Spectral colour and correlation of noise in subdivided populations
F. Lögdberg and U. Wennergren
Abstract
Introduction
Environmental variation influences many characteristics of populations and communities. Even though
this is studied a lot there are still many uncertainties. Here we focus on the effects of some
environmental variation properties, which are spectral colour of noise and correlation of noise in space,
on population variation for a single species.
One way of studying environmental variation is through the power spectrum analysis of time series data
(see for example Halley, 1996). The time series are then described as a set of sine waves with different
frequencies (Cuddington and Yodzis, 1999). An equal mix of all frequencies is called a white noise, as an
analogy to white light. A dominance of low frequencies corresponds to time series with long-term cycles,
so called red noise. A dominance of high frequencies is thus blue noise.
Our main focus is the effects of spectral colour of environmental noise on the spectral colour of
population density variation. Except for environmental noise, many other factors influence the spectral
colour of population density, for example intrinsic population dynamics such as density dependence (???
In Vasseur, 2007) or time delays (Kaitala and Ranta, 1996 check also Ranta et al., 1995), and the spatial
structure (ref? kolla Steele, 1985, White, 1996 and Kaitala and Ranta, 1996). What is interesting is to
investigate how much of the environmental disturbance, described as spectral colour, that is picked up
by the population density variation. Knowing about a species power spectrum is important since
populations are embedded in food web structures, and theirs variation and spectral properties will
constitute parts of other species environmental conditions (Zu and Li, 2003; Vasseur, 2007).
Important to handle dispersal, metapopulation concept (hur formulera?). Therefore we use a subdivided
species population perspective. With the subdivided population comes an environmental variation in
space. When studying effects of environmental noise we then also have the noise property of
correlation in space to take into account, i.e. affects of environmental noise synchronizing population
dynamics (e.g. Moran effect). For a subdivided population, synchrony between patches (or
subpopulations) is an important part of the variation. Synchrony among populations is mainly caused by
1) spatially-correlated environmental variation (Moran, 1953; Hudson and Cattadori, 1999); 2) dispersal
(Heino, 1999; Lande et al., 1999; Ripa, 2000); and 3) trophic interactions (Koenig, 2001; Liebhold et al.,
2004; Ranta et al., 2006). Thus there is an effect of environmental variation on both spectral
components and synchrony. The spectral colour and the synchrony also affect each other (Gao et al.,
2007; Vasseur, 2007b; kolla ref!).
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Adding variation to a model will have a large effect on population properties; maybe extinction risk is
the most obvious (ref general). In their study Inchausti and Halley (2003) showed that the amount of
variation is much more important for the extinction risk than the reddening of the variation. Spectral
colour of noise and correlation are, in some way, variance distributed in time and space. Keeping the
overall variance, i. e. over time and space, constant we can explore the effects of spreading the variance
in different dimensions, i.e. effects of colour and correlation, more explicit.
For this we need a method that makes it possible to generate noise and control for these different
properties at the same time. It is a challenge modelling subdivided populations with certain coloured
noise and desired degree of synchrony (or cross-correlation). There is a risk of introducing large
variability into the desired synchrony when creating coloured cross-correlated time series with either
spectral synthesis or autoregressive methods; or vice-versa introduce large variability into the desired
colour when synchronizing coloured noise (Vasseur, 2007b). Since there is a methodological problem
when modelling coloured noise in spatially-structured populations it is important to analyze actual value
of spectral colour and synchrony (or cross-correlation) that is used in the model. According to our
knowledge few studies have actually measured the values that are directly used in the model (ref).
Here we present a method of 1/f-noise where both colour and correlation can be controlled for at the
same time. The model can handle all different amounts of subpopulation for each global population, and
still keep desired values of colour and correlation. Our aim was also to generate noise with the same
method as noise were analyzed. We have investigated effects of environmental noise colour and
correlation on spectral colour of population density, on both global and local level. Beside noise
properties, effects of number of subpopulations were explored.
Method
The model describes a global population consisting of a number of subpopulations (or patches)
connected by dispersal. Dispersal is density independent and a mass-action mixing process, i.e. all
patches are equally connected. Within a patch we assume the well-known discrete-time Ricker dynamics,
generalized to the form used by Petchey et al. (1997) for control of over and under-compensatory
dynamics. Dispersal occurs first, then reproduction, and finally census:
N i (t  1)  N i* (t ) exp( r (1  ( N i* (t ) / K ) b ))
n
n
j i
j i
N i* (t )  N i (t )(1   d ji )   d ij N j (t )
d ji , d ij  d /( n  1)
where Ni(t) is local population density at time t, r is per capita rate of increase, K is carrying capacity, d is
per capita dispersal rate, and n is number of patches. The degree of over- or under-compensatory
dynamic is given by the exponent b. The range of b is from 0 to 1, and b=1 indicates over-compensatory.
2
The compensatory dynamics is also affected by the size of intrinsic growth rate, r. We have kept r = 1.5
throughout the total simulation, and therefore we only had the parameter b to take into account when
comparing the compensatory dynamics.
Environmental noise enters the model either indirect in growth rates (as in Kaitala et al., 1997) or in
carrying capacity (as in Ruokolainen and Vasseur, 2008):
N i (t  1)  N i* (t ) exp( r (1  ( N i* (t ) / K ) b ))(1   env )
N i (t  1)  N i* (t ) exp( r (1  ( N i* (t ) / K (1   env )) b ))
where εenv is the environmental noise parameter. The environmental noise is generated by spectral
methods and fast Fourier transformation (FFT), with a similar technique used by Halley et al. (2004).
Two-dimensional 1/|f|γ noise is created, where γ is the spectral colour parameter. One dimension
corresponds to time and the other to space and degree of synchrony. By changing the variance along
the space-axes, in the plane of frequencies (?), we change the degree of synchrony. Time series for each
patch are picked from the two-dimensional FFT, and adjusted to maintain desired γ and variance. We
keep the overall variance constant, and examine values of γ and synchrony before incorporate them into
the simulation. The synchrony among patches is an analogue to spatial heterogeneity as it is described
in Gao et al. (2008) and Petchey et al. (1997). We use the same spectral methods for both generating
and analyzing the results.
Motivate why we use 1/f-noise instead of any autoregressive method!
Motivate why we only use white and red noise!
Initial number in each patch is a Gaussian random number from a distribution with mean=0.5K and
standard deviation = ?. The global population size is calculated each time step as the sum of all
subpopulations. Any population sizes below one are set to zero. The length of the time series simulated
are 1000, and number of replicates are 200. The global population sizes were n = [5 10 50 100].
Result
The γ–value measures the spectral colour; the larger the γ the redder is the noise colour. Spectral colour
of global population density were the same or larger than spectral colour of noise entering each local
subpopulation (entering in carrying capacity) for uncorrelated or “half”-correlated noise (ρ 0.1-0.6)
(fig.1A). For correlated noise (ρ 0.8-0.9) the global spectral colour of density was smaller than for the
noise colour values. When local noise was almost totally uncorrelated (ρ 0.1) the spectral colour of the
density seemed to be saturated at a gamma value around one (fig. 1A).By just increasing noise
correlation a bit (ρ 0.2, 0.4) the trend towards some saturation value could be imagined. Turning again
to almost totally correlated noise (ρ 0.9), the global population behaved as a single population or as one
local population (compare fig. 1A and fig. 1B). In general the result analyzed at local level showed
smaller values of spectral colour of density compared both to the result of spectral colour at the global
level and to the spectral colour of the noise entering the model. There was also a smaller effect of
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correlation on the local level compared to the global. The effect of correlation increased when spectral
values decreased, approaching zero and a white noise.
Effects of correlation in fig. 1 can also include effects of dispersal, i.e. the dispersal (even though kept
constant for all different parameter combinations) will affect the spectral colour of the densities
differently according to the degree of noise correlation entering the local populations.
In fig. 2 the differences are summarized as the product of spectral colour of the response variable (i.e.
the densities, global and local) divided by spectral colour of (environmental) noise. Spectral colour of
density analyzed on global population level increased when noise colour increased, but is overall smaller
than noise colour (fig. 2D). On local level (fig. 2A) low spectral noise values were spread out according to
different correlations but in all cases were below one, this means that γ of the response variable
(density) is smaller than γ of the noise. The differences in the response variable for different correlation
decreased when increasing γ value for environmental noise. Analyzing the response variable as the
spectral colour of local densities showed a clear effect of correlated noise; increased correlation resulted
in decreased difference quote (response/noise) but also decreased difference for increased γ of noise.
Analyzed result of γ for the environmental noise on a global level showed that the result of response γ
agreed with γ of global noise (fig. 1A and fig. 3). Fig 3 also shows that there is a big difference of global
noise γ for different amount of subpopulations in a global population. Different number of local patches
also had an effect on the variance for the global population (fig 4). The more local patches the more
variance in the global population, but only when there noise was (almost) totally correlated among the
patches.
The general picture(fig. 1), when comparing result of global level and local level, held for a more
pronounced under-compensatory Ricker dynamics (b=0.1, eq. z), but the effects of noise correlation
were smaller (fig. 1c, d). The main difference, though, were that the overall values of spectral colour of
densities were much larger. So the feedback in the system, the return path to carrying capacity, had an
effect on spectral colour of densities. When adding noise indirect into the growth (eq. z) resulted in the
same qualitative response as when noise entered in carrying capacity, but γ of the response variable
(both global and local density) were smaller. Putting noise in the growth rate instead of carrying capacity
affects the variance of the population density; hence these two systems cannot be compared
quantitatively. The same goes for the explicit under compensatory model.
We have checked for keeping total variance of noise constant and keeping the right 1/f-noise
characteristics for the environmental noise (maybe this should be removed to the method section?).
Discussion
Our model and results can be seen as a baseline when concerning space and landscape characteristics.
We want to explore the effects of environmental noise in a spatially structured single species model,
without dealing with landscape characteristics (e.g. aggregation of patches) and distance dependent
dispersal because that will blur the result in an already large parameter range. This means that dispersal
is a mass-action mixing process and individuals will spread out according to a uniform distribution. This
in turn means that the landscape is as simple as possible, but still allows effects of dispersal and
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synchrony among patches. The effects of totally correlated environmental noise will be at its maximum,
because as soon as landscape and distance dependent dispersal will come into the model, the effects of
synchrony together with dispersal will probably be smaller.
Since variance has a large impact on population dynamics (see for example ref), we chose to keep the
overall mean and variance of environmental noise constant, to be able to explore the effects of variance
distributed in different ways. The magnitude of the variance itself will not influence the result. This
means that for each individual subpopulation the mean and variance for the environmental noise time
series are constant regardless of the colour and correlation of the noise. We just re-arrange the total
variance into different pattern of variation (/correlation) in time and space, to be able to investigate the
effects of the structure of variation. By keeping it constant at the local level, it will influence the result
on the global level. So far, most studies introduce a certain noise colour into the local patches and
analyze population properties (e.g. extinction risk and spectral colour) on the global population level
(ref). Being aware of both the spectral colour of noise and the spectral colour of density on both the
local and the global level will increase the knowledge of the system. It is important to know if a species
in a food web structure will interact with other species on a local or a global level. Knowing spectral
colour of density on a global level also means that one has to know the colour of noise on a global level,
if analyzing spectral colour transferring through the system properly. Noise at subpopulation level is
whiter than noise at global population level, if noise is not totally correlated between all the patches
because then it is the same spectral colour value.
To conclude: a divided population is more red then a single population, if there is a moderate synchrony
(totally synchronous subpopulations will behave as a single population), and if there are no landscape
properties to take into account. This effect will probably disappear when increasing number of
subpopulations to infinity, and the dynamics will be whiter than a corresponding single population.
Furthermore a single population within a global divided population is less red than the global one….
The most common way of modelling environmental noise is to let one parameter reflect the noise. The
parameter can be for example growth rate or carrying capacity. Noise can enter the model additively or
multiplicatively. Here the noise enters as a multiplicative process in either carrying capacity or growth
rate. According to our results, there is no qualitative difference between these two when concerning
spectral colour of the simulated time series of population density. There is though a quantitative
difference that is probably due to the fact that how the models are set up will affect the variance of the
response variable, i.e. the population density. (jag tror det även blir på input level eftersom det ena är r
och det andra är K/N)
To restrict the parameter space we modelled parameters in “the biological feasible range” (Gao et al.,
2007). We only model red noise (not blue, and actually not really white). We also use a fix value or
growth rate, 1.5 (ref).
Compare the result with others similar work. Maybe compare with Ruokolainen and Fowler, even
though this is a food web.
5
It can be hard to distinguish between environmental variation and dispersal causing synchrony among
subpopulations (ref?). In our model, we put in desired degree of synchronized environmental variation.
We also have to deal with the indirect effects of dispersal. In the same manner as synchrony affects the
spectral colour, the environmental spectral colour can affect the synchrony. For the two-population
model examined by Vasseur (2007), reddened environmental noise colour increased the synchrony.
Since the correlation of noise has an effect on spectral colour of population densities there could be a
double effect of reddening the noise colour; i.e. effect of the red colour itself, effect of possible
correlation and effect more correlation/synchrony caused by the red noise. Exploring effects of
environmental noise on synchrony among subpopulation densities is beyond the scope of this paper.
Although it is interesting that it seems to be impossible to separate noise colour and
correlation/synchrony. This “critical value” of correlation in noise that will alter the result of either
extinction risk (Petchey et al) or spectral colour of density (Gao), maybe does not exist. It is probably
only an effect of the model. Decreasing noise correlation (/increase spatial heterogeneity) means adding
a white noise on top of the environmental noise in their model. The more spatial heterogeneity, the
more white is the noise colour that is put into the model, regardless of the colour of the original noise
time series (and this has nothing to do with the parameter β as it should according to Gao et al).
According to number of subpopulations: A global population, not that well synchronized, will be affected
by the number of subpopulations at least when studying spectral colour. Variation in K, overcomp:
comparing 5 subpopulations to 100 subpopulations. On subpopulation level there is no difference. On
metapopulation level there is. The max value of how red the system (out) can be change when changing
number of subpopulations and degree of synchrony. This will probably change when adding landscape
properties.
Is there an effect of the dynamic behaviour? Remember that the type of dynamics is affected by both
growth rate and the (density dependent) feedback (i.e. both r and b). What can be seen in figure 1 is
thus that overcompensation affects the system such that noise colour cannot be kept, and that under
compensatory dynamics has its own red dynamics. Summary: For under compensation the intrinsic
dynamics dominate, and for overcompensation the noise colour dominate. Under compensatory
dynamics seems to have a redder intrinsic dynamics, probably due to slower changes when tracking
carrying capacity. For the combination of noise colour and noise correlation there seem to be a point
where the colour of N cannot be more reddened (saturated redness). Hence the feedback dynamics in
the population model has an effect on the outcome. (Vi kanske inte ska prata om over and under comp
utan istället om två olika komp-dynamiska modeller ?)
In the future: We want to continue by adding a more realistic landscape that will affect how to handle
the dispersal and the noise correlation/synchrony among patches.
From Gao: The dispersal cannot redden the spectral colour of density (the result holds for subpop=2, 3,
5 and 10). The spectral colour can be reddened by the environmental noise. For every constant α,
increasing correlation leads to increasing spectral colour of N.
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