrees with having constant mean and variance of
noise at the local scale, independent of noise color and synchrony.
We used three different values of the carrying capacity (K): 20, 50, and 100. Initial population density for each
subpopulation N0i was a Gaussian random number with mean 0.5K and standard deviation 0.125K. The local
5
dispersal rate di was a Gaussian random number set at the start of each simulation with mean dm and standard
deviation 0.25dm. Values of dm were 0.1, 0.2, or 0.3. Population sizes below one were set to zero. Time series length
was 1000, and the number of replicates was 100. The number of subpopulations n was 5, 10, 50, or 100. Inasmuch
as density regulation dynamics set by parameter b (Eq. 1) can also be influenced by the growth rate r, we kept r
constant throughout the complete simulation. We chose a low value, r = 1.5, to generate stable population dynamics.
For each time step, the regional population size was calculated as the sum of all subpopulations. Regional population
extinction occurred when N was equal to zero for all local populations. Extinction risk was calculated as the
proportion of extinct regional populations out of all replicates. Furthermore, we also calculated mean population
density (i.e., the mean of regional density over the time period) for all replicates, and we checked the mean
population variance in the same manner. The coefficient of variation (CV) was calculated as standard deviation
divided by density mean, and the mean values for all replicates are presented in the figures.
3.
Results
Initially, we arrived at the results summarized in Fig. 2 of model case 1, which were derived from the ordinary
Ricker equation (i.e., b = 1) with more or less over-compensating density dependence and noise entering the
carrying capacity K. As can be seen, increased spectral color (i.e., higher reddening) led to greater density that
approached the overall mean carrying capacity (Kmean* number of patches). The increase in mean density, together
with decreased variance with increasing spectral color, resulted in a small drop in stability (CV) and a lower
extinction risk. An increase in synchrony had the opposite effect: decreased density and increased variance led to an
increased CV and a higher extinction risk. For medium to low values of synchrony (ρ < 0.8), there was no extinction
risk at all. Combinations of parameters that led to low extinction risk (e.g., low Kmean, few subpopulations, and low
degree of synchrony) also reduced the effects of spectral color (i.e., the otherwise steep curve flattened; Fig. 2).
The same system comprising noise entering K but with under-compensatory dynamics (model case ii) resulted in a
lower extinction risk due to a strong reduction in the variance of population density (and dynamics with no crashes).
To facilitate comparison of the over- and under-compensatory dynamics, we increased environmental variance by a
factor of 2.0 for the under-compensatory Ricker model, and the results are presented in Fig. 3. Increased spectral
color led to an increase in mean population density, as in the case described above (model case i). On the other hand,
there was also an increase in population variance. Moreover, both CV and extinction risk reached their maxima in
relation to intermediate spectral color values, which means that the rather low mean densities at these γ values could
not compensate for the increasing variances, which in turn increased the extinction risk. Also, in agreement with the
model case i, the effects of increased synchrony led to decreases in density, variance, CV, and extinction risk. The
most synchronous cases (dotted curves in Fig. 3) did not follow this general pattern, and this can be explained by
extinctions that were too numerous and too rapid, resulting in time series that were too short to provide valid results
(lengths of time series down to 5% of the maximum simulated time).
Over compensatory dynamics together with noise entering the growth rates also resulted in low variance in
population density, and thus in this case (iii) we multiplied the variance of the environmental noise by a factor of
1.3. The results are summarized in Fig. 4. As in the case with under-compensatory dynamics (ii), there was a
maximum in extinction risk for intermediate values of γ. The effects of increased γ on population variance and CV
were the same as in the first model case (i), seen as decreasing values with increasing γ. For all four output
measurements, the effects of synchrony were the same as in both the other cases (i and ii).
Increased dispersal rate (from 0.1 to 0.3), increased value of carrying capacity (from 20 and 50 to 100), and
increased number of subpopulations/patches (from 5, 10, and 50 to 100) all resulted in decreased extinction risk but
no qualitative differences (these specific results are not presented in any figure). Table 1 summarizes the effects of
noise color on the various output parameters, i.e., mean population density, density variance, CV, and extinction
risk. The arrows in the table indicate increases or decreases in output values upon increase in the spectral color (i.e.,
6
moving from almost white noise [γ = 0.2] to red noise) in the three model cases (over- or under-compensatory
dynamics and noise in K or r).
4. Discussion
Colored noise such as variation in resources or growth rates is a persistent component of the population dynamics of
natural communities (e.g., Steele 1985). Another component is the spatial aspect seen either as a set of patches
(Gilpin and Hanski 1991; Kareiva and Wennergren 1995) or as a continuous landscape (Driscoll 2005). In this
study, we introduced a novel method that combines these two components (time and space), and we also conducted
a detailed analysis of the effects of colored noise and synchrony. The most notable finding is that the influence of
colored noise is straight-forward and easy to interpret. Both variance and means of densities have very simple
relationships with the reddening of noise, regardless of inherent dynamics or synchrony between patches. The effect
of noise color on means and variances are monotonic, yet the dynamics and whether the noise enters in K or in r
determines whether the means and variances will decrease or increase with reddened noise (see table 1). When a
population experiences noise, it can do one of the following: (1) track the noise almost exactly, because the
autocorrelation of the noise is so high that the mean and the variance of the population are directly related to the
noise; (2) track the noise even if it is over-compensating or has a delayed response and thus leading to variance in
population densities that is larger than the variance directly related to noise, which will also reduce the mean; (3) be
slow in reacting and not track the noise completely, thereby resulting in less variance than that directly related to
actual noise, although the slow reactivity will also reduce the mean. Whether a population has over- or undercompensatory dynamics and whether the noise acts on the growth rate or on the resources (the r or the K) defines
which color of noise will be associated with any of these three phenomena. We also found that the stability measure
CV cannot capture the whole picture. To some extent, CV does go hand in hand with the extinction risk, but, to fully
understand the conditions, it is also necessary to include knowledge about the mean and the variance of the
population density. Degree of synchrony is most important for predicting extinction risk, but it will not change the
general trends of the effects of noise color or vice versa. Increasing the mean value of carrying capacity along with
an increased number of subpopulations will preserve the qualitative results, despite the quantitative consequences
resulting in reduced extinction risks.
A population that has over-compensatory dynamics and noise affecting carrying capacity (model case i) will never
experience the slow-reactive phase. Such dynamics move from increased variance and reduced mean during white
noise to an almost exact tracking of red noise, which results in a continuously reduced risk of extinction during
increased reddening of the noise (Fig. 2). The under-compensatory dynamics and noise in carrying capacity (model
case ii) follow a different path that involves slow reaction to white noise. Hence, as the noise becomes redder, the
dynamics move from reduced variance to the variance of exact tracking of noise. The mean will also increase with
the reddening of noise, which will in turn reduce the risk of extinction; in contrast, the increase in variance will raise
the extinction risk. These two counteracting forces will result in an initial increase to a maximum extinction risk
when γ is close to 0.7, and from there the mean density will reach such values that the risk of extinction decreases
with further reddening of noise (Fig. 3). Other parameters in the system (e.g., dispersal and growth rates) will
determine the exact shape of these two patterns (i.e., the over-compensatory dynamics leading to continuously
reduced risk and the under-compensatory dynamics causing a maximum risk at medium red noise), and yet the
qualitative results may hold given that the system is kept outside inherent oscillatory or chaotic regimes.
Extinction risk of model case iii, noise affecting the growth rate in an over-compensatory dynamics; show a similar
type of relationship with colored noise as of model case ii, noise impacting the carrying capacity in an undercompensatory dynamics. This is somewhat surprising but both cases have two counteracting forces, population
means and variances, resulting in raised extinction risk for intermediate noise color. Although, in model case iii
population means (most pronounced when larger degree of synchrony) and variances decrease with reddening noise
color (they both increase in model case ii). This pattern can be explained by following reasoning. Over a specified
time interval the variance of the noise decreases with reddening of noise. Density regulation then has a chance to
7
keep the population density closer to the carrying capacity. It then follows that reddening of noise causes a decrease
in population density variance. As the variance decreases both the high and low density extremes are reduced. Thus,
the otherwise asymmetric distribution around the mean caused by truncation of negative density values will be more
symmetric in the red noise environment due to the smaller variance, which in its turn can be seen as a small
reduction of population density mean with reddening of noise color. This reasoning is confirmed by measuring skew
of density distribution around its mean; a positive value indicates a skew distribution (e.g., caused by the fact that
the negative density values has been truncated) and a zero measure indicates a symmetric distribution. For the most
synchronized population the skewness value is 3.5 for white noise and -0.5 for red noise, leading to a smaller density
mean value for the latter (Fig. 4). When having a low degree of synchrony the distribution tends to a more
symmetric one, because of less need of truncations and thus a smaller (or none) difference in density mean between
white and red noise environment (Fig. 4). This is probably dependent on the value of K and other parameters
affecting the density mean so that if there is a symmetric distribution already for white noise there will be no
difference in density mean between white and red noise. So to sum up: the redder the noise, the smaller the variance
and the mean, most pronounced for a synchronized population. In general, noise affecting r or K reflects different
kinds of population responses to variations in their environment. The influence of noise is exerted mainly on the
growth rate in some cases and chiefly on the carrying capacity at other time, but noise can also have a major impact
on both those aspects (Roughgarden 1979). Therefore, the next step might be to introduce noise in both r and K, for
example, to ascertain whether r in combination with under-compensatory dynamics can generate even more
pronounced maxima in extinction risk.
From the discussion above, it can be concluded that reddening of noise does not necessarily lead to a reduction in
extinction risk. Certainly, it will be reduced for over-compensatory dynamics with noise in carrying capacity as
reported in previous studies (Ripa and Lundberg 1996; Petchey et al. 1997; Cuddington and Yodzis 1999). In the
other two cases it will depend on the initial color of noise; the risk increases when the initial color is white, whereas
it decreases if the initial color is medium red. This hump-shaped response in extinction risk for under-compensatory
dynamics is not previously explained, although it appears in, for example, the studies of Cuddington and Yodzis
(1999 [Fig. 2a]) and Schwager et al. 2006 [Fig. 3a]). The hump-shaped response in the case noise affects the growth
rate (Fig. 4d) is not reported earlier. A closer look reveals that the mean density for dynamics where noise enters in
K is lowest in white noise. Clearly, a population with low densities is more fragile and sensitive to other stochastic
occurrences, for example catastrophes, demographic stochasticity, and other less predictable events that were not
included in our stochastic process. Given this, we argue from a more general ecological perspective that, when
resources are scarce and the mean density becomes severely reduced, white noise entering in K will generate larger
extinction risk than those reported in our study. Red noise will not necessarily result in lower densities, but it will
cluster them. Accordingly, a period like this will be more susceptible to catastrophic events and demographic
stochasticity, since several events may occur within a prolonged period of low densities. From a theoretical
standpoint, there is a threshold where these rare events may interact with periods of low density. Hence there is also
a general ecological perspective at the red end of the spectra; in red noise and low densities one may expect higher
extinction risk than those reported in our study and also notably that there is an even larger change than of those in
white noise given the same mean density.
Introducing the spatial dimension together with the temporal dimension entails fairly complex methodology, which
nonetheless provides results that are easy to interpret. Earlier studies have shown that regional extinction risk is
coupled to the degree of synchrony (Heino et al. 1997; Palmqvist and Lundberg 1998; Amritkar and Rangarajan
2006; Ruokolainen and Fowler 2008). Our findings demonstrate that the synchrony of noise between patches has a
major effect on extinction risks, and a system with patches of unsynchronized noise will have extremely low
extinction risks. Hence, the rescue effect of other patches is evident, regardless of the color of the noise. There is no
complex interaction between color and synchrony, other than synchrony being such a strong force that the effect of
color is almost eradicated in unsynchronized systems. This does not agree with previous studies showing that the
effects of noise color differ markedly between homogeneous and heterogeneous landscapes (i.e., synchronized and
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unsynchronized patches; see review by Ruokolainen et al. 2009). However, it is highly likely that the explanation for
this discrepancy is that the earlier studies could not achieve the same results due to incorrect introduction of
synchrony into the one-dimensional noise method. For example, in Petchey et al. (1997) the more heterogeneous
(unsynchronized) the more white noise is added to the initial color, i.e., the most reddened noise is not that red as
denoted in the unsynchronized system. From our study, we conclude that synchrony between patches is much more
important than color as a component of noise, and it does not change the qualitative effects of noise color.
We used a mass-action mixing model that is not spatially explicit and involves no distance dependence in either
dispersal or synchrony. Relaxing those assumptions will change the effect of synchrony, whereas the overall results
will probably be the same. The more synchrony there is, the higher the extinction risk will be, because it will make
the system more similar to a single-population system and thus there will be no rescue effect. We used a system
comprising 100 subpopulations, which can be regarded as a cluster of patches with fairly homogeneous
environment; this implies synchronized noise, or it may represent more heterogeneously distributed patches that are
located within a larger area and are thus experiencing a less synchronized environment. To obtain a clear
understanding of the results, we added an implicit landscape in this first step, which in turn implies that we had to
start with an extremely simplified dispersal rule: the mass-action mixing process. From an empirical point of view,
this may be a special case scenario. However, from a theoretical perspective, we need some sort of base-line to
understand the more complex system, and we argue that mass-action mixing can serve that purpose when
investigating general effects of dispersal per se, without including confounding effects of aspects such as distance
dependence and aggregation patterns. Our approach can be further developed towards this type of more specific
spatiotemporal systems with heterogeneous landscape configurations and dispersal kernels. The next level of our
research regarding the effect of noise on extinction risks may include a study of the relationship between spatial
patterns (e.g., aggregations) and the distance dependence of synchrony and dispersal (Lindström et al. 2008). An
investigation of that type constitutes a larger methodological challenge, but it will allow us to further expand our
two-dimensional spectral representation by adding higher dimensions.
The results of the current study indicate that, despite the decrease in risk of extinction with reddening of noise, it is
more important to determine correlation–synchrony–of noise between patches than to ascertain the color of the
noise. From an empirical standpoint it may be necessary to consider how to distribute the finite amount of measures.
A long time series in a single patch will assess the color more correctly, whereas shorter time series in a set of
patches may reveal the synchrony of noise. Pondering the question of whether it is better to use red noise instead of
white when studying population dynamics, we can conclude that it is more import to use correct synchrony for a
spatially subdivided setting. If there is actually a substantial extinction risk, it may be under estimated if overlay red
noise is assumed, and yet the error becomes much larger if we assume an excessively high degree of synchrony.
Incorrect use of white noise will instead over-estimate the extinction risk. This will be the case for overcompensatory population dynamics and noise affecting carrying capacity. For under-compensatory dynamics,
incorrect use of both white and red noise used will result in under-estimations of the extinction risk, if the true noise
is somewhere in between. When noise is affecting the growth rate, the color is not that important and cannot lead to
inferior predictions. Notwithstanding, it is necessary to be aware that in a situation where red noise is lowering the
extinction risk by increasing the mean over the time period, the population will be concomitantly exposed to fewer
but longer periods of poor conditions. Hence, when population densities are generally low, catastrophic events or
demographic stochasticity will have an important impact on the extinction risk, especially if there is synchronized
red environmental noise.
Acknowledgements
We thank W. Fagan for comments on an earlier version of the manuscript.
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Figure legends
Fig. 1 Illustration of a two-dimensional spectral method for generating noise with both temporal (spectral color) and
spatial (synchrony) properties. In this example, noise color is red, γ = 1.2 (slope of cone) (top), while degree of
synchrony is changed from low, ρ = 0.1 (left), to high, ρ = 0.9 (right). Top panels: Amplitudes and frequencies are
generated for the spectral representation of the time series and are adjusted for specified noise color, synchrony and
variance. Log(amplitudes) are plotted against log(frequencies). Middle panels: The dataset is inversely transformed,
creating a noise “landscape”. Bottom panels: Times series for (ten) local populations. Note: the exact values on the
axes are not important given the illustrative purpose.
Fig. 2 Diagrams considering the degree of synchrony among subpopulations and showing the effects of spectral
color of environmental noise on the following: mean population density (A), variance of population density (B),
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coefficient of variation (C), and extinction risk calculated as proportion of extinct replicates out of 100 at time T =
1000 (D). Results are shown for Ricker population dynamics (i.e., oscillatory dynamics) with r = 1.5, number of
subpopulations = 10, Kmean = 100, and dispersal = 0.1 (mass-action mixing). Environmental noise was generated by
a two-dimensional FFT method and entered the model in K.
Fig. 3 Diagrams illustrating different degrees of synchrony among subpopulations and showing the effects of
spectral color of environmental noise on the following: mean population density (A), variance of population density
(B), coefficient of variation (C), and extinction risk calculated as proportion of extinct replicates out of 100 at time T
= 1000 (D). Results are shown for Ricker population dynamics with density dependence corrected for undercompensation (i.e., monotone dynamics), with r = 1.5, number of subpopulations = 10, Kmean = 100, and dispersal =
0.1 (mass-action mixing). Environmental noise was generated by a two-dimensional FFT method and entered the
model in K. Here, magnitude of noise was multiplied by 2.0 to obtain variances in population density of the same
magnitude as of oscillatory dynamics (Fig. 2).
Fig. 4 Diagrams illustrating different degrees of synchrony among subpopulations and showing the effects of
spectral color of environmental noise on the following: mean population density (A), variance of population density
(B), coefficient of variation (C), and extinction risk calculated as proportion of extinct replicates out of 100 at time T
= 1000 (D). Results are shown for ordinary Ricker population dynamics (i.e., monotone dynamics), with r = 1.5,
number of subpopulations = 10, K = 100, and dispersal = 0.1 (mass-action mixing). Environmental noise was
generated by a two-dimensional FFT method and entered the model in the growth rate r. Here, magnitude of noise
was multiplied by 1.3 to obtain variances in population density of the same magnitude as of oscillatory dynamics
(Fig. 2).
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