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 (e.g., 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. 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 (see Hanski 1998
for a review). 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 in theoretical studies 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, in analogy to white light, it is termed white noise. A dominance of low frequencies is
denoted red noise, which has positive autocorrelation. According to Halley (1996), red noise is generally a
reasonable 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 people 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 of one-patch single species populations have shown that red noise has
positive effects on persistence time or extinction risk when population dynamics is over-compensatory (Petchey et
al. 1996; Ripa and Lundberg 1996; Cuddington and Yodzis 1999). When population dynamics is undercompensatory the result is altered in most studies (e.g. Petchey et al. 1996; Cuddington and Yodzis 1999) but not in
all (Heino et al. 2000). The under-compensatory dynamics actually has a hump-shaped response which is not
discussed but can be seen in for example Cuddington and Yodzis (1999) for 1/f noise. Schwager et al. (2006) show
the same result for AR-noise and discuss that the hump-shaped response depends on the minimum value that is
allowed for the carrying capacity. We argue that there is a hump but that this is not always visible in the studied
spectral region (especially for AR-noise that has a limited range compared to 1/f-noise). Whether previous studies
show the hump, show only the left part of the hump (i.e., increasing extinction risk with increasing spectral value) or
the right part (i.e., decreasing extinction risk) is due to different model set-ups (e.g., scaling of variance of red noise
Heino et al. 2000 and how noise is incorporated into the model Mutshinda and O’Hara 2010) and different values of
inherent model parameters (e.g., the growth rate and the spectral color are interact and hence effect the position of
the hump, Johst and Wissel 1997; Heino 1998). In the present study we applied a large enough spectral region when
investigating 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 realized growth rates.
1
Adding space is not a simple task, because it is not easy to assess the degree of synchrony/correlation between local
populations, which should nonetheless be chosen and tested if space is introduced to a stochastic spatial system.
Complete synchrony of patches will have an effect that is quite straight-forward, because the populations will
behave as a single population system. On the other hand, a completely unsynchronized system will have dynamics
involving a high probability of rescue effects 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. Note
that we did not study how synchrony between populations may occur as a result of dispersal (referens) or by the
environmental spatiotemporal noise (Liu et al 2009) even though we are aware of the mechanism (referens). Instead,
we focused on how synchrony of environmental noise affect the extinction risk of populations.
Any environmental noise can be characterized both by its synchrony and its noise color. Hence, the two are coupled
to each other and when generating environmental noise 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 noise
and synchrony of generated values, but it was applied to a set of species (a food web), not a set of patches. Vasseur
(2007) has demonstrated a solution also for 1/f methods, applied to a two-patch system but it is hard to expand to a
larger system of more than two patches. Yet, 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 an 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 instead of space. 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 studies (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
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 excluding zero frequencies.
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
2
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
fi
i
, f t )2 

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
the origin A(0,0). Consequently, the mean of the time series of each of the patches, env t (i ) , is then represented by
the amplitudes along the axis of f i, A(f i ,0), and the variance of these means is the sum of the amplitudes along this
axis.


var envt (i)  
fi
1
A( f i ,0) 2
2
M
(Eq. 3)
The mean over patches at any specific time, env i (t ) , is represented along the axis of ft , A(0,ft), and the variance of
these means is the sum of the amplitudes along this axis.


var envi (t )  
ft
1
A(0, f t ) 2
M2
(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.


var envi (t )   
ft
1
A(0, f t ) 2
M2
(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
3
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:
Data for amplitudes of the frequencies A(fi , ft) 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, ft)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, ft) and A(fi ,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.
a)
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) The time series used for simulations are randomly chosen from the set of series (see step f).
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)
The code for how to generate the noise is presented in Appendix A.
2.2.
Applying the novel noise method in a spatially implicit model
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 changing the parameter
b:
Ni ,t 1  Ni ,t e
r (1( Ni ,t / Ki ,t )b )
(Eq. 6)
where Ni,t is population density of patch 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., global dispersal) and all subpopulations were therefore equally connected. Dispersal occurred
first, then reproduction, and finally census:
n
 
  n

N i ,t 1   N i ,t 1   d ji     dij N j ,t  e


j i
  j i

 
 

 
r  1  Ni ,t  1

  

 


d ji 






dij N j ,t  / Ki ,t 




b





(Eq. 7)
where dij and dji denote dispersal rate between patches. Since we apply mass-action mixing they are calculated as
dij/(n-1) where dij and dji , respectively, are random numbers from the same distribution.
4
Environmental variation was a 1/|f|γ -noise with spectral exponents ranging from 0.2 to 1.2. From a twodimensional 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 affecting realized growth rate (as described by Kaitala et al. 1997). :
N t 1  N t e r (1( Nt / K (1 t ))
b
)
(Eq.8)
N t 1  N t e r (1( Nt / K ) ) (1   t )
b
(Eq. 9)
In total, we simulated three model cases: (i) noise entering K (Eq. 8) and over-compensatory dynamics, (ii) noise
entering K (Eq. 8) and under-compensatory dynamics, and (iii) noise affecting growth rate (Eq. 9) 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), but of course the magnitude of the noise was kept constant for all
simulations within each case. The equation of population dynamics (Eq. 6) 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 global 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 Ni ,0 was a Gaussian random number with mean K/2 and standard deviation K/8. The local dispersal
rates dji and dij were random numbers from a Gaussian distribution with mean dm and standard deviation dm/4, set at
the start of each simulation. 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 500. 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 global population size was calculated as the sum of all subpopulations. Global population
extinction occurred when N was equal to zero for all local populations. Extinction risk was calculated as the
proportion of extinct global populations out of all replicates. Furthermore, we also calculated mean population
density (i.e., the mean of global 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.
3.
Results
Increased synchronization (ρ from 0.1 to 0.9) of environmental noise resulted in increased extinction risk
independent of population dynamics (over- or under-compensatory), how noise entered the model, or color of noise
(Fig. 2). Thus, there were no interaction effects between synchrony and noise color. Effects of noise color on
extinction risk were however more complex.
For case (i) (over-compensatory and noise entering in K) increased reddening i.e., increased spectral color γ,
resulted in decreased extinction risk (Fig. 2) as a result of the effects on population density. The mean of population
5
density increased when increasing γ and the variance of population density decreased when increasing γ (Fig. 3).
Note that the mean and variance of the entered noise were not changed when changing γ. The increase in mean
density, together with decreased variance with increasing γ, resulted in a small drop in stability (CV), reflecting the
extinction risk. 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., 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). In Fig. 3, mean and variance of population
density is shown only for ρ=0.6 but all values of ρ showed similar results within each model case.
To facilitate comparison of the over- and under-compensatory dynamics, we increased environmental variance by a
factor of 1.8 for the under-compensatory Ricker model (ii). By this, population variances in case (i) and (ii) were of
the same magnitude (Fig. 3). As in case (i), increased γ led to an increase in mean population density for case (ii)
(Fig. 3, blue lines) but on the other hand, there was also an increase in population variance (Fig. 3) and both CV and
extinction risk reached their maxima in relation to intermediate γ values (Table 1, Fig. 2).
For case (iii), over-compensatory and noise affecting realized growth, we multiplied the variance of the
environmental noise by a factor of 1.3 to achieve a variance of population density of the same magnitude as in case i
(Fig. 3). As in case (ii), with under-compensatory dynamics, there was a maximum in extinction risk for
intermediate values of γ. The effects of increased γ on population variance (Fig. 3) and CV (Table 1) were the same
as in case (i), seen as decreasing values with increasing γ. Mean of population density decreased with increasing γ.
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).
4.
Discussion
Colored noise such as variation in resources or growth rates is an expected component of 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 our analysis show
that the influence of colored noise is more straight-forward and easy to interpret compared to what has been
presented in previous studies. Both variances 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 of densities are monotonic, yet the dynamics and how noise enters the model determines
whether the means and variances will decrease or increase with reddened noise (see table 1). When a population
experiences noise that affects K, 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. It is necessary to include knowledge about the
mean and the variance of the population density separately and not combined in a CV measure when studying
population dynamics affected by colored noise. We also studied higher moments of the population density
distribution, i. e. skewness and kurtosis, yet we excluded them in our presented results since we found no
correlations or trends that may explain the results.
6
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 in
our setup of parameters 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 a system (e.g., dispersal and
growth rates) will determine the specific shape of any 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.
In case (iii) the population dynamics is the same as in case (i), i.e., over-compensatory, but environmental noise is
added to the model outside the population regulation part in the equation and thus affecting the realized population
growth while the carrying capacity is constant. The two main differences between case (i) and (iii) are that for case
(iii) the impact of environmental noise is more direct and not filtered through intrinsic population regulation
mechanisms, and K is also constant over time. When environmental noise is white, extreme density values are
amplified due to the over-compensation, resulting in population variances larger than that directly related to the
noise and also the mean of population density becomes smaller than K for both cases (Fig. 3). In the red region,
population density will better track the noise but is also being down-regulated by (the constant) K when density is
larger than K, causing a decrease in population variance but no increase in population density mean as in case (i).
Compare population dynamics in a white versus a red environment in a single-patch system of case (iii) in figure 4.
So, for case (iii) there will not be a decrease in extinction risk when increasing γ as in case (i), for the spectral
interval tested here, but rather a small increase in extinction risk for γ close to one, and followed by a small
decrease. In the zone around γ =1 the noise is a mix of fast fluctuations and the better-tracking fluctuations causing a
larger variance between replicates and by that, a larger extinction risk when extinction risk is a measure of number
of extinct replicates. Cuddington and Yodzis (1999) discussed that this increased variation can be explained by the
fact that intermediate noise color has an equal influence of short- and long-term cycles (Keshner 1982 in Cuddington
and Yodzis 1999) and thus more possible outcomes of the total noise signal (compare to white noise dominated only
by short-term cycles and red noise dominated by long-term cycles). 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 K and the
realized growth. It is also 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 concomitantly be 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 (referens) , and these mechanisms may be included
in future studies.
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 global 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 also 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. Our results
stress that 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
7
studies showing that the effects of noise color differ markedly between homogeneous and heterogeneous landscapes
(i.e., synchronized and 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 most of all, it does not change the qualitative
effects of noise color. Fler referenser till detta? Finns inte så många; Heino 98 (resultat som är tvärtom), kanske en
av artiklarna som editor hänvisar till.
To obtain a clear understanding of the spatial 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 (Lindström et al in press) and
dispersal kernels (Lindström et al. 2008).
Acknowledgements
We thank two anonymous reviewers for valuable comments and W. Fagan for comments on an earlier version of the
manuscript. This work was supported by the Linköping University, Sweden.
References
Amritkar RE, Rangarajan G (2006) Spatially synchronous extinction of species under external forcing. Phys.Rev.
Lett. 96, 258102
Bjørnstad ON, Ims RA, Lambin X (1999) Spatial population dynamics: analyzing patterns and processes of
population synchrony. Trends Ecol. Evol. 14, 427-432
Cuddington KM, Yodzis P (1999) Black noise and population persistence. Proc. R. Soc. Lond. B 266, 969-973
Driscoll DA (2005) Is matrix a sea? Habitat specificity in a naturally fragmented landscape. Ecol. Entomol. 30, 8-16
Gilpin M, Hanski I (1991) Metapopulation dynamics: empirical and theoretical investigations. London: Academic
Press
Halley JM (1996) Ecology, evolution and 1/f-noise. Trends Ecol. Evol. 11, 33-37
Halley JM, Hartley S, Kallimanis AS, Kunin WE, Lennon JJ, Sgardelis SP (2004) Uses and abuses of fractal
methodology in ecology. Ecol. Lett. 7, 254-271
Hanski I (1998) Metapopulation dynamics. Nature 396, 41-49
Heino M (1998) Noise colour, synchrony, and extinctions in spatially structured populations. Oikos 83, 368-375
Heino M, Kaitala V, Ranta E, Lindström J (1997) Synchronous dynamics and rates of extinction in spatially
structured populations. Proc. R. Soc. Lond. B 264, 481-486
8
Heino M, Ripa J, Kaitala V (2000) Extinction risk under colored environmental noise. Ecography 23, 177-184
Inchausti P, Halley J (2002) The long-term temporal variability and spectral colour of animal populations. Evol.
Ecol. Res. 4, 1033-1048
Inchausti P, Halley J (2003) On the relation between temporal variability and persistence time in animal populations.
J. Anim. Ecol. 72, 899-908
Johst K, Wissel C (1997) Extinction risk in a temporally correlated fluctuating environment. Theor. Popul. Biol. 52,
91-100
Kaitala V, Ylikarjula J, Ranta E, Lundberg P (1997) Population dynamics and the colour of environmental noise.
Proc. R. Soc. Lond. B 264, 943-948
Kareiva P, Wennergren U (1995) Connecting landscape patterns to ecosystem and population processes. Nature 373,
299-302
Keitt TH (2000) Spectral representation of neutral landscapes. Landscape Ecol. 15, 479-493
Keshner M (1982) 1/f noise. Proc. IEEE 70, 212-218
Lindström T, Håkansson N, Westerberg L, Wennergren U (2008) Splitting the tail of the displacement kernel shows
the unimportance of kortosis. Ecology 89, 1784-1790
Mode CJ, Jacobson ME (1987) A study of the impact of environmental stochasticity on extinction probabilities by
Monte Carlo integration. Math. Biosci. 83, 105-125.
Mutshinda CM, O’Hara RB (2010) On the setting of environmental noise and the performance of population
dynamical models. BMC Ecology 10:7
Palmqvist E, Lundberg P (1998) Population extinctions in correlated environments. Oikos 83, 359-367
Petchey OL, Gonzales A, Wilson HB (1997) Effects on population persistence: the interaction between
environmental noise colour, intraspecific competition and space. Proc. R. Soc. Lond. B 264, 1841-1847
Pimm SL (1991) The balance of nature? Ecological issues in the conservation of species and communities. The
University of Chicago Press, Chicago and London
Pimm SL, Redfearn A (1988) The variability of population densities. Nature 334, 613-614
Ripa J, Lundberg P (1996) Noise colour and the risk of population extinctions. Proc. R. Soc. Lond. B 263, 17511753
Roughgarden J (1979) Theory of population genetics and evolutionary ecology: an introduction. Macmillan
Publishing Co., Inc., New York
Ruokolainen L, Fowler MS (2008) Community extinction patterns in colored environments. Proc. R. Soc. Lond. B
275, 1775-1783
Ruokolainen L, Lindén A, Kaitala V, Fowler MS (2009) Ecological and evolutionary dynamics under coloured
environmental variation. Trends Ecol. Evol. 24, 555-563
Schwager M, Johst K, Jeltsch F (2006) Does red noise increase or decrease extinction risk? Single extreme events
versus series of unfavorable conditions. Am. Nat. 167, 879-888
9
Steele JH (1985) A comparison of terrestrial and marine ecological systems. Nature 313, 355-358
Vasseur DA (2007) Environmental colour intensifies the Moran effect when population dynamics are spatially
heterogeneous. Oikos 116, 1726-1736
Vasseur DA, Yodzis P (2004) The color of environmental noise. Ecology 85, 1146-1152
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 The degree of synchrony among subpopulations showing the effects of spectral color of environmental noise
on global extinction risk for three different cases: over compensatory dynamics and noise affecting K (i), under
compensatory and noise affecting K (ii), and over compensatory and noise affecting realized growth (iii). Extinction
risks were calculated as proportion of extinct replicates out of 500 at the end of simulated time period T = 1000.
Local population dynamics followed the Ricker equation with density dependence corrected for either over or undercompensation (i.e., oscillatory or monotone dynamics) and with r = 1.5, number of subpopulations = 10, K mean =
100, and dispersal = 0.1 (mass-action mixing). Environmental noise was generated by a two-dimensional FFT
method. For case (ii) and case (iii) the magnitude of noise was multiplied by 1.8 and 1.3, respectively, to obtain
variances in population density of the same magnitude as of case (i). Note: some lines are identical and overlapping
(for zero extinction risk in (i) and (iii) and for 100 per cent extinction risk in case (ii)).
Fig. 3 The effects of spectral color of environmental noise on the mean of the global population density (left) and
variance of global population density (right) for three different model cases. Case (i): over compensatory
(oscillatory) dynamics and noise affecting K (magenta colored), case (ii): under compensatory (monotone) dynamics
and noise affecting K (blue), and case (iii): over compensatory (oscillatory) dynamics and noise affecting realized
growth (red). Number of subpopulations was 10, r = 1.5, Kmean = 100, and dispersal = 0.1 (mass-action mixing).
Environmental noise was generated by a two-dimensional FFT method. For case (ii) and case (iii) the magnitude of
noise was multiplied by 1.8 and 1.3, respectively, to obtain variances in population density of the same magnitude as
of case (i). Shadows behind lines show 95 per cent confidence intervals.
Fig. 4 Simulated population dynamics in a one-patch system obeying Ricker population dynamics (r=1.2 and
K=100), i.e., over compensatory (oscillatory) dynamics. Environmental noise had a direct effect on population
density and thus affecting the realized population growth. Environmental noise was generated by a FFT method for
either white (γ=0) or red (γ=2.5) noise. Lines are showing the value of K and the mean of population density over
the simulated time period (T=1000) for the two cases where environmental noise is white or red.
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Appendix A. Programming code: FFT noise in two dimensions
Simulations were performed in MATLAB 7.5.0 (R2007b, The Mathworks, Natick, MA, USA). Since there are no
simple relationships between the spectral representation and synchrony measured as pair-wise cross-correlation we
first generated a large dataset from which one may determine what values of α to use for different noise colors γ to
achieve a specific synchrony ρ at a given variance.
Function [NoiseLandscape] = Generate2DNoise (Time, Subpop, Gamma, Alfa, Mean, Variance)
Kernel = ones (Time, Subpop); % Make a Kernel matrix of amplitudes, all amplitudes=1.
Kernel = Kernel_synch (Kernel, Time, Subpop, Alfa, Mean); % Filter Kernel row-axis according to scalevariance
(value set by interpolation from given dataset for setting right synchrony). Note: all other amplitudes=1.
Frequencies = Frequencies_matrix (Time, Subpop); % Frequencies in 2dim for Kernel.
Kernel = Kernel_gamma (Kernel, Frequencies, Time, Subpop, Gamma, Mean); % Filter Kernel to 1/f^gamma.
Kernel = Kernel_variance (Kernel, Time, Subpop, Mean, Variance); % Rescale to specified variance and specified
mean-value (Fig. 1, Top panels).
Kernel = Generate_Landscape_Kernel (Kernel, Time, Subpop, Mean, replicate_variation, phase_shift);
% Make variation for replicates by using phases.
NoiseLandscape = ifft2(ifftshift(Kernel),'symmetric'); % Invers fourier-transforming: getting a 'noise-landscape'
with specified gamma, variance and synchrony (Fig. 1, Middle panels).
End
Function [Kernel] = Kernel_synch (Kernel, row, col, Alfa, Mean)
% Filter Kernel row-axis (Time) according to α; re-distribution of variance of row-mean-values will synchronize
columns.
Kernel(:,fix(col/2)+1)=Alfa*Kernel(:,fix(col/2)+1);
Kernel(fix(row/2)+1,fix(col/2)+1)=row*col*Mean;
End
Function [Frequencies] = Frequencies_matrix (row, col)
% Frequencies in 2dim for Kernel. Make a matrix; matrix-numbers are Euclidean distances to Origo.
centerpoint = [fix(row/2)+1,fix(col/2)+1];
[xpos,ypos] = meshgrid(1-centerpoint(2):col-centerpoint(2),1-centerpoint(1):row-centerpoint(1));
Frequencies=(xpos.^2+ypos.^2).^(1/2);
End
Function [Kernel] = Kernel_gamma (Kernel, Frequencies, row, col, Gamma, Mean_value)
% Filter Kernel to 1/f^gamma.
Kernel=(1./Frequencies.^(Gamma/2)).*Kernel;
Kernel(fix(row/2)+1,fix(col/2)+1)=row*col*Mean_value;
End
Function [Kernel] = Kernel_variance (Kernel, row, col, Mean_value, Variance)
% Rescale to specified variance and specified mean-value, by changing sum and centerpoint (DC) of Kernel. Note:
This will not affect gamma, i.e. the slope of regression-line in the log(A)-log(1/f) plot, since Origo does not count
and c is +log(c) in the log(A)-log(1/f) plot.
% Calculate variance of inherent Kernel, and the 'mean-amplitude'.
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Kernel(fix(row/2)+1,fix(col/2)+1)=0;
SumSumKernelNotOrigo=sum(sum(abs(Kernel).^2));
c=SumSumKernelNotOrigo/((row*col)^2);
% Adjust to specified 'mean-amplitude' (Mean-value).
Kernel=sqrt(Variance/c)*Kernel;
Kernel(fix(row/2)+1,fix(col/2)+1)=row*col*Mean_value;
End
Function [Kernel] = Generate_Landscape_Kernel (Kernel, row, col, Mean_value);
% Make variation for replicates by using phases, phase-shift should be set to 1.
% Uniform random numbers for frequensis are picked at an interval of 2pi (or shorter if phase-shift<1):
phase_shift = 1;
phase_values=2*pi*phase_shift*rand(fix(row/2)+1,col);
% Setting amplitude values for the frequencies:
replicate_variation = 0; % This value may be adjusted to give normal distribution for very large Gamma.
amplitide_values=exp(replicate_variation*randn(fix(row/2)+1,col)); % Note: Variation in amplitudes, i.e. variation
in Kernel, ought to be < 1. exp(..) brings in summing in log(A)/log(1/f) when added multiplicatively.
% Make a variation-kernel for real-numbers (upper and lower half must be reflecting and shifted over y-axis):
Kernel_variation=zeros(row,col);
Kernel_variation(1:fix(row/2)+1,1:end)=amplitide_values.*complex(cos(phase_values),sin(phase_values));
Kernel_variation(fix(row/2)+1:end,1:end)=fliplr(flipud(amplitide_values.*complex(cos(phase_values),sin(phase_values))));
% x-axis is the reflecting line. Sides should be shifted and conjugate to each other over the break-even point:
phase_values=2*pi*phase_shift*rand(1,fix(col/2));
amplitide_values=exp(replicate_variation*randn(1,fix(col/2)));
xaxis=[fliplr((amplitide_values.*complex(cos(phase_values),-sin(phase_values))))
(complex(row*col*Mean_value,0)) (amplitide_values.*complex(cos(phase_values),sin(phase_values)))];
Kernel_variation(fix(row/2)+1,:)=xaxis;
% The variation and its phases is multiplied to the fourier Kernel. This is equivalent to summerizing in
log(A)/log(1/f):
Kernel=(Kernel_variation).*Kernel;
Kernel(fix(row/2)+1,fix(col/2)+1)=row*col*Mean_value;
End
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