EFFECTS OF NOISE COLOUR AND SYNCHRONY ON EXTINCTION RISK IN DIFFERENT
LANDSCAPE TYPES
ABSTRACT
INTRODUCTION
Environmental stochasticity is an important part of the fluctuation of population dynamics and also for species
extinction risk. In parallel, metapopulation theory (ref) has opened up for the importance of space when studying
population dynamics (Kareiva and Wennergen ?). Adding space means that several local populations can be
connected by dispersal. By dispersal local extinct populations can be rescued and recolonized, increasing survival
for the whole regional population (ref? någon Hanski?). This rescue effect assumes some kind of variation over
time, either external (environmental variation) or internal, or both. Most important for population dynamics is of
course mean and variance of environmental variation, but also its autocorrelation (noise colour), and assuming a
spatial dimension results in the fact that this noise can be more or less synchronized in space over time. The
degree of synchrony has a large effect on extinction risk (ref) and therefore it is important to investigate the
effects of synchronized, coloured noise in landscapes with spatially explicit arrangements of patches. Effects of
synchronized, colured noise on extinction risk have been studied in spatially explicit random landscapes (Petchey
et al. 1997 no! Not Petchey! Kolla, ev stryk meningen). We have, in an earlier study, developed a method with
correct handling of synchrony and noise colour in a spatially implicit population model (Lögdberg and Wennergren
submitted). In this study we will combine this method of generating environmental noise for both time (colour)
and synchrony (space) with yet another of our methods of generating spatially explicit patches with different
arrangements (Lindström et al. accepted), i.e. different landscape characteristics.
The characteristic of a landscape, for example the clumping of habitat patches, is here measured by its continuity
and its contrast. The landscape itself is modelled as a set of discrete patches surrounded by a hostile matrix
(metapopulation). The continuity describes how large aggregates of patches there is and the contrast describes the
difference between sparse and dense areas and can in some way be explained as the parameter deciding upon
how many isolated patches or “stepping stone” there is between the aggregates. Landscapes of combinations of
different values of these parameters are shown in Fig. 1. Low values of both continuity and contrast results in
landscapes with random placed patches. The opposite extreme will result in one single aggregate where all
patches are clumped together. These different types of arrangement thus result in different spatial distributions of
resources affecting population growth (ref?).
Variation in resources over time, spread over the landscape implies that local patches or local environmental
stochasticity will vary more or less in synchrony with the other patches. The degree of synchrony among local
populations is important for survival of the regional population (Bjørnstad et al. 1999; Greenman and Benton 2001;
Engen et al. 2002; Liebhold et al. 2004). Here we use a kind of null model for incorporating synchrony where the
degree of synchrony is a measure of the whole landscape and does not include any distances between individual
patches. A combination of for example a high contrast landscape (i.e. all patches are placed within aggregates) and
a totally synchronized environmental noise is most related to a species with long distance dispersers. All
combinations of different landscape characteristics and different degrees of environmental synchrony are not that
relevant but included for the sake of completeness and the difficulties to judge because it depends on scale (which
is set by for example dispersal ability, how environmental variation affects the population, and mean and variance
of resources).
Studies of natural time series, abiotic and biotic, showed autocorrelated or red variation (Steele 1985; Pimm and
Redfearn 1988; Halley 1996; Inchausti and Halley 2002; Vasseur and Yodzis 2004). The effect of red noise on
persistence time or extinction risk, compared to white (non-correlated), has been studied intensively for enclosed
single-species population models (e.g. Ripa and Lundberg 1996; Johst and Wissel 1997; Kaitala et al. 1997;
Cuddington and Yodzis 1999; Halley and Kunin 1999). The results are contradictive and show both postive and
negative effects on population survival when comparing red (autocorrelated) versus white (random) noise (Ripa
and Lundberg 1996; Johst and Wissel 1997; Petchey et al. 1997; Cuddington and Yodzis 1999; Schwager et al.
XXXX). Analyzing the result has shown the importance of for example scaling the noise (when generating noise
with AR-methods, Heino et al. 2000), how noise is incorporated into the model (i.e. affecting K or r, Mutshinda and
O’Hara 2010) and maybe most important: the population dynamics and how the density dependent growth is
regulated, whether it is over- or under-compensating (Petchey et al. 1997; Heino et al. 2000; Ruokolainen et al.
2009). When the population is overcompensating a red noise increases the possibility to track the changes (given
that the noise is most affecting the population via changes in carrying capacity) and thus decreases the extinction
risk (by decreasing the risk of density crashes) compared to white noise (Ruokolainen et al. 2009, Lögdberg and
Wennergren, submitted). For under compensating populations there is not the same risk of crashes so the
variation of the population density will actually increase (and density mean decrease) with reddening of noise
causing an increase in extinction risk (Ruokolainen et al. 2009). Extreme values of red noise (extremely high
autocorrelation) will result in perfect tracking of the noise and decreased extinction risk for all population
dynamics, independent of type of density regulation, as can be seen in Cuddington and Yodzis (1999). This causes a
humped shape response to reddening noise for under compensatory dynamics (Lögdberg and Wennergren
submitted) that can also be seen in Cuddington and Yodzis (1999, Fig. 2a) and in Schwager et al. (2006, Fig.X).
It is a challenge to generate coloured noise in a spatial setting when also handles synchrony (Vassuer 2007). In an
earlier study (Lögdberg and Wennergren submitted), we investigated the extinction risk in a model with coloured,
synchronized noise where space was modelled implicit allowing a mass-action mixing dispersal rule. We concluded
that there was no complex interaction effects between synchrony and colour, not in line with earlier studies by for
example Petchey et al. (1997) who included a spatially explicit random landscape and concluded that there is a
difference of the effect of colour for synchronized versus unsynchronized environmental noise. This is probably
because they unintentionally introduced larger variance (Vassuer 2007) into the unsynchronized noise (Petchey et
al. 1997). Nevertheless, the underlying spatially explicit landscape is an important part when modeling the
population dynamics. Petchey et al. (1997) only used random placed (or regular?) patches but the spatially explicit
arrangement of patches do have importance for species invasion speed (Lindström et al. accepted), species
distribution (Lindström et al. 2008) and population existence, at least for low amounts of habitats (With et al.
1997; Söndgerath and Schröder 2000). We will here combine a spatially explicit arrangement of patches,
generated as Neutral Point Pattern Landscapes (NPPL) (Lindström et al. accepted) with our earlier two-dimensional
noise-synchrony method (Lögdberg and Wennergren submitted). Thus, we will investigate the effects of how
resources are distributed in space (landscape structure and synchrony) and time (noise colour) on extinction risk.
The aim of this study is to investigate the effects of coloured and synchronized environmental noise on extinction
risk for spatially subdivided populations in random and non-random patch-landscapes. The amount of resources
(mean and variance) is of course most important for population survival, but the resource distribution per se can
increase extinction risk from zero to hundred per cent.
METHOD
We have modeled a spatially subdivided population affected by environmental noise. Local populations occurred
in patches surrounded by a non-habitat matrix, so called Neutral Point Pattern Landscapes NPPL (Lindström et al.
xxxx), generated by a spectral method. Dispersal between local populations was distance dependent and modelled
by a kernel function. Environmental noise was generated with a two-dimensional spectral method (Lögdberg and
Wennergren submitted) similar to the method of generating the NPPL. Thus, we have used two different spectral
methods and to facilitate for the reader all parameters and simulated values are summarized in table 1. The
different components of the model are presented in the following sections. For running the simulations we used
Matlab 7.8. As response variable we analyzed the extinction risk, i.e. the proportion of extinct regional populations
at the end of simulated time T out of all replicates. We also checked mean and variance of population density over
T. We first generated a set of parameter combinations (colour and synchrony of noise, over- and undercompensatory, noise entering in K or in r and different landscape configurations), and then we also used real
landscapes consisting of trees in the County of Östergötland, Sweden. These trees serve as habitat/host for many
organisms (e.g. insect and lichens) and can be modeled as discrete patches surrounded by a matrix. Landscape
characteristics of the real tree-landscapes were compared with landscape characteristics of the NPPLs.
LOCAL DYNAMICS
Local population dynamics was governed by the Ricker equation.
N i ,t 1  N i ,t e
r (1( Ni , t / Ki , i )b )
(Eq. 1)
Ni,t is the population density at patch i and time t, r is per capita rate of increase, K i,t is the carrying capacity at
patch i and time t. Parameter b controls the dynamic behaviour by changing from over compensating density
regulation (b=1) to under compensating (b=0.1) (Petchey et al. 1997). We simulated both since earlier studies
(Petchey et al. 1997; Cuddington and Yodzis 1999; Heino et al. 2000; Ruokolainen et al. 2009; Lögdberg and
Wennergren submitted) have shown the importance of population dynamics when concerning the effects of
coloured noise on population density and extinction risk.
LANDSCAPE
Landscapes can be generated by spectral synthesis methods (Halley ?; Keitt 2000 ). A common method is the fast
Fourier transform (FFT) that represents pattern, both for example one-dimensional time series and twodimensional lattice data (or higher dimensions), with sine and cosine functions. In a 1/|f|γ-noise landscape the
amplitudes A(f) of the sine functions are coupled to the frequencies f by a power-law, and the γ is the slope of the
curve in a log(amplitude) versus log(frequencies) plot.
A( f )  1 / f 
f 0
(Eq. 2)
Since we are interested in simulating metapopulation dynamics with defined patches we need a point pattern
landscape. We used a method of simulating Neutral Point Pattern Landscapes (NPPL) as described by Lindström et
al. (accepted). These landscapes are defined by three parameters; the number of patches (N), landscape Continuity
(γ), and landscape Contrast (δ).
Continuity is calculated from the spectral representation of the patch distribution and based on the analysis
presented in Mugglestone and Renshaw (1996). It is a scale free measure of spatial autocorrelation and measured
as the negative slope of a linear regression fitted to a plot of log(amplitude) vs. log(frequency). Hence it is based on
the assumption of self similarity over multiple scales and in Lindström et al. (accepted) it is shown that this
assumption holds for different biologically relevant patch distributions. Readers familiar with time series analysis
may find it useful to interpret Continuity as a point pattern equivalent of the color of 1/f noise (see Halley 1996 for
a review of 1/f noise). Contrast is also calculated from the spectral representation of the patch distribution and is a
scale free measure of density dispersion. Further details on how to calculate Continuity and Contrast and how to
generate landscapes with given values of Continuity and Contrast are provided in Lindström et al (accepted). Some
examples of landscape with different characteristics (i.e. different values of Continuity and Contrast) are shown in
Fig. 1. Throughout this study we used a constant number of patches for all simulated NPPLs (for parameter value
see Table 1).
Fig. 1. Neutral Point Pattern Landscapes (NPPL): landscape characteristic are set by Continuity and Contrast. Lower
left panel is a random NPPL with Continuity = 0 and Contrast = 1. Increasing Contrast towards right (3 and 5
respectively) and increasing Continuity from bottom and up (1 and 2 respectively).
DISPERSAL
Dispersal was distance dependent and modeled as an exponential decay displacement function:
P ( D) 
e  dA
S
,
A  1/ a
(Eq. 3)
where P(D) is the proportion of dispersing individuals in patch i ending up in patch j at distance d after 1 time step.
The dispersal distance, here denoted as ν, set the value of the parameter a (in Eq. 3) through a series of equations
presented in Lindström et al. (accepted), to get two-dimensional properties. S is a scaling factor dependent on the
assumption on absolute versus relative distance dependence. Here we used relative distance dependence. To still
get effects of the landscape characteristics, if any, the dispersing individuals can spread to other patches if close or
stay in their home patch if there are no other patches nearby. Dispersal mortality was not included to keep the
parameter span as low as possible. To minimize effects of the edges the NPPLs were modeled as periodic when
applying the dispersal kernel.
ENVIRONMENTAL NOISE
Also for the environmental noise we used spectral synthesis methods to generate a two-dimensional 1/|f|γ-noise.
Here, the two dimensions correspond to time and space, instead of only space as in the case of landscapes and
generating of point patterns (see Section xx). The time dimension set the autocorrelation in time, i.e. the noise
colour and the spatial dimension set the correlation or the degree of synchrony among patches. In short, a “noiselandscape” was obtained by FFT consisting of time series with specified colour along the x-axis and specified
degree of synchrony along the y-axis. The time series were picked randomly for each patch, which means that all
patches had the same noise colour over the simulated time period T and synchrony ρ was a “region-wide
synchrony” measured by pair-wise cross-correlation (Bjørnstad et al. 1999). This method, of generating twodimensional noise, was first presented in Lögdberg and Wennergren (xxxx). We here give a brief summation but
for more details see Lögdberg and Wennergren (xxxxx). As in the case of generating the landscape the amplitudes
A(f) increases with decreasing frequency f at a rate determined by the spectral exponent γ (i.e. the noise colour in
this case) according to Eq. 1. The spectral exponent is thus the slope of the curve in the log(amplitudes) versus
log(frequencies) plot or the slope of the cone in the two-dimensional case (where log(frequencies) are a plane) but
not including zero frequencies. It is only the amplitudes at the zero frequencies that determine the variance of the
time series and thus the variance can be set by multiplying all these frequencies by the parameter β without
changing the γ.
 1
var  (i, t )     2
M

 A( f
ft
i
, f t )2 
fi

1
1
1
2
i 2
t
2
A
(
0
,
0
)

A
(
0
,
f
)

A
(
f
,
0
)


2
2

M2
fi M
ft M

(Eq. 4)
A(fi,ft) are the amplitudes representing the time series εi,t in the Fourier transform, where fi and ft are the
frequencies of sine functions in the space/patch and time dimensions. M2 is the number of grid points in the twodimensional space. The mean of the time series of each patch
 i (t ) is now represented by the sum of the
amplitudes along the axis of fi and the mean over patches at any given time
 t (i) is thus represented by the sum
of the amplitudes along the axis of ft.


var  i (t )  
ft


var  t (i)  
fi
1
A( f t ,0) 2
2
M
(Eq. 5)
1
A(0, f i ) 2
2
M
(Eq. 6)
Now, the means along any of these axes can be adjusted by multiplying the amplitudes by a parameter a to make
the time series more or less correlated. This will not change the overall variance, the mean, or the slope (i.e. γ).


var  i (t )   
ft
1
A( f t ,0) 2
2
M
(Eq. 7)
So, by this procedure the degree of synchrony can be set without changes in the noise color or the overall variance
and mean of the environmental noise. Since there are no simple relationships between the measured synchrony
and the spectral representation a numerical method has to be applied to determine the value of β given the
overall variance and the noise colour to achieve the expected values of synchrony. We measured ρ and γ in all runs
to ensure correct values. To reduce the variance between replicates we used the random phase-shift method
before inverse Fourier transforming the data set to the real time series (Vasseur 2007 and general transform
theory) and to ensure a correct 1/|f|γ-noise we checked for linearity in the power spectrum. Environmental noise
εi,t was affecting the local dynamics either in carrying capacity K or (indirect) in growth rate r.
N i ,t 1  N t e
r (1( Nt / K (1 i , t ))b )
N i ,t 1  N t e r (1( Nt / K ) ) (1   i ,t )
(Eq. 8)
b
(Eq. 9)
We thus, had three different model cases; over compensatory dynamics with noise in K, over compensatory
dynamics with noise affecting growth, and under compensatory dynamics with noise in K. Due to model
formulation noise per se has not the same impact on population density for the different cases, for example the
mathematical formulation of changing from over to under compensatory response results in a weaker control by K
and hence also a smaller impact of noise. The variance of the noise was therefore corrected between the different
model cases by a scaling factor to approximately give the same order of population variance. Mean and variance of
environmental noise is of course most important for population density but since these were not the main focus of
this study we wanted them to be as constant parameters as possible. In a sense, the scaling factor can be seen as a
measure of how large impact noise will have on population density due to the way the population dynamics is
formulated. Variance of environmental noise was kept constant during the generating of the noise and just scaled
before put into the population simulation, thus not affecting the noise colour and the degree of synchrony. We
used the model case of over compensatory and noise entering in K as the base line and corrected the other two
according to get the same overall variance of population density.
Table 1. Parameters; explanation and values used in simulations.
DATA OF TREE-LANDSCAPES IN ÖSTERGÖTLAND
Hollows with wood mould (i.e. loose wood and fungi) often form inside the trunks when old deciduous trees age
(ref). A specialized insect fauna, mainly beetles and flies, harbours this type of habitat (Dajoz, 2000). Due to
changed land use and abandoned management habitats of old trees that used to be widespread in pasture
woodlands and wooded meadows have now been severely reduced (Nilsson 1997; Kirby and Watkins 1998). In
Sweden, the most important tree for this type of fauna is the oak (Quercus robur) (Palm 1959) and several oakdependent saproxylic beetles are today threatened and on the red-list (ref). Something about lichens also! For the
oak-dependent species the oaks compose clearly defined patches, and the whole system can be easily modeled as
a spatially subdivided population in a point-pattern landscape (metapopulation).
In this part of the study we have used the same population model and environmental noise as for the generated
NPPLs above (simulating a hypothetic oak-dependant beetle species) but the underlying landscape was real oak
data from the County Administration Board in Östergötland Sweden, composed by information about x- and y
coordinates, and hollow stage (ref). Landscape characteristics were analyzed according to the method presented in
Lindström et al. (accepted) and for a set of trees (513 trees in hollow stages 4-6) equal to number of patches in the
NPPL simulations the values of Continuity and Contrast were 1.0 and 3.5 respectively. We analyzed the extinction
risk and the aim was to compare the extinction risk from a real landscape simulation with our generated NPPLs to
investigate the value of using Continuity and Contrast as measures of landscape characteristics.
We also used xxx-tree data (elm, pines, spruce other???).
Note that the values of Continuity and Contrast are within the range of parameter values of the NPPL generated in
the first part of our study (Table 1).
RESULTS AND DISCUSSION
First we present and discuss results from random NPPL. These landscapes are most in line with earlier studies on
coloured noise affecting population density in a spatial setting (e.g. Petchey et al. 1997; Lögdberg and Wennergren
submitted). We then show the result from non-random NPPL for different values on Continuity and Contrast.
Finally are the results from simulations with real tree data landscapes.
RANDOM LANDSCAPES
For Continuity = 0 (as for white noise in the time series analogue) and Contrast = 1 the patches are randomly
distributed over the landscape. A large enough dispersal distance will mean that migrants can reach all other
patches in the landscape with the same probability like a mass-action mixing (MAM) dispersal rule. For much
shorter dispersal distances as in this study (variance of dispersal kernel about xx% of landscape width) the explicit
landscape structure becomes more important. Still, the general result with random NPPL is similar to our earlier
implicit landscape studies, see result and discussion below. Of course the quantitative results differ, with larger
extinction risk for the case with shorter dispersal distances.
Reddening of noise decreased the extinction risk for the over compensatory dynamics when noise entered in K
(Fig. 2a). In a white noise environment the population cannot track the carrying capacity very well, while a red
noise environment with more slow changes facilitates the tracking and thus decreases the risk of population
crashes (Ruokolainen et al. 2009). Increasing the degree of synchrony of the noise increased the extinction risk.
There were no interaction effects of noise colour and synchrony, in line with an earlier study (Lögdberg and
Wennergren submitted) but contradicting the result by Petchey et al. (1997) who probably unintentionally
changed the red colour when increasing degree of synchrony (see discussion in Lögdberg and Wennergen
submitted).
Still over compensatory dynamics, environmental noise affecting the growth needed to be multiplied with a factor
1.3 to get about the same variance of population density as in the case when noise entered in K. The overall
extinction risk, then, was larger (for the random NPPL) but a reddening of noise still lowered extinction risk in
general (Fig. 2). Thus, for over compensatory dynamics reddening of noise will (in general) decrease the extinction
risk independent of noise in K or in the growth, as most coloured noise studies predict (Ruokolainen et al. 2009).
There are exceptions however, when adding space and unsynchronized environmental noise a peak in extinction
risk occurred for intermediate noise colour values (Fig. 2b). Why this response for unsynchrony? En titt på variance
och medel visar att dessa inte uppträder på samma sätt som vi förklarar I förra pappret. Vid en närmare titt på fig
2b kan man fundera på om humpen finns för totalsynk om man skulle sänka utdöenderisken generallt eller om den
faktiskt bara uppträder när man har rum/osynk? Jag hoppas mina pågånde körningar kan ge en vink om vilket
strax. The effects of increasing the degree of synchrony brought in an increase in extinction risk besides the effects
of removal of the hump-shaped response of noise colour.
For the under compensatory case where noise affected K, to get a similar variance of population density as in the
case for over compensatory dynamics, environmental noise was multiplied with a factor 1.8. Also in this case there
was a hump-shaped response, i.e. an increase of extinction risk for intermediate reddening of noise. The humpshaped response are not discussed but shown in some one-patch single species models (Cuddington and Yodzis
1999; Schwager et al. 2006?) Utveckla, se förra manuset. Eller bara hänvisa till förra manuset?
Increasing the variance of the dispersal kernel (i.e. increase of the possible dispersal distance) decreased the
extinction risk (result not shown) for all three cases above. The quantitative result is much dependent on the
interplay between number of patches, carrying capacity K, and size of landscape grid contra dispersal distances.
Fig. 2. Random landscapes. Effects on extinction risk for A) over compensating and noise entering in K, B) over
compensating and noise affecting r, and C) under compensating and noise entering in K.
Table 2. Results of simulations for random landscapes.
NON-RANDOM LANDSCAPES
Continuity for random NPPL is zero. Increasing the Continuity will clump patches together in aggregates; the larger
the Continuity the larger becomes the aggregates (Fig. 1). The Contrast is a measure of the difference between
sparse and dense areas. For the random NPPL Contrast is one and increasing the Contrast will increase the
difference between aggregates and the “matrix” (Fig. 1). Thus, for a large Continuity value and distinct aggregated
areas a low Contrast result in patches acting as stepping-stones between the aggregates. The aggregates
themselves are not that dense as they would have been with a large Contrast value. For a large Contrast value, on
the other hand, almost all patches will be inside an aggregate with no stepping-stones in between. A low
Continuity value together with a large Contrast means that patches randomly put nearby by the Continuity
measure will be assembled even more by the Contrast measure and in practice act as one patch but with twice as
large carrying capacity (that is why this type of NPPL in Fig. 1 seems to have less number of patches).
Results from over compensatory dynamics and noise entering in K are presented in Fig. 3. An increase in
Continuity, i.e. forming of aggregates, increased the extinction risk while an increase in Contrast had the opposite
effect. There were no complex interaction effects between the Continuity and Contrast or between the landscape
structure and the environmental noise (synchrony and colour). Thus, the landscape structure had an effect on the
extinction risk for a spatially divided population exposed to environmental variation. The same effects on
extinction risk by increasing Continuity (increased risk) and Contrast (decreased risk) as in the case above also
passed for over compensatory dynamics and noise affecting growth (Fig. 4) and under compensatory dynamics
with noise entering in K (Fig. 5).
Something about the synchrony and that our model can be seen as a synchrony null model. And also, some
explanation of the result!
Of course the dispersal rule, both shape and variance of displacement kernel plays a major role for when the
landscape characteristics have an effect on population dynamics.
Fig. 3. Environmental noise is affecting carrying capacity and population dynamics is over compensatory. Lower left
panel is a random NPPL with Continuity = 0 and Contrast = 1. Increasing Contrast towards right (3 and 5
respectively) and increasing Continuity from bottom and up (1 and 2 respectively).
Fig. 4 Over comp noise in r.
Fig. 5 Population dynamics is under compensatory and environmental noise is affecting the carrying capacity, K.
TREE-DATA LANDSCAPES
An intermediate value of Continuity (1.0) and a medium to large Contrast (3.5) for the old oak landscape gives
rather good conditions for an oak-dependent species if only concerning the landscape structure assuming that
these landscape characteristics is a good description of landscape structure when coupled to population extinction
risk. Both in the case of over and under compensatory this seem to work (Fig. 6).
Fig. 6 Result from oak landscape simulation.
CONCLUSION
We started out with modeling a random landscape that depending on dispersal distance can be similar to earlier
spatially implicit studies, our own (Lögdberg and Wennergen submitted) and others (Petchey et al. 1997, Schwager
?, andra?). For over compensating population dynamics when noise affects the carrying capacity the extinction risk
will decrease if reddening the and decrease the degree of synchrony. This is in line with most one-patch and
several-patches single species models presented so far (e.g. Ruokolainen et al. 2009). When noise instead affects
the growth or when it affects carrying capacity in an under compensating dynamics there is a peak in extinction
risk for intermediate reddening of noise. In all three model cases, an increase in degree of synchrony results in
increase in extinction risk, and for the latter two cases the hump-shaped response to colouring noise can be
masked.
Also, we showed that the spatial structure of patches does matter. Landscape structure are here measured by two
landscape characteristics; Continuity and Contrast. Low values of both measure result in random Point Pattern
Landscapes (NPPL) while increasing the values will form non-random NPPL with different types of aggregates. An
increase in Continuity brings larger aggregates and also an increased extinction risk, independent on underlying
population dynamics or how noise is incorporated into the model. The Contrast is a measure of the difference
between sparse and dense areas and an increase in Contrast results in more patches packed into aggregates and
no stepping-stone patches in between and also a decrease in population extinction risk.
The landscape measures; Continuity and Contrast, can well capture the landscape structure coupled to its effects
on population density and extinction risk. A simulation with real old oak data showed results in line with the
generated NPPLs. What does our result imply? The landscape is pretty good yet to improve it one should focus
on…. In our study we have not studied lifehistory properties of species, and especially not dispersal kernels. From
our previous studie ecology and proc there is an indication that it’s mainly the variance that is important yet if the
extinction patterns becomes apperaent the system may become more close to invasion dynamics and then the
shape, kurtosis is more important. Our study do also include the density dependence and the colour of noise and
our results may then guide an applied study on oaks with some further insights. First of all it points out the
importance of synchrony, measures of environmental synchrony over the landscape is vital. Any actions that may
decrease synchrony is crucial. Secondly it’s important to asses the density dependence since the effect of colour
differs markedly between under and ….. . If it’s possible to switch from effects on carrying capacity onto growth
rates it can be beneificial since the sensitivity is much lower for systems with varaivle growth rates. Finaly if it’s
possible to asses the density dependence it’s time to investigate the colour. Whether it’s better to find measures
that increase the autocorrelation of environmental noise, or the appearance of it, (increase redness), or that
reduces it, dependence on the autocorrelation in the original system and what kind of density dependent
regulation that mainly effects the species.
Secondly knowledge on
ACKNOWLEDGEMENT
We thank the County Administration Board of Östergötland for tree-data.
REFERENCES
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
Engen
Greenman JV, Benton TG (2001) The impact of stochasticity on the behaviour of nonlinear population models:
synchrony and the Moran effect. Oikos 93, 343-351
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
Inchausti, Halley
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
Liebhold
Lindström T, Håkansson N, Wennergren U (accepted) Proceedings
Lögdberg F, Wennergren U (submitted). Theor. Ecol.
Mutshinda CM, O’Hara RB (2010) On the setting of environmental noise and the performance of population
dynamical models. BMC Ecology 10:7
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
Ripa J, Lundberg P (1996) Noise colour and the risk of population extinctions. Proc. R. Soc. Lond. B 263, 1751-1753
Roughgarden J (1975) A simple model for population dynamics in stochastic environments. Am. Nat. 109, 713-736
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
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
With et al. 1997