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The shape of the spatial kernel and its implications for biological invasions in patchy
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environments.
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Tom Lindström, Nina Håkansson and Uno Wennergren
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Abstract
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Ecological and epidemiological invasions occur in a spatial context. In the study presented we
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tested how these processes relate to the distance dependence of spread or dispersal between
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spatial entities such as habitat patches or infective units. The distance dependence was
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described by a spatial kernel which can be characterized by its shape, quantified by kurtosis, and
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width, quantified by the kernel variance. We also introduced a method to analyze or generate
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non randomly distributed infective units or patches as point pattern landscapes. The method is
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based on Fourier transform and consists of two measures in the spectral representation;
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Continuity that relates to autocorrelation and Contrast that refers to difference in density of
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patches, or infective units, in different areas of the landscape. The method was also used to
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analyse some relevant empirical data where our results are expected to have implications for
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ecological or epidemiological studies. We analyzed the distributions of large old trees (Quercus
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and Ulmus) as well as the distributions of farms (both cattle and pig) in Sweden. We tested the
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invasion speed in generated landscapes with different amount of Continuity and Contrast. The
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results showed that kurtosis, i.e. the kernel shape, was not important for predicting the invasion
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speed in randomly distributed patches or infective units. However, depending on the
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assumptions of dispersal, it may be highly important when the distribution of patches or
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infective units deviates from randomness, in particular when the Contrast is high. Hence speed
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of invasions and spread of diseases depends on its spatial context through the spatial kernel
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intertwined to the spatial structure. This implies high demands on the empirical data; it
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requires knowledge of both shape and width of the spatial kernel as well as spatial structure of
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patches or infective units.
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1. Introduction
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Both ecological and epidemiological studies are concerned with invasion of organisms. The
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mechanism and dynamics of invasion are essential components in numerous specific topics.
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These include recolonization of habitats (Lubina & Levin 1988, Seabloom et al. 2003), migration
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in response to climate variations (Clark 1998, Walters et al. 2006), spread of human and
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livestock diseases (Fergusson 2001, Boender et al. 2007) and invasion of alien species (Skellam
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1951, Urban 2008). The rate of the invasion will largely be determined by the dispersal or spatial
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contact pattern that allows for transmission. Commonly, this is described with a spatial kernel
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(Clark 1998, Tildesley et al. 2008). In this paper we investigate the role of kernel characteristics
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and how this is affected by the spatial arrangement of the habitats or infective units.
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The kernel may be characterized by the 2nd and 4th moment (Clark 1998, Mollison 1991,
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Lindström et al. 2008). The 2nd moment is more commonly known as variance (ν) or squared
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displacement and is a measure of the width of the kernel. Kurtosis (κ), a dimensionless quantity
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defined as the 4th moment dived by the square of the 2nd moment, describes the shape. For
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animal and plant dispersal a random walk or correlated random walk might be assumed which
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will result in a kernel according to a Gaussian distribution (Turchin 1998), where κ=3 or κ=2 for
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one and two dimensional kernels respectively. In this study we will consider two dimensional
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kernels since most ecological an epidemiological dynamics occur in at least two dimensional
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landscapes. These kernels are also often denoted dispersal kernels in ecological studies yet we
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will use the notation spatial kernel throughout the paper.
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For epidemiological studies, assumptions regarding the kurtosis of the kernel should be made
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from knowledge of how transmission occurs. If transmission arises through direct contact the
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kernel should be based on the movement behaviour of the hosts. For many pathogens however,
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transmission is mediated via a vector. If the movement of the vector resembles a random walk
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it may be a fair assumption to model transmission with a Gaussian kernel (Gerbier et al. 2008)
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but outbreak data (Fergusson et al. 2001) and studies of pathways that may mediate
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transmission (Lindström et al. 2009) often reveal highly leptokurtic distributions. Empirical
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studies show that dispersal most commonly deviate from Gaussian distributions. Usually a
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leptokurtic (κ>2) distribution is observed for both plants (Kot et al. 1996, Skarpaas & Shea 2007)
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and animals (Schweiger et al. 2004, Walters et al. 2006), implying a peak in density at short
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distances but at the same time a fat tail, indicating fairly frequent long distance dispersers. A
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number of explanations has been proposed that explain leptokurtic dispersal, including
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population differences in dispersal abilities (Fraser & Bernatchez 2001), temporal variation in
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the diffusion constant (Yamamura et al. 2007) and loss of individuals during dispersal (Schneider
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1999).
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If a Gaussian dispersal in a homogenous and continuous space is assumed, the invasion can be
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modeled as a reaction diffusion process and the speed of the invasion will be proportional to
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the variance (Skellam 1951). Deviations from Gaussian kernels may still tend to the same speed,
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i.e. determined by the variance of the kernel, as long as the tail is exponentially bounded
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(Mollison 1977, Clark 1998). If however the density in the distributions tail is higher than an
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exponentially decreasing function (for which κ>4 in two dimensions) the invasion speed is
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expected to accelerate. Recent work has moved further from invasion speed assuming
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homogenous and continuous space and turned the attention to heterogeneous landscapes
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(Smith et al. 2002, Urban et al. 2008). In this paper we take another step and focus on invasion
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of organisms in environments where the habitats or infective units are best represented as
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discrete entities with a fixed spatial location. Examples of this are studies of livestock epidemics
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(Keeling 2001, Boender et al. 2007) and ecological invasions where habitats and infective units
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are considered as isolated patches surrounded by a hostile matrix (as is done in metapopulation
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studies). Throughout the paper we will refer to such habitats and infective units as patches.
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Our aim in this paper is to explore the role of kurtosis, κ, and variance, ν, of spatial kernels on
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the speed of biological invasion in patchy environments. Such results may support studies and
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the predictive power of estimated speed of invasions. We expect that this may also depend on
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the spatial pattern of focal entities. We therefore introduce a method to incorporate spatial
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aggregation in point patterns using spectral density. Hence we test whether the role of kurtosis
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and variance is dependent on spatial structures as aggregation in patchy landscapes. To
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exemplify what spatial patterns may be found, we also analyze relevant point pattern data with
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a method developed from the analysis given by Mugglestone & Renshaw (2001).
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2. Materials and Methods
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2.1 Kernel variance and kurtosis
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In this study we modeled the spatial kernel with a generalized normal distribution (Nadarajah
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2005). In Lindström et al. (2008) this is extended to two dimensions for symmetrical kernels.
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Kernel density is given by
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P D  
e
 a
d
b
S
(2.1)
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Where d is the distance and S is a normalizing constant which in two dimensions is given by
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S
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Parameters a and b determines ν and κ of the kernel. For two dimensions these are given by
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4
 
b
  a2  
2
 
b
b
.
2a1 b 
(2.2)
(2.3)
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6 2
    
b
b
     2
  4 
   
  b 
(2.4)
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Hence the kernel density P(D) can be completely defined by kurtosis and variance; examples are
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given in figure 1.
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2.2 Generating and analyzing neutral point pattern landscapes
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Keith (2000) defined neutral landscapes for lattices as models where the value at any point in
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the landscape can be considered random and pointed out that this does not exclude models
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with spatial autocorrelation. Such landscapes are then intrinsically stationary while completely
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random surfaces with no autocorrelation are second-order stationary (Cressie, 1993). This
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neutral landscape definition may also be applied to point pattern landscapes where the
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distribution of points may deviate from random as long as the exact position of a point cannot
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be predicted. We used a set of such landscapes to test the effect of kernel characteristic under
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different patterns of spatial aggregation. We refer to these as Neutral Point Pattern Landscapes
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(NPPL). We will use spectral density functions to handle the autocorrelation and in the first
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section we show how spectral density functions can be applied to point pattern.
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2.2.1 Spectral density and point pattern
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Spectral density has been used frequently for time series and lattice data (see Mugglestone &
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Renshaw 1996 for relevant references). The basis of the spectral method follows from the
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Fourier theorem where it is stated that continuous time series or surfaces (and also higher
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dimensions); can be represented by a combination of sine waves with different frequencies and
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amplitudes. Analysis of the time series or surfaces can then instead be performed on the sine
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waves. Here we present two measurements required to capture the spatial point pattern.
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Continuity (γ) is a measure of spatial autocorrelation over multiple scales. It is a measure of the
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Power Spectral Density Function (PSDF). Large values of γ means that nearby areas have similar
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density. It is a measure of the relationship between frequencies and amplitudes assuming a
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linear relationship after logarithmic transformation of frequencies and amplitudes. Hence γ is
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given by the slope from a linear regression fitted to the log(frequency) vs. log(amplitude). In the
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analysis of time series this measure is termed 1/f noise. We are interested in analyzing and
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generating point pattern data and hence there is a methodological part for the transformation
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between point pattern and the continuous representation. Mugglestone & Renshaw (1996)
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have described an efficient way to calculate the PSDF for point pattern data (equation 2.5).
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
 

PSDFPP    cos( K  p xi )     sin( K q yi ) 
 i
  i

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(μp,μq) are frequencies for Fourier transform of grid data (Mugglestone & Renshaw 1996), (xi,yi)
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are coordinates in the point pattern, and K is a constant that determines the number of
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2
(2.5)
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frequencies. Mugglestone and Renshaw (1996) argue that no more frequencies should be used
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than there are points in the point pattern, N, to keep them independent. Therefore we chose K
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to be
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γpp for the point pattern using the same method as for continous data, but with the power
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spectral density function for point pattern data PSDFPP (see Mugglestone & Renshaw 1996).
N p 2 rounded down. And the number of frequencies used is nF =2K×2K. We measure
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The second measure is Contrast (δ), which is a normalized measure of density dispersion. Large
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values of δ reflect a large difference between sparse and dense areas. We measure δ in the
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frequency domain as the coefficient of variation, CV, over point patterns. To formulate an
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equation for that measure we first have to relate it to variance in the continuous case. That is
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how variance in a continuous landscape can be measured using spectral representation. Note
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that when time series and surfaces are represented by sine functions it is solely the amplitudes
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of the sine functions that determines the variance, as shown in equation 2.6 below. As an
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example we use a time series α(t) with its Fourier transform A(μ) where μ is the frequency of a
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sine function.
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var  (t )  
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The mean is represented by the amplitude in origin and hence the coefficient of variation,
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standard deviation divided by the mean, is
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CV 
1
M2
 | A( ) |
2

1
1
A2 (0) 
mean( PSDF  PSDF (origo ))
2
M
M
M
1
mean( PSDF  PSDF (origo ))
PSDF (origo ) M
(2.6)
(2.7)
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Switching to point pattern by equation 2.5 we end up with an equation of the Contrast
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measure:
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δ = CVPP 
NF
PSDFPP (origo )
1
mean( PSDFPP  PSDFpp (origo ))
NF
(2.8)
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2.2.2 Generating neutral point pattern landscapes
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To get NPPL with given characteristics we generated lattice landscapes of size m×m. The density
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defines the probability of a point in the landscapes. We first generated 2-dimensional 1 / f
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noise (denoted LG) using a method similar to that presented by Halley et al. (2004). Hence this is
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still a representation of a lattice landscape not a point pattern. The values in LG are normally
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distributed and since this may include negative values it is not suitable for describing
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probabilities. While this could be solved by truncating we found that it would not allow for
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generation of sufficiently high values of δ. We therefore transformed LG using spectral mimicry.
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This method is defined by Cohen et al. (1999) and has been used when applying Fourier series
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to time series analysis. Cohen et al. presents the method for transformation to a series with
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normally distributed values with a specific mean and variance. We instead transformed LG to LΓ
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using a Γ distribution (which contains no values <0) with mean=1/m2 and coefficient of variation
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δL. Point locations were distributed according to the probabilities given by LΓ. Examples of the
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method are given in figure 2.
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L
-
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While γ and δ of the spectral point pattern is determined by γL and δL, they are altered by both
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the Γ-transformation of the grid values and the distribution of points. Hence we measured these
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quantities in the generated landscape (see method given above). The relationship between
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spectral point pattern values of γ and δ used in the study and the γL and δL required to generate
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them was found iteratively. Furthermore, we found that the linear relationship in the power
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spectra was maintained better for large grids (values of m) and we used m=2000. The
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autocorrelation parameter Continuity generates a general aggregation pattern while the
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variance within the system is reflected by the Contrast parameter. Some examples of the NPPL
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generated with the method can be found in figure 2. High Contrast parameter will impose more
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isolated clusters of aggregated points onto the aggregation structure defined by the Continuity
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parameter; compare the two rightmost examples in figure 2.
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2.2.3 Analysing neutral point pattern landscapes
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We also tested the NPPL model by analyzing empirical point pattern data. The empirical data
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consists of tree distributions and the distributions of farms in Sweden. The tree data was
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provided by the Östergötland County Administrative Board. It is the result of a massive
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inventory of large and old trees, (Länsstyrelsen Östergötland 2009).The locations and
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production type of farms was supplied by the Swedish Board of Agriculture, and more details on
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the data can be found in Nöremark et al. (2009) and Lindström et al. (2009).
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2.3 Simulation
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The effect of κ and ν on invasion was estimated by simulating invasions in NPPL with discrete
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time and the combination of parameters given by table 1. Some combinations of δ and γ were
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not possible to generate (see figure 3-5). Starting at a random patch, we simulated invasions
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with 200 replicates of each parameter combination, for both absolute and relative distance
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dependence. To reduce edge effects, we arranged the landscape such that the starting point
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was located in the centre of the NPPL, which is possible due the periodic nature of the Fourier
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transform.
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2.3.1 Probability of colonization
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The probability of colonization from one occupied patch to an unoccupied one can be modeled
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differently, corresponding to different assumptions regarding dispersal and contact. These
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probabilities are used for simulation studies of invasion and spread of disease. First, one may
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assume that the probability is only dependent on the distance between the two patches, dij, in
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which case the probability, P, of patch i becoming occupied by dispersal from patch j within one
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time step is given by
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P(Ot+1(i) = 1| Ot(j) = 1, Ot(i)= 0) = RP(dij)
(2.9)
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where Ot(k) is equal to one if patch k is occupied at time t and equal to zero if it is unoccupied,
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and R is a measure of growth rate. This modeling approach assumes that the probability of
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colonization from one occupied patch to an unoccupied one is independent of the existence and
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position of other patches. We will refer to this as absolute distance dependence. Alternatively
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one may assume that the colonization potential of all occupied patches is the same. In that case,
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equation 2.9 and also equation 2.1, for colonization from patch j is normalized by summation
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over all patches k≠j:
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S  e
N 1
( d kj / a )b
(2.10)
k 1
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where N is the number of patches. We refer to this as relative distance dependence and all
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patches will have the same colonizational potential regardless whether it’s an isolated patch or
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positioned within a dense area.
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2.3.2 Simulation outputs and analysis
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Our interest was to estimate the importance of κ for biological invasions. Two measurements of
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invasion speed were analyzed. First we investigate the time, Τl, to reach fixed proportions, pl, of
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occupied patches. We used pl = 10%, 50% and 90 %, to get estimates at different stages of the
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invasion. Secondly, we also analyzed the speed, Ψ, of spatial spread, defined as
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  dl t
(2.11)
l
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where dl is a fixed distance and tl is the number of time steps required to reach that distance. In
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this paper we present the results for dl=0.25 (given relative to the unit square). At this distance,
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the influence of the edge effect is considered very small. For Ψ, we analyzed the results of both
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absolute and relative distance dependence.
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The results were analyzed with an ANOVA (type three) for each combination of landscape
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parameters, with the output parameters as dependent variable and ν and κ as categorical
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predictors. Since the outputs showed non normal residuals, a Box-Cox transform (Box & Cox
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1964) was performed for each analysis. The exact value of γ and δ varies between replicates and
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therefore were included as continuous co-variables. The relative effect of kurtosis was
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calculated Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the mean sum of squares of κ and ν,
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respectively.
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3. Results
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3.1 Simulations of invasion in neutral point pattern landscapes
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Our results show that kurtosis of the dispersal kernel is generally a factor that has significant
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effect on the speed of invasion (figure 3) compared to its variance, but the effect varied
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dependent on the spatial structure of patches. Black areas in figures 3-5 indicate low
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importance of kurtosis, and this is consistently found for random NPPLs (δ=1 and γ=0),
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indicating that the shape of the kernel is of little importance in when patches are randomly
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distributed. The trend was found for both densities tested but more prominent for dense NPPLs.
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Also, the relative importance changes during the course of invasion (figure 5) with the most
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prominent effect found during the initial phases of invasion. The general pattern is that the
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Contrast was the characteristic that mainly shifted the importance of kurtosis while Continuity
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had less effect. This can be seen in figure 3 and 5 as a more evident shift left-right than up-
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down.
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Figure 4 show that kurtosis had less importance when invasion was modeled with relative
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distance dependence (as described in section 2.3.1). We have analyzed invasions with relative
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distance dependence for both speed and times to fixed proportions, yet all these results also
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showed no effect of kurtosis and hence are only represented by figure 4. The results suggest
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that in studies of organisms corresponding to these assumptions it may be sufficient to estimate
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the variance and disregard the kernel kurtosis.
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3.2 Examples of real neutral point pattern landscapes (NPPL)
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To demonstrate the use of NPPL and what characteristics may be found in areas where our
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results have impact we analyzed relevant data with the method given in section 2.2, figure 6.
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We analyzed the distribution of two tree species, oak (Quercus) and elm (Ulmus). Especially old
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trees of these species are important habitats for saproxylic insects. Many of these are
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endangered and limited dispersal has been proposed to be a major explanation (Ranius 2006,
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Hedin et al. 2008). Both tree species are also host for many lichens (Jüriado et al. 2009) and
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Ulmus is in addition relevant for epidemiological studies because of the spread of Dutch elm
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disease (Ophiostoma ulmi) (Gilligan & van den Bosch 2008). We also examined the spatial
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distribution of pig and cattle farms in southern Sweden. The spatial distributions of farms are
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known to be essential for possible outbreak of livestock diseases (Boender et al. 2007). The
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distribution of the analyzed data and their estimated values of Contrast (δ) and Continuity (γ)
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(figure 6) indicate that the NPPL model is applicable also for analyzing empirical data. All the
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relations, 1/fγ, in the spectral representation of the point patterns are consistent with the linear
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assumptions of the assumptions of γ. The Continuities in the point patterns are all fairly close to
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one, but the Contrast measures are more variable ranging from 1.29 for cattle farms to 4.9 for
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elm trees.
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Discussion
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Ecological and epidemiological processes occur in a spatial context. Our understanding of, and
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possibility to predict and control, those processes are dependent on how well we may describe
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this spatial context. This includes both the spatial environment and the spatial behaviour of the
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process itself. In our work we have studied invasion, using a patchy landscape as the spatial
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environment and a family of spatial kernels for modeling of the spatial behaviour. The novel
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part of our study is to release commonly used assumption of homogeneous and continuous
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spatial structures and instead focusing on the process in patchy landscapes. We thereby include
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the interplay between the spatial kernel and patchy landscapes. Our result indicates that
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depending on the assumptions of distance dependence, this interplay may be very strong and
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whether kurtosis has evident effect on the invasion speed depends on the spatial structure of
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the patches. More specifically the importance of kurtosis of the spatial kernel is measured
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relative the variance of the spatial kernel. Kurtosis is a measure of the shape of the spatial
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kernel and thereby our results emphasize the importance of correct representation of this
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kernel feature. Since a vast area of topics such as colonization of habitats, migrations in
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response to climate variations, and spread of diseases occurs in a spatial context where spatial
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structure is an obvious component (Kareiva & Wennergren 1995) we expect that our results
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may have implications on direct applications and on future research and investigations. That the
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importance of kurtosis differs depending on landscape structure implies that both speed of
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invasion, and the methodology to estimate it, may differ between landscapes. In some
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landscapes it may suffices with the variance of the spatial kernel while other landscapes enforce
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assessments of the kurtosis of the kernel. Furthermore it also stresses the importance of
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developing empirical methods that correctly captures landscape structure. In this study we
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evaluated both a direct spatial measure of speed and time to a specified proportion colonized,
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which corresponds to slightly different questions regarding invasions. The trends are similar
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(figures 3 and 5) and hence our results have implications for studies focusing on either of these
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measures.
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In this study we have used an admittedly simplified colonization model to represent both
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ecological invasions and spread of disease. That analogy between colonization in a
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metapopulation and spread of disease has been discussed and used in disease modeling, for
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example by Vernon & Keeling (2009) in their study on spread of disease in a network
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representation. As pointed out in their study, the assumptions of a simplified colonization
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model may be to crude to capture the dynamics of any real invasion, but it allows for testing the
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effect contact of the contact structure. Our aim was to reduce the system such that the main
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characteristics in the study was landscape and dispersal and we excluded recovery/extinction
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and within patch dynamics such as density dependence. We argue that our results regarding the
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importance of kurtosis and the interaction with landscape features would hold also for more
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realistic models.
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Both variance and kurtosis relates to long distance dispersal (LDD). Studies of LDD commonly
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defines this (see e.g. Nathan 2006) as either dispersal events beyond some fixed distance or
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some percentile of the tail. As these distances or percentiles are chosen by the researchers, the
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measures of LDD are to some extent subjective and comparison between studies may be
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problematic. We argue that dispersal is better described by analysis of the spatial kernel and its
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characteristics. From analysis of dispersal in continuous space it has been shown in several
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studies (for example Yamamura 2004, van den Bosch et al. 1990, Kot et al. 1996) that the fat tail
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of the spatial kernel, reflected by kurtosis in our study, has an impact on invasion speed. To our
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knowledge this is however the first study that focuses on the importance of the kernel
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characteristics for invasions in patchy environments, using both random and non random
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distribution of patches. By describing the kernel by variance and kurtosis, and test the effect of
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these, it is possible to analyze if and when these characteristics are important to estimate. This
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is made possible by the use of a kernel function where these characteristics are possible to
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control. In previous studies (Lindström et al. 2008) we have studied the effect of kernel
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characteristics on the population distribution and found that kurtosis was not important and
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hence did not have to be estimated. Here we find that the importance is dependent on the
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spatial characteristics.
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An invasion in a patchy landscape may spread over the landscape fairly different compared to
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the diffusive processes in a more homogenous landscape. In a patchy landscape, as illustrated in
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figures 2 and 6, some of the last patches to be colonized are not necessarily the most distant to
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the initial point. This is why we chosen to evaluate both a direct spatial measure of speed and
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time to specified proportion colonized. For example, the spatial speed measure applies to when
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a disease will reach a specific area or country, while the proportion colonized applies to how
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much will be infected within an area.
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We also used two different assumptions regarding colonization from an occupied patch,
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referred to as absolute or relative distance dependence. These correspond to different
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assumptions of the organisms dispersal pathways. Absolute distance dependence would best
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describe a colonization process of an organism with large amount of propagules and passive
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dispersal (e.g. by wind). In such case, the probability of one patch colonizing another is
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independent of the probability of colonization of other patches. The relative distance
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dependence corresponds to colonization by actively dispersing individuals without mortality or
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disease spread between farms via human activities if the number of contacts of infected
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premises is independent of its location (e.g. the number of animal transports may be expected
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to be the same for geographically isolated farms and those in dense areas – Lindström et al.
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2009). Many colonization processes would be a mixture of the two mechanisms.
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The review of Hawkes (2009) introduces a set of principles regarding the relation between
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movement behaviour, dispersal and population processes. It is pointed out that a more complex
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spatial kernel is expected when there is individual variation in movement behaviour within the
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population and such variation also promotes leptokurtic spatial kernels. Another principle of
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Hawkes (2009) is that for some species we may even anticipate changes of movement
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behaviour during dispersal as a result of spatial structure. Such a feedback will interweave the
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landscape and the spatial kernel even more. This feedback principle somewhat twist the
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question into what movement behaviour is optimal in a given spatial structure? Our results may
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then give some guidance since the importance of kurtosis reflects possible selection pressure in
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relation to invasion.
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The interaction between landscape structure and spatial kernel emphasizes the need of reliable
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estimates of these features. Kernels with variable kurtosis and variance, such as the two
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dimensional generalized normal distribution given by Lindström et al. (2008), may form a basis
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for assessing the spatial kernel while the Neutral Point Pattern Landscape method that we
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introduced here may capture these structures for point patterns representation of non random
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landscapes. Spectral representation has become increasingly important to spatial data analysis.
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It is especially advantageous when studying spatial dependence in point pattern processes since
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it may capture more complex dependencies than other techniques, even anisotropy
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(Schabenberger & Gotway 2005). Keitt (2000) introduced spectral methods to landscape
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ecology and presented neutral landscapes for lattice models. By developing the point pattern
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representation by Mugglestone & Renshaw (1996) and the spectral mimicry of time series by
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Cohen et al. (1999) we introduce the neutral point pattern landscape model (NPPL). The
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methodology includes both a continuity measure related to autocorrelation and a contrast
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measure that relates to the variance in the landscape. The contrast measure can be viewed as a
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measure of proportion of points within aggregates, see rightmost examples in figure 2 while
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continuity is a measure of spatial autocorrelation. This effectively means that the structure is
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locally similar and may explain the decrease in the effect of kurtosis with higher continuity. For
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random landscapes, the effect of kurtosis is very small and for high continuity the distribution of
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patches may locally resemble a random distribution patches. High contrasts result in groups of
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locally connected but regionally isolated patches and colonization between such isolated groups
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are rare when dispersal is limited (i.e. low variance). The occurrence of rare but long distant
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events (described by the tail of the leptokurtic kernels) enables such events.
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The distributions of trees and Swedish farms indicate that the NPPL analyses may capture
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important structures in vast areas of empirical data. The analysis reveals contrast measures in
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the interval 1.3-4.9 which indicate landscape structures where kurtosis clearly matters in our
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sumulations. The analyses also show that the distribution of trees seem to have a higher
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contrast measure than farms. While it is not the aim of this paper to compare these two
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systems, our results does suggest that kurtosis is more important in studies focusing on
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dispersal between trees than for epidemiological studies on disease spread between farms.
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Also, the analyzed examples show that the assumptions of a linear relationship between
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log(frequency) and log(amplitude) appears to be a god fit. This means that there is a spatial self
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similarity over scales, which is the definition of a fractal process (Halley et al. 2004). There are
21
403
however many underlying processes for the distributions of these point patterns. Because of
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this, and reminded of Bakers lemma given in Halley et al. (2004) as “Even an elephant appears
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linear if plotted on log–log axes”; we refrain from drawing conclusions on the fractal properties
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of these distributions. Instead we conclude that the analyzed patterns justify the assumptions of
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the NPPL used in this study.
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The impact of the spatial aspect in ecological and epidemiological theory is especially apparent
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and obvious in the light of invasions and spread of disease. The spatial aspect has two
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components, the landscape and the dispersal of organisms. We have showed that these two are
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entwined when the landscape structure is complex and it’s usually not enough to assess the
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variance of the dispersal kernel. Instead the specific shape of the spatial kernel becomes
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important. Yet, its importance is dependent on the landscape structure and thereby there is a
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need to measure this structure. These theoretical results point out that studying ecological and
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epidemiological spread in a spatial context puts a lot of pressure on empirical details on
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dispersal, contact patterns and landscape structures.
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Acknowledgement
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We thank the Swedish Contingency Agency (MSB) for funding and also both the County Administration
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Board of Östergötland and the Swedish Board of Agriculture for supplying data.
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Table 1. Input parameters of the simulations and values used
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Figure 1. (a) Probability densities at distance from source for κ=4 and ν=0.0025 (dashed),
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ν=0.005 (solid) and ν=0.01 (dotted) respectively. (b) Probability densities at distance from source
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for ν=0.005 and κ=2 (dashed), κ=4 (solid) and κ=6 (dotted) respectively. Embedded axis’ shows
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same as major axes but at larger distances and with logarithmic y-axis.
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Figure 2. Examples of (first row) spatial distributions of patches used in the simulation study and
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(second row) their corresponding spectral densities with estimated Continuity (γ) and Contrast
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(δ).
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Figure 3. The relative importance of κ for the speed of spatial spread with absolute distance
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dependence under different landscape parameters (Contrast - δ and Continuity - γ). Black
527
indicates that κ is unimportant and instead the variance of the dispersal kernel determines the
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speed while white areas indicate that κ is highly important. The relative importance was
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calculated as Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the mean sum of squares of κ and ν,
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respectively, from ANOVAs for each combination of δ and γ. Areas where the grid appears (for
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low δ and high γ) are point pattern landscape not possible to generate with present method.
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Figure 4. The relative importance of kurtosis, κ, for the speed of spatial spread with relative
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distance dependence under different landscape parameters (Contrast - δ and Continuity - γ).
535
Black indicates that κ is unimportant and instead the variance of the dispersal kernel determines
536
the speed while white areas indicate that κ is highly important. The relative importance was
537
calculated as Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the mean sum of squares of κ and ν,
538
respectively, from ANOVAs for each combination of δ and γ. Areas where the grid appears (for
539
low δ and high γ) are point pattern landscape not possible to generate with present method.
540
541
Figure 5. The relative importance of κ for time of invasion to reach proportions (pl=0.1, 0.5 and
542
0.9) of occupied patches with absolute distance dependence under different landscape parameters
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(Contrast - δ and Continuity - γ). Black indicates that κ is unimportant and instead the variance of
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the dispersal kernel determines the speed while white areas indicate that κ is highly important.
545
The relative importance was calculated as Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the
546
mean sum of squares of κ and ν, respectively, from ANOVAs for each combination of δ and γ.
547
Areas where the grid appears (for low δ and high γ) are point pattern landscape not possible to
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generate with present method.
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Figure 6. Observed spatial distribution of Np patches of (top row, left to right) Quercus and
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Ulmus large trees and pig and cattle farms. Large tree defined as more than 0.7 m in breast height
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diameter or with visible holes. In the second row we show corresponding speqtral densities and
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the regression line of 1/fγ given the estimated Conitnuity (γ) and Contrast (δ). X and Y axis of top
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row are given as 106 m.