The tail of the spatial kernel and its implications for biological invasions in patchy environments.
Lindström Håkansson Wennergren
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 and Levin 1988, Seabloom et al. 2003), migration in response to
climate variations (Clark 1998, Walters et al. 2006), 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), a dimensionless quantity defined as the 4th moment
dived by the square of the 2nd moment, describes the shape. 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 contact 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.
Empirical studies show that dispersal most 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 and 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 proportional to the variance
(Skellam 1951). Deviations from Gaussian kernels may still tend to the same speed, i.e. determined by
the variance of the kernel, as long as the tail is exponentially bounded (Mollison 1977, Clark 1998). If
however the density in the distributions tail is higher than an exponentially decreasing function (for
which κ>4 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 are considered as
isolated patches surrounded by a hostile matrix (as is done in metapopulation studies).
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 and Renshaw (2001).
2. Material and Method
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
S
Eq 1
Where d is the distance and S is a normalizing constant which in two dimensions is given by
S
b
.
2a1 b 
Eq 2
Parameters a and b determines ν and κ of the kernel. For two dimensions these are given by
4
 
b
  a2  
2
 
b
Eq 3
6 2
   
b
b
     2
  4 
   
  b 
Eq 4
Hence the kernel density P(D) can be completely 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.
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, Ch. 2.2.1). 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 and
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 and Renshaw (1996) have described an efficient way to calculate the PSDF for point
pattern data (eq 5).
2

 

PSDFPP    cos( K  p xi )     sin( K q yi ) 
 i
  i

2
(5)
(  p ,  q ) are frequencies for fourier transform of grid data, see Mugglestone and Renshaw (1996).
(xi,yi) are coordinates in the point pattern.
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
√𝑁
in the point pattern, N, to keep them independent. Therefore we chose K to be
2
rounded down. And the number of frequencies used is NF =2K×2K. We measure gammapp for the point
pattern using the same method as for continous data, but with the power spectral density function for
point pattern data PSDFPP (see Mugglestone 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
eq. 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
(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
(7)
Switching to point pattern by equation 5 we end up with an equation of the Contrast measure:
δ = CVPP 
NF
1
mean( PSDFPP  PSDFpp (origo )
PSDFPP (origo ) N F
(8)
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 2-dimensiona1/|𝑓|𝛾𝐿 -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 the method are given in
Figure 2.
2.2.2 Analysing neutral point pattern landscapes
While γ and 𝛿 of the spectral point pattern is determined by γL and δL, they are altered by both the Γtransformation 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. 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, (nån REF!). 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.
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).
2.3 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.
Table 1. Input parameters of the simulations and values used
Parameter
δ
γ
N
ν
κ
Explanation
Contrast of patch density
Continuity of patch density
Number of patches
Variance of kernel
Kurtosis of kernel
Parameter values
1, 2, 3, 4, 5
0, 0.5, 1, 1.5, 2
500, 10000
0.0025, 0.005, 0.01
2, 4, 6
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
of patch i becoming occupied within one time step is given by
𝑃(𝑂𝑡+1 (𝑖) = 1|𝑂𝑡 (𝑗) = 1, 𝑂𝑡 (𝑖) = 0) = 𝑅𝑃(𝑑𝑖𝑗 )
Eq 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 9 and also equation 1, for
colonization from patch j is normalized by summation over all patches 𝑘 ≠ 𝑗:
𝑏
−(𝑑𝑘𝑗 ⁄𝑎 )
𝑆 = ∑𝑁−1
𝑘=1 𝑒
Eq 10
where N is the number of patches. We refer to this as relative distance dependence and all patches will
have the same colonizational potential regardless whether it’s an isolated patch or positioned within a
dense area.
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
Ψ = 𝑑𝑙 ⁄𝑡𝑙
Eq 11
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 a categorical predictors. Since the
outputs showed non normal residuals, a Box-Cox transform 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 little
importance in 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 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 4 show that kurtosis had less importance when invasion was modeled with relative distance
dependence (as described in section 2.3.1). We have analyzed invasions with relative distance
dependence for both speed and times to fixed proportions, yet all these results also showed no effect of
kurtosis and hence are only represented by figure 4. The results suggest that in studies of organisms
corresponding to these assumptions it may be sufficient to estimate the variance and disregard the
kernel kurtosis.
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.
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 have
impact 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 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 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 linear relations, 1/f γ, in the
spectral representation of the point patterns are clearly significant. The Continuities in the point patterns
are all fairly close to one yet 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.
The spatial context includes both the spatial environment and the spatial behaviour 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 behaviour. The novel part of our study is to release previous
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 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. A high importance, shown as white areas in figure 3-5, implies
that kurtosis is more important than the variance. 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
and 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 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 to crude to capture the dynamics of any real
invasion, but it allows for testing the effect contact of the contact structure. Our aim was to reduce the
system such that the main characteristics in the study was landscape and dispersal and 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 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, 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 and hence did not have to be
estimated. Here 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
behaviour, dispersal and population processes. It is pointed out that a more complex spatial kernel is
expected when there is individual variation in movement behaviour 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 behaviour during dispersal as a result of spatial
structure. Such a feedback will interweave the landscape and the spatial kernel even more. This feedback
principle somewhat twist the question into what movement behaviour 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 introduced 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 and Gotway 2005, Ch. 2.5.7). Keitt (2000) introduced
spectral methods to landscape ecology and presented neutral landscapes for lattice models. By
developing the point pattern representation by Mugglestone and 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 distribution 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 matters. 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 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 Barkers dilemma 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.
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Refs jag saknar
Cressie, 1993, Ch. 2.2.1
Kareiva & Wennergren, 1995 (har den bara i pappersform)
Schabenberger and Gotway, 2005