1
The shape of the spatial kernel and its implications for biological
invasions in patchy environments.
Manuscript
Tom Lindström, Nina Håkansson and Uno Wennergren
IFM Theory and Modelling, Linköping University, 581 83 Linköping, Sweden
Keywords: kurtosis, spread of disease, point patterns, spectral density, dispersal, invasion
Abstract: Ecological and epidemiological invasions occur in a spatial context. In the study presented we
tested how these processes relate to the distance dependence of spread or dispersal between spatial entities
such as habitat patches or infective units. The distance dependence was described by a spatial kernel
which can be characterized by its shape, quantified by kurtosis, and width, quantified by the kernel
variance. We also introduced a method to analyze or generate non randomly distributed infective units or
patches as point pattern landscapes. The method is based on Fourier transform and consists of two
measures in the spectral representation; Continuity that relates to autocorrelation and Contrast that refers
to difference in density of patches, or infective units, in different areas of the landscape. The method was
also used to analyse some relevant empirical data where our results are expected to have implications for
ecological or epidemiological studies. We analyzed two distributions of large old trees (Quercus and
Ulmus) as well as the distributions of farms (both cattle and pig) in Sweden. We tested the invasion speed
in generated landscapes with different amount of Continuity and Contrast. The results showed that
kurtosis, i.e. the kernel shape, was not important for predicting the invasion speed in randomly distributed
patches or infective units. However, depending on the assumptions of dispersal, it may be highly
important when the distribution of patches or infective units deviates from randomness. In particular it
becomes evident when the Contrast is high, for example as high as in the two spatial distributions of large
old trees. We conclude that speed of invasions and spread of diseases depends on its spatial context
through the spatial kernel intertwined to the spatial structure. This implies high demands on the empirical
data; it requires knowledge of both shape and width of the spatial kernel as well as spatial structure of
patches or infective units.
1. Introduction
Both ecological and epidemiological studies are concerned with invasion of organisms.
The mechanism and dynamics of invasion are essential components in numerous specific
topics. These include recolonization of habitats (Lubina & Levin 1988, Seabloom et al.
2003), migration in response to climate variations (Clark 1998, Walters et al. 2006),
2
spread of human and livestock diseases (Fergusson 2001, Boender et al. 2007) and
invasion of alien species (Skellam 1951, Urban 2008). The rate of the invasion will
largely be determined by the dispersal or spatial contact pattern that allows for
transmission. Commonly, this is described with a spatial kernel (Clark 1998, Tildesley et
al. 2008). In this paper we investigate the role of kernel characteristics and how this is
affected by the spatial arrangement of the habitats or infective units.
The kernel may be characterized by the 2nd and 4th moment (Clark 1998, Mollison 1991,
Lindström et al. 2008). The 2nd moment is more commonly known as variance (ν) or
squared displacement and is a measure of the width of the kernel. Kurtosis (κ) describes
the shape and is a dimensionless quantity defined as the 4th moment dived by the square
of the 2nd moment. For animal and plant dispersal a random walk or correlated random
walk might be assumed which will result in a kernel according to a Gaussian distribution
(Turchin 1998), where κ=3 or κ=2 for one and two dimensional kernels respectively. In
this study we will consider two dimensional kernels since most ecological an
epidemiological dynamics occur in at least two dimensional landscapes. These kernels are
also often denoted dispersal kernels in ecological studies yet we will use the notation
spatial kernel throughout the paper.
For epidemiological studies, assumptions regarding the kurtosis of the kernel should be
made from knowledge of how transmission occurs. If transmission arises through direct
contacts, the kernel should be based on the movement behaviour of the hosts. For many
pathogens however, transmission is mediated via a vector. If the movement of the vector
resembles a random walk it may be a fair assumption to model transmission with a
Gaussian kernel (Gerbier et al. 2008) but outbreak data (Fergusson et al. 2001) and
studies of pathways that may mediate transmission (Lindström et al. 2009) often reveal
highly leptokurtic distributions. Also in ecological setting, empirical studies show that
dispersal commonly deviate from Gaussian distributions. Usually a leptokurtic (κ>2)
distribution is observed for both plants (Kot et al. 1996, Skarpaas & Shea 2007) and
animals (Schweiger et al. 2004, Walters et al. 2006), implying a peak in density at short
distances but at the same time a fat tail, indicating fairly frequent long distance dispersers.
A number of explanations has been proposed that explain leptokurtic dispersal, including
population differences in dispersal abilities (Fraser & Bernatchez 2001), temporal
variation in the diffusion constant (Yamamura et al. 2007) and loss of individuals during
dispersal (Schneider 1999).
If a Gaussian dispersal in a homogenous and continuous space is assumed, the invasion
can be modeled as a reaction diffusion process and the speed of the invasion will be given
by the diffusion constant (Skellam 1951), which may be calculated from the variance.
3
Deviations from Gaussian kernels may still tend to a constant speed as long as the tail is
exponentially bounded (Mollison 1977, Kot et al. 1996. Clark 1998). If however the
density in the distributions tail is higher than an exponentially decreasing function (for
which κ>3.33 in two dimensions) the invasion speed is expected to accelerate. Recent
work has moved further from invasion speed assuming homogenous and continuous space
and turned the attention to heterogeneous landscapes (Smith et al. 2002, Urban et al.
2008). In this paper we take another step and focus on invasion of organisms in
environments where the habitats or infective units are best represented as discrete entities
with a fixed spatial location. Examples of this are studies of livestock epidemics (Keeling
2001, Boender et al. 2007) and ecological invasions where habitats and infective units are
considered as isolated patches surrounded by a hostile matrix (as is done in
metapopulation studies). Throughout the paper we will refer to such habitats and infective
units as patches.
Our aim in this paper is to explore the role of kurtosis, κ, and variance, ν, of spatial
kernels on the speed of biological invasion in patchy environments. Such results may
support studies and the predictive power of estimated speed of invasions. We expect that
this may also depend on the spatial pattern of focal entities. We therefore introduce a
method to incorporate spatial aggregation in point patterns using spectral density. Hence
we test whether the role of kurtosis and variance is dependent on spatial structures as
aggregation in patchy landscapes. To exemplify what spatial patterns may be found, we
also analyze relevant point pattern data with a method developed from the analysis given
by Mugglestone & Renshaw (2001).
2. Materials and Methods
2.1 Kernel variance and kurtosis
In this study we modeled the spatial kernel with a generalized normal distribution
(Nadarajah 2005). In Lindström et al. (2008) this is extended to two dimensions for
symmetrical kernels. Kernel density is given by
P D  
e
 a
d
b
(2.1)
S
where d is the distance and S is a normalizing constant which in two dimensions is given
by
4
S  2a 2 2 b  b .
(2.2)
Parameters a and b determines ν and κ of the kernel. For two dimensions these are given
by
4
 
b
  a2  
2
 
b
6 2
   
b
b
     2
  4 
   
  b 
(2.3)
(2.4)
Hence the kernel density P(D) can be defined by kurtosis and variance; examples are
given in figure 1.
Figure 1. (a) Probability densities at distance from source for κ=4 and ν=0.0025 (dashed), ν=0.005 (solid) and
ν=0.01 (dotted) respectively. (b) Probability densities at distance from source for ν=0.005 and κ=2 (dashed),
κ=4 (solid) and κ=6 (dotted) respectively. Embedded axis’ shows same as major axes but at larger distances
and with logarithmic y-axis.
5
2.2 Generating and analyzing neutral point pattern landscapes
Keith (2000) defined neutral landscapes for lattices as models where the value at any
point in the landscape can be considered random and pointed out that this does not
exclude models with spatial autocorrelation. Such landscapes are then intrinsically
stationary while completely random surfaces with no autocorrelation are second-order
stationary (Cressie, 1993). This neutral landscape definition may also be applied to point
pattern landscapes where the distribution of points may deviate from random as long as
the exact position of a point cannot be predicted. We used a set of such landscapes to test
the effect of kernel characteristic under different patterns of spatial aggregation. We refer
to these as Neutral Point Pattern Landscapes (NPPL). We will use spectral density
functions to handle the autocorrelation and in the first section we show how spectral
density functions can be applied to point pattern.
2.2.1 Spectral density and point pattern
Spectral density has been used frequently for time series and lattice data (see
Mugglestone & Renshaw 1996 for relevant references). The basis of the spectral method
follows from the Fourier theorem, where it is stated that continuous time series or
surfaces (and also higher dimensions); can be represented by a combination of sine waves
with different frequencies and amplitudes. Analysis of the time series or surfaces can then
instead be performed on the sine waves. Here we present two measurements required to
capture the spatial point pattern.
Continuity (γ) is a measure of spatial autocorrelation over multiple scales. It is a measure
of the Power Spectral Density Function (PSDF). Large values of γ means that nearby
areas have similar density. It is a measure of the relationship between frequencies and
amplitudes assuming a linear relationship after logarithmic transformation of frequencies
and amplitudes. Hence γ is given by the slope from a linear regression fitted to the
log(frequency) vs. log(amplitude). In the analysis of time series this measure is termed 1/f
noise. We are interested in analyzing and generating point pattern data and hence there is
a methodological part for the transformation between point pattern and the continuous
representation. Mugglestone & Renshaw (1996) have described an efficient way to
calculate the PSDF for point pattern data (equation 2.5).
6
2

 

PSDFPP    cos( K  p xi )     sin( K q yi ) 
 i
  i

2
(2.5)
Here, (μp,μq) are frequencies for Fourier transform of grid data (Mugglestone & Renshaw
1996), (xi,yi) are coordinates in the point pattern, and K is a constant that determines the
number of frequencies. Mugglestone and Renshaw (1996) argue that no more frequencies
should be used than there are points in the point pattern, N, to keep them independent.
Therefore we chose K to be N p 2 rounded down. And the number of frequencies used
is nF =2K×2K. We measure γpp for the point pattern using the same method as for
continuous data, but with the power spectral density function for point pattern data
PSDFPP (see Mugglestone & Renshaw 1996).
The second measure is Contrast (δ), which is a normalized measure of density dispersion.
Large values of δ reflect a large difference between sparse and dense areas. We measure δ
in the frequency domain as the coefficient of variation, CV, over point patterns. To
formulate an equation for that measure we first have to relate it to variance in the
continuous case. That is how variance in a continuous landscape can be measured using
spectral representation. Note that when time series and surfaces are represented by sine
functions it is solely the amplitudes of the sine functions that determines the variance, as
shown in equation 2.6 below. As an example we use a time series α(t) with its Fourier
transform A(μ) where μ is the frequency of a sine function.
var  (t )  
1
M2
 | A( ) |
2

1
1
A2 (0) 
mean( PSDF  PSDF (origo ))
2
M
M
(2.6)
The mean is represented by the amplitude in origin and hence the coefficient of variation,
standard deviation divided by the mean, is
CV 
M
1
mean( PSDF  PSDF (origo ))
PSDF (origo ) M
(2.7)
Switching to point pattern by equation 2.5 we end up with an equation of the Contrast
measure:
  CVPP 
NF
PSDFPP (origo )
1
NF
mean( PSDFPP  PSDF pp (origo ))
(2.8)
7
2.2.2 Generating neutral point pattern landscapes
To get NPPL with given characteristics we generated lattice landscapes of size m×m. The
density defines the probability of a point in the landscapes. We first generated 2dimensional 1 / f
L
-noise (denoted LG) using a method similar to that presented by Halley
et al. (2004). Hence this is still a representation of a lattice landscape not a point pattern.
The values in LG are normally distributed and since this may include negative values it is
not suitable for describing probabilities. While this could be solved by truncating we
found that it would not allow for generation of sufficiently high values of δ. We therefore
transformed LG using spectral mimicry. This method is defined by Cohen et al. (1999) and
has been used when applying Fourier series to time series analysis. Cohen et al. presents
the method for transformation to a series with normally distributed values with a specific
mean and variance. We instead transformed LG to LΓ using a Γ distribution (which
contains no values <0) with mean=1/m2 and coefficient of variation δL. Point locations
were distributed according to the probabilities given by LΓ. Examples of NPPL generated
with the method are given in figure 2.
While γ and δ of the spectral point pattern is determined by γL and δL, they are altered by
both the Γ-transformation of the grid values and the distribution of points. Hence we
measured these quantities in the generated landscape (see method given above). The
relationship between spectral point pattern values of γ and δ used in the study and the γL
and δL required to generate them was found iteratively. Furthermore, we found that the
linear relationship in the power spectra was maintained better for large grids (values of m)
and we used m=2000. The autocorrelation parameter Continuity generates a general
aggregation pattern while the variance within the system is reflected by the Contrast
parameter. Some examples of the NPPL generated with the method can be found in figure
2. High Contrast parameter will impose more isolated clusters of aggregated points onto
the aggregation structure defined by the Continuity parameter; compare the two rightmost
examples in figure 2.
8
Figure 2. Examples of (first row) spatial distributions of patches used in the simulation study and (second
row) their corresponding spectral densities with estimated Continuity (γ) and Contrast (δ).
2.2.3 Analysing neutral point pattern landscapes
We also tested the NPPL model by analyzing empirical point pattern data. The empirical
data consists of tree distributions and the distributions of farms in Sweden. The tree data
was provided by the Östergötland County Administrative Board. It is the result of a
massive inventory of large and old trees, (Länsstyrelsen Östergötland 2009).The locations
and production type of farms was supplied by the Swedish Board of Agriculture, and
more details on the data can be found in Nöremark et al. (2009) and Lindström et al.
(2009).
Table 1. Input parameters of the simulations and values used
Parameter
Explanation
Parameter values
δ
Contrast of patch density
1, 2, 3, 4, 5
γ
Continuity of patch density
0, 0.5, 1, 1.5, 2
n
Number of patches
500, 10000
ν
Variance of kernel
0.0025, 0.005, 0.01
κ
Kurtosis of kernel
2, 4, 6
9
2.3 Simulation
The effect of κ and ν on invasion was estimated by simulating invasions in NPPL with
discrete time and the combination of parameters given by table 1. Some combinations of δ
and γ were not possible to generate (see figure 3-5). Starting at a random patch, we
simulated invasions with 200 replicates of each parameter combination, for both absolute
and relative distance dependence. To reduce edge effects, we arranged the landscape such
that the starting point was located in the centre of the NPPL, which is possible due the
periodic nature of the Fourier transform.
2.3.1 Probability of colonization
The probability of colonization from one occupied patch to an unoccupied one can be
modeled differently, corresponding to different assumptions regarding dispersal and
contact. These probabilities are used for simulation studies of invasion and spread of
disease. First, one may assume that the probability is only dependent on the distance
between the two patches, dij, in which case the probability, P, of patch i becoming
occupied by dispersal from patch j within one time step is given by
P(Ot+1(i) = 1| Ot(j) = 1, Ot(i)= 0) = RP(dij)
(2.9)
where Ot(k) is equal to one if patch k is occupied at time t and equal to zero if it is
unoccupied, and R is a measure of growth rate. This modeling approach assumes that the
probability of colonization from one occupied patch to an unoccupied one is independent
of the existence and position of other patches. We will refer to this as absolute distance
dependence. Alternatively one may assume that the colonization potential of all occupied
patches is the same. In that case, equation 2.9 and also equation 2.1, for colonization from
patch j is normalized by summation over all patches k≠j:
N 1   d kj 
a 

S  e
b
(2.10)
k 1
where N is the number of patches. We refer to this as relative distance dependence and all
patches will have the same colonization potential regardless whether it’s an isolated patch
or positioned within a dense area.
10
2.3.2 Simulation outputs and analysis
Our interest was to estimate the importance of κ for biological invasions. Two
measurements of invasion speed were analyzed. First we investigate the time, Τl, to reach
fixed proportions, pl, of occupied patches. We used pl = 10%, 50% and 90%, to get
estimates at different stages of the invasion. Secondly, we also analyzed the speed, Ψ, of
spatial spread, defined as
  dl t
(2.11)
l
where dl is a fixed distance and tl is the number of time steps required to reach that
distance. In this paper we present the results for dl=0.25 (given relative to the unit square).
At this distance, the influence of the edge effect is considered very small. For Ψ, we
analyzed the results of both absolute and relative distance dependence.
The results were analyzed with an ANOVA (type three) for each combination of
landscape parameters, with the output parameters as dependent variable and ν and κ as
categorical predictors. Since the outputs showed non normal residuals, a Box-Cox
transform (Box & Cox 1964) was performed for each analysis. The exact value of γ and δ
varies between replicates and therefore were included as continuous co-variables. The
relative effect of kurtosis was calculated Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the
mean sum of squares of κ and ν, respectively.
3. Results
3.1 Simulations of invasion in neutral point pattern landscapes
Our results show that kurtosis of the dispersal kernel is generally a factor that has
significant effect on the speed of invasion (figure 3) compared to its variance, but the
effect varied dependent on the spatial structure of patches. Black areas in figures 3-5
indicate low importance of kurtosis, and this is consistently found for random NPPLs
(δ=1 and γ=0), indicating that the shape of the kernel is of slight importance when patches
are randomly distributed. The trend was found for both densities tested but more
prominent for dense NPPLs. Also, the relative importance changes during the course of
invasion (figure 5) with the most prominent effect found during the initial phases of
invasion. The general pattern is that the Contrast was the characteristic that mainly shifted
11
the importance of kurtosis while Continuity had less effect. This can be seen in figure 3
and 5 as a more evident shift left-right than up-down.
Figure 3. The relative importance of κ for the
speed of spatial spread with absolute distance
dependence under different landscape parameters
(Contrast - δ and Continuity - γ). Black indicates
that κ is unimportant and instead the variance of
the dispersal kernel determines the speed while
white areas indicate that κ is highly important.
The relative importance was calculated as
Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the
mean sum of squares of κ and ν, respectively, from
ANOVAs for each combination of δ and γ. Areas
where the grid appears (for low δ and high γ) are
point pattern landscape not possible to generate
with present method.
Figure 4. The relative importance of kurtosis, κ,
for the speed of spatial spread with relative
distance dependence under different landscape
parameters (Contrast - δ and Continuity - γ).
Black indicates that κ is unimportant and instead
the variance of the dispersal kernel determines the
speed while white areas indicate that κ is highly
important.
The relative importance was
calculated as Eκ=MSκ/(MSκ + MSν) where MSκ and
MSν are the mean sum of squares of κ and ν,
respectively, from ANOVAs for each combination
of δ and γ. Areas where the grid appears (for low δ
and high γ) are point pattern landscape not
possible to generate with present method.
12
Figure 5. The relative importance of κ for time of invasion to reach proportions (pl=0.1, 0.5 and 0.9) of
occupied patches with absolute distance dependence under different landscape parameters (Contrast - δ and
Continuity - γ). Black indicates that κ is unimportant and instead the variance of the dispersal kernel
determines the speed while white areas indicate that κ is highly important. The relative importance was
calculated as Eκ=MSκ/(MSκ + MSν) where MSκ and MSν are the mean sum of squares of κ and ν, respectively,
from ANOVAs for each combination of δ and γ. Areas where the grid appears (for low δ and high γ) are point
pattern landscape not possible to generate with present method.
3.2 Examples of real neutral point pattern landscapes (NPPL)
To demonstrate the use of NPPL and what characteristics may be found in areas where
our results are implications, we analyzed relevant data with the method given in section
2.2, figure 6. We analyzed the distribution of two tree species, oak (Quercus) and elm
(Ulmus). Especially old trees of these species are important habitats for saproxylic insects.
Many of these are endangered and limited dispersal has been proposed to be a major
explanation (Ranius 2006, Hedin et al. 2008). Both tree species are also host for many
lichens (Jüriado et al. 2009) and Ulmus is in addition relevant for epidemiological studies
because of the spread of Dutch elm disease (Ophiostoma ulmi) (Gilligan & van den Bosch
2008). We also examined the spatial distribution of pig and cattle farms in southern
Sweden. The spatial distributions of farms are known to be essential for possible outbreak
13
of livestock diseases (Boender et al. 2007). The distribution of the analyzed data and their
estimated values of Contrast (δ) and Continuity (γ) (figure 6) indicate that the NPPL
model is applicable also for analyzing empirical data. All the relations, 1/fγ, in the spectral
representation of the point patterns are consistent with the linear assumptions of the
assumptions of γ. The Continuities in the point patterns are all fairly close to one, but the
Contrast measures are more variable ranging from 1.29 for cattle farms to 4.9 for elm
trees.
Figure 6. Observed spatial distribution of N patches of (top row, left to right) Quercus and Ulmus trees and
pig and cattle farms and (second row) corresponding speqtral densities with estimated Conitnuity (γ) and
Contrast (δ).
Discussion
Ecological and epidemiological processes occur in a spatial context. Our understanding
of, and possibility to predict and control, those processes are dependent on how well we
may describe this spatial context. This includes both the spatial environment and the
spatial behavior of the process itself. In our work we have studied invasion, using a
patchy landscape as the spatial environment and a family of spatial kernels for modeling
of the spatial behavior. The novel part of our study is to release commonly used
assumption of homogeneous and continuous spatial structures and instead focusing on the
process in patchy landscapes. We thereby include the interplay between the spatial kernel
and patchy landscapes. Our result indicates that depending on the assumptions of distance
dependence, this interplay may be very strong and whether kurtosis has evident effect on
14
the invasion speed depends on the spatial structure of the patches. More specifically the
importance of kurtosis of the spatial kernel is measured relative the variance of the spatial
kernel. Kurtosis is a measure of the shape of the spatial kernel and thereby our results
emphasize the importance of correct representation of this kernel feature. Since a vast
area of topics such as colonization of habitats, migrations in response to climate
variations, and spread of diseases occurs in a spatial context where spatial structure is an
obvious component (Kareiva & Wennergren 1995) we expect that our results may have
implications on direct applications and on future research and investigations. That the
importance of kurtosis differs depending on landscape structure implies that both speed of
invasion, and the methodology to estimate it, may differ between landscapes. In some
landscapes it may suffices with the variance of the spatial kernel while other landscapes
enforce assessments of the kurtosis of the kernel. Furthermore it also stresses the
importance of developing empirical methods that correctly captures landscape structure.
In this study we evaluated both a direct spatial measure of speed and time to a specified
proportion colonized, which corresponds to slightly different questions regarding
invasions. The trends are similar (figures 3 and 5) and hence our results have implications
for studies focusing on either of these measures.
In this study we have used an admittedly simplified colonization model to represent both
ecological invasions and spread of disease. That analogy between colonization in a
metapopulation and spread of disease has been discussed and used in disease modeling,
for example by Vernon & Keeling (2009) in their study on spread of disease in a network
representation. As pointed out in their study, the assumptions of a simplified colonization
model may be too crude to capture the dynamics of any real invasion, but it allows for
testing the effect of the contact structure. Our aim was to reduce the system such that the
main characteristics in the study was landscape and dispersal, hence we excluded
recovery/extinction and within patch dynamics such as density dependence. We argue that
our results regarding the importance of kurtosis and the interaction with landscape
features would hold also for more realistic models.
Both variance and kurtosis relates to long distance dispersal (LDD). Studies of LDD
commonly defines this (see e.g. Nathan 2006) as either dispersal events beyond some
fixed distance or some percentile of the tail. As these distances or percentiles are chosen
by the researchers, the measures of LDD are to some extent subjective and comparison
between studies may be problematic. We argue that dispersal is better described by
analysis of the spatial kernel and its characteristics. From analysis of dispersal in
continuous space it has been shown in several studies (for example Yamamura 2004, van
den Bosch et al. 1990, Kot et al. 1996) that the fat tail of the spatial kernel, reflected by
15
kurtosis in our study, has an impact on invasion speed. To our knowledge this is however
the first study that focuses on the importance of the kernel characteristics for invasions in
patchy environments, using both random and non random distribution of patches. By
describing the kernel by variance and kurtosis, and test the effect of these measures, it is
possible to analyze if and when these characteristics are important to estimate. This is
made possible by the use of a kernel function where these characteristics are possible to
control. In previous studies (Lindström et al. 2008) we have studied the effect of kernel
characteristics on the population distribution and found that kurtosis was not important do
not have to be estimated with regard to distribution. Here, when studying invasion, we
find that the importance is dependent on the spatial characteristics.
An invasion in a patchy landscape may spread over the landscape fairly different
compared to the diffusive processes in a more homogenous landscape. In a patchy
landscape, as illustrated in figures 2 and 6, some of the last patches to be colonized are
not necessarily the most distant to the initial point. This is why we chosen to evaluate
both a direct spatial measure of speed and time to specified proportion colonized. For
example, the spatial speed measure applies to when a disease will reach a specific area or
country, while the proportion colonized applies to how much will be infected within an
area.
We also used two different assumptions regarding colonization from an occupied patch,
referred to as absolute or relative distance dependence. These correspond to different
assumptions of the organisms dispersal pathways. Absolute distance dependence would
best describe a colonization process of an organism with large amount of propagules and
passive dispersal (e.g. by wind). In such case, the probability of one patch colonizing
another is independent of the probability of colonization of other patches. The relative
distance dependence corresponds to colonization by actively dispersing individuals
without mortality or disease spread between farms via human activities if the number of
contacts of infected premises is independent of its location (e.g. the number of animal
transports may be expected to be the same for geographically isolated farms and those in
dense areas – Lindström et al. 2009). Many colonization processes would be a mixture of
the two mechanisms.
The review of Hawkes (2009) introduces a set of principles regarding the relation
between movement behavior, dispersal and population processes. It is pointed out that a
more complex spatial kernel is expected when there is individual variation in movement
behavior within the population and such variation also promotes leptokurtic spatial
kernels. Another principle of Hawkes (2009) is that for some species we may even
anticipate changes of movement behavior during dispersal as a result of spatial structure.
16
Such a feedback will interweave the landscape and the spatial kernel even more. This
feedback principle somewhat twist the question into what movement behavior is optimal
in a given spatial structure? Our results may then give some guidance since the
importance of kurtosis reflects possible selection pressure in relation to invasion.
The interaction between landscape structure and spatial kernel emphasizes the need of
reliable estimates of these features. Kernels with variable kurtosis and variance, such as
the two dimensional generalized normal distribution given by Lindström et al. (2008),
may form a basis for assessing the spatial kernel while the Neutral Point Pattern
Landscape method that we introduced here may capture these structures for point patterns
representation of non random landscapes. Spectral representation has become increasingly
important to spatial data analysis. It is especially advantageous when studying spatial
dependence in point pattern processes since it may capture more complex dependencies
than other techniques, even anisotropy (Schabenberger & Gotway 2005). Keitt (2000)
introduced spectral methods to landscape ecology and presented neutral landscapes for
lattice models. By developing the point pattern representation by Mugglestone &
Renshaw (1996) and the spectral mimicry of time series by Cohen et al. (1999) we
introduce the neutral point pattern landscape model (NPPL). The methodology includes
both a continuity measure related to autocorrelation and a contrast measure that relates to
the variance in the landscape. The contrast measure can be viewed as a measure of
proportion of points within aggregates, see rightmost examples in figure 2 while
continuity is a measure of spatial autocorrelation. This effectively means that the structure
is locally similar and may explain the decrease in the effect of kurtosis with higher
continuity. For random landscapes, the effect of kurtosis is very small and for high
continuity the distribution of patches may locally resemble a random distribution patches.
High contrasts result in groups of locally connected but regionally isolated patches and
colonization between such isolated groups are rare when dispersal is limited (i.e. low
variance). The occurrence of rare but long distant events (described by the tail of the
leptokurtic kernels) enables such events.
The distributions of trees and Swedish farms indicate that the NPPL analyses may capture
important structures in vast areas of empirical data. The analysis reveals contrast
measures in the interval 1.3-4.9 which indicate landscape structures where kurtosis clearly
matters in our simulations. The analyses also show that the distribution of trees seem to
have a higher contrast measure than farms. While it is not the aim of this paper to
compare these two systems, our results does suggest that kurtosis is more important in
studies focusing on dispersal between trees than for epidemiological studies on disease
spread between farms. Also, the analyzed examples show that the assumptions of a linear
17
relationship between log(frequency) and log(amplitude) appears to be a god fit. This
means that there is a spatial self similarity over scales, which is the definition of a fractal
process (Halley et al. 2004). There are however many underlying processes for the
distributions of these point patterns. Because of this, and reminded of Bakers lemma
given in Halley et al. (2004) as “Even an elephant appears linear if plotted on log–log
axes”; we refrain from drawing conclusions on the fractal properties of these
distributions. Instead we conclude that the analyzed patterns justify the assumptions of the
NPPL used in this study.
The impact of the spatial aspect in ecological and epidemiological theory is especially
apparent and obvious in the light of invasions and spread of disease. The spatial aspect
has two components, the landscape and the dispersal of organisms. We have showed that
these two are entwined when the landscape structure is complex and it’s usually not
enough to assess the variance of the dispersal kernel. Instead the specific shape of the
spatial kernel becomes important. Yet, its importance is dependent on the landscape
structure and thereby there is a need to measure this structure. These theoretical results
point out that studying ecological and epidemiological spread in a spatial context puts a
lot of pressure on empirical details on dispersal, contact patterns and landscape structures.
Acknowledgement
We thank the Swedish Contingency Agency (MSB) for funding and also both the County
Administration Board of Östergötland and the Swedish Board of Agriculture for
supplying data.
References
Boender, G. J., Meester, R., Gies, E. & De Jong, M. C. M. 2007 The local threshold for geographical
spread of infectious diseases between farms. Prev. Vet. Med. 82, 90-101.
Box, G. E. P. & Cox, D. R. 1964 An analysis of transformations. J. Roy. Stat. Soc. B. 26, 211-252.
Clark, J. S. 1998 Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord.
Amer. Nat. 152, 204-224.
Cohen J. E., Newman C. M., Cohen A. E., Petchey O. L. & Gonzalez A. 1999 Spectral mimicry: A
method of synthesizing matching time series with different Fourier spectra. Circ. Syst. Signal Process. 18,
431-442.
18
Cressie N. 1993 Statistics for spatial data revised edition. Chapter 2.2.1. USA: John Wiley & Sons, Inc.
Fraser, D. J. & Bernatchez, L. 2001 Adaptive evolutionary conservation: towards a unified concept for
defining conservation units. Mol. Ecol. 10, 2741-2752.
Ferguson, N. M., Donnelly, C. A. & Anderson, R. M. 2001 The foot-and-mouth epidemic in Great Britain:
Pattern of spread and impact of interventions. Science 292, 1155-1160.
Gerbier, G., Baldet, T., Tran, A., Hendrickx, G., Guis, H., Mintiens, K., Elbers, A. & Staubach, C. 2008
Modelling local dispersal of bluetongue serotype 8 using Random walk. Prev. Vet. Med. 87, 119–130.
Gilligan, C. A. & van den Bosch, F. 2008 Epidemiological models for invasion and persistence of
pathogens. Annu. Rev. Phytopathol. 46, 385-418.
Halley, J. M., Hartley, S., Kallimanis, A. S., Kunin, W. E., Lennon, J. J. & Sgardelis, S. P. 2004 Uses and
abuses of fractal methodology in ecology. Ecol. Lett. 7, 254-271.
Hawkes, C. 2009 Linking movement behaviour, dispersal and population processes: is individual variation
a key? J. Anim. Ecol. 78, 894-906.
Hedin, J., Ranius, T., Nilsson, S. G. & Smith, H. G. 2008 Restricted dispersal in a flying beetle assessed
by telemetry. Biodivers. Conserv. 17, 675–684.
Jüriado, I., Liira, J. & Paal, J. 2003 Epiphytic and epixylic lichen species diversity in Estonian natural
forests. Biodivers. Conserv. 12, 1587–1607.
Kareiva, P. & Wennergren, U. 1995. Connecting landscape patterns to ecosystem and population
processes. Nature 373, 299-302.
Keeling, M. J., Woodhouse, M. E., Shaw, D. J. & Matthews, L. 2001 Dynamics of the 2001 UK foot and
mouth epidemic: Stochastic dispersal in a dynamic landscape. Science 294, 813-817.
Keitt, T. H. 2000 Spectral representation of neutral landscapes. Landscape Ecol. 15, 479-493.
Kot, M., Lewis, M. A. & van den Driessche, P. 1996 Dispersal data and the spread of invading organisms.
Ecology 77, 2027-2042.
Lindström, T., Håkansson, N., Westerberg, L. & Wennergren, U. 2008 Splitting the tail of the
displacement kernel shows the unimportance of kurtosis. Ecology 89, 1784–1790.
Lindström, T., Sisson, S. A., Nöremark, M., Jonsson, A. & Wennergren, U. 2009 Estimation of distance
related probability of animal movements between holdings and implications for disease spread modeling.
Prev. Vet. Med.91, 85-94.
Lubina, J. A. & Levin S. A. 1988 The Spread of a Reinvading Species: Range Expansion in the California
Sea Otter. Amer. Nat. 131, 526-543.
Länsstyrelsen Östergötland 2009. Skyddsvärda träd i Östergötland – 1997-2008. Rapport 2008:13
19
Mollion, D. 1977 Spatial contact models for ecological and epidemic spread. J. Roy. Stat. Soc. B. 39, 283326.
Mollion, D. 1991 Dependence of epidemic and population velocities on basic parameters. Math. Biosci.
107, 255-287.
Mugglestone, M. A. & Renshaw, E. 1996 A practical guide to the spectral analysis of spatial point
processes. Comput. Stat. Data An. 21, 43-65.
Mugglestone, M. A. & Renshaw, E. 2001 Spectral tests of randomness for spatial point patterns. Environ.
Ecol. Stat. 8, 237-251.
Nadarajah, S. 2005 A generalized normal distribution. Appl. Statist. 32, 685-694.
Nathan, R. 2006 Long-distance dispersal of plants. Science 313, 786-788.
Nöremark, M., Håkansson, N., Lindström, T., Wennergren, U. & Sternberg Lewerin, S. 2009 Spatial and
temporal investigations of reported movements, births and deaths of cattle and pigs in Sweden. Acta Vet.
Scand. 51:37.
Ranius, T. 2006 Measuring the dispersal of saproxylic insects: a key characteristic for their conservation.
Popul. Ecol. 48, 177–188.
Schabenberger O. & Gotway C. 2005. Statistical methods for spatial data analysis. Chapter 2.5.7.
London: Chapman & Hall.
Schneider, J. C. 1999 Dispersal of a highly vagile insect in a heterogeneous environment. Ecology 80,
2740-2749.
Schweiger, O., Frenzel, M. & Durka, W. 2004 Spatial genetic structure in a metapopulation of the land
snail Cepaea nemoralis (Gastropoda: Helicidae). Mol. Ecol. 13, 3645–3655.
Seabloom, E. W., Borer, E. T., Boucher, V. L., Burton, R. S., Cottingham, K. L., Goldwasser, L., Gram,
W. K., Kendall, B. E. & Micheli, F. 2003 Competition, seed limitation, disturbance, and reestablishment
of California native annual forbs. Ecol. Appl. 13, 575-592.
Skarpaas, O. & Shea, K. 2007 Dispersal patterns, dispersal mechanisms, and invasion wave speeds for
invasive thistles. Am. Nat. 170, 421-430.
Skellam, J. G. 1951 Random dispersal in theoretical populations. Biometrika 38, 196-218.
Smith, D. L., Lucey, B., Waller, L. A., Childs, J. E. & Real, L. A. 2002 Predicting the spatial dynamics of
rabies epidemics on heterogeneous landscapes. Proc. Natl. Acad. Sci. 99, 3668-3672.
Tildesley, M. J., Deardon, R., Savill, N. J., Bessell, P. R., Brooks, S. P., Woolhouse, M. E., Grenfell, B. T.
& Keeling, M. J. 2008 Accuracy of models for the 2001 foot-and-mouth epidemic. Proc. R. Soc. B. 275,
1459-1468.
Turchin, P. 1998 Quantitative Analysis of Movement, Sinauer Associates, Sunderland, MA.
20
Urban, M. C., Phillips, B. L., Skelly, D. K. & Shine, R. 2008 A toad more traveled: The heterogeneous
invasion dynamics of cane toads in australia. Am. Nat. 171, E134-E148.
Van den Bosch, F., Metz, J. A. J. & Diekmann O. 1990 The velocity of spatial population expansion. J.
Math. Biol., 28, 529-565.
Vernon, M. C. & Keeling, M. J. 2009 Representing the UK’s cattle herd as static and dynamic networks.
Proc. Roy. Soc. B. 276, 469-476.
Walters, R. J., Hassall, M., Telfer, M. G., Hewitt, G. M. & Palutikof, J. P. 2006 Modelling dispersal of a
temperate insect in a changing climate. Proc. R. Soc. B., 273, 2017-2023.
Yamamura K. 2004 Dispersal distance of corn pollen under fluctuating diffusion coefficient. Popul. Ecol.
46, 87-101.
Yamamura K., Moriya, S., Tanaka, K., & Shimizu, T. 2007 Estimation of the potential speed of range
expansion of an introduced species: characteristics and applicability of the gamma model. Popul. Ecol. 49,
51-62.